1 Introduction

Optical fiber sensors, with the advantages of small size, electromagnetic immunity and good corrosion resistance, had become excellent choices for measurements in harsh environments [1, 2]. With the rapid development of optical fiber sensors, many kinds of sensing structures based on optical fiber had been proposed, such as optical fiber Fabry–Perot (F–P) cavity, fiber Bragg grating (FBG), photonic crystal fiber (PCF) sensors and so on [3,4,5,6]. To improve the measurements, except developing better sensing structure, one should also consider applying novel interrogation method of the optical fiber sensing system. The sensing signals from these optical fiber structures mentioned above were always analyzed by intensity and wavelength interrogation methods. Generally speaking, the wavelength interrogation method was more preferred in optical fiber sensing technology, because the influence of the light power fluctuation in the sensing system could be significantly reduced. In this case, the resolution of the measurement depended on the wavelength resolution of the spectrometer. For precision measurements based on optical fiber sensors, a high-performance spectrometer is indispensable, which increased the cost of the sensing system largely. Other kinds of interrogation methods were still necessary to be explored, in order to give another option to analyze the sensing signal from optical fiber.

In this work, based on fork interference pattern of orbital angular momentum (OAM) beam and plane wave, the phase interrogation method for fiber Mach–Zehnder interferometers (MZI) is proposed. The feature of the fork interference pattern is extracted by principal component analysis (PCA) method, and the relationship between the phase difference and the variation of the fork pattern is investigated. In this method, the power of incident light does not affect the main feature of the interference pattern, so the influence of the light power fluctuation can be eliminated. Moreover, compared with the wavelength interrogation systems by employing spectrometers, the requirement of the incident light power is rather low in the proposed method, and the system is simple and low cost. The theoretical resolution of this method is estimated to be 0.0636 µε, which depends on the parameters of the imaging system and the fiber properties. The experimental system is built to verify our design. In the reference path of the fiber interferometer, the OAM beam is emitted from the two-mode fiber (TMF) after mode conversion processes [7]. The plane wave is generated by collimating the emitted light from single mode fiber (SMF) in the measuring path. The interference pattern of the two beams is recorded by the charge-coupled device (CCD) camera. In order to characterize the sensing performance of the system, strain is applied on sensing fiber in the measuring path to change the phase delay in the MZI. The features of the fork interference patterns under different strain are extracted. It should be noticed that, except strain, other parameters with the ability to modify the light phase delay in fiber can also be measured by this interrogation method.

2 Theoretical analysis

2.1 System design

To realize the phase interrogation, the fiber MZI is designed as shown in Fig. 1. By the 3 dB optical coupler (OC), the light emitted from laser source is divided into two parts which are transmitted in the measuring path and reference path, respectively. In the reference path, the OAM beam is generated in two modes fiber by the mode converter, whose working principle will be discussed in Sect. 3.1. In the measuring path, the plane wave can be approximately obtained by collimating the light emitted from the end of SMF. By using a beam-splitting cube, the interference pattern of the light beams from the two paths can be formed and recorded by the CCD camera. The angle between the two beams can be adjusted by rotating the reflector. When the incident light passes through the sensing fiber in the measuring path, the phase delay can be modified by the variation of the stress on fiber. The relationship between the interference pattern and the phase delay is investigated. After that, the phase interrogation method by analyzing the digital interference pattern is verified, when applied in sensing system. The principles of interference pattern and the phase interrogation will be explained in the next two sections.

Fig. 1
figure 1

The schematic diagram of the phase interrogation method based on the OAM beam

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2.2 Fork interference pattern

OAM beam is characterized by a helical phase front of exp(ilφ), in which φ refers to the azimuth angle and l is the topological charge [8, 9]. Because of the helical phase structure, it has the doughnut shape intensity profile with a phase singularity point at the beam center. Driven by the unique properties, OAM beam has been widely used in particle manipulation, optical tweezers and optical communications [10,11,12]. In this work, the distinctive interference pattern of OAM beam and plane beam is employed to demodulate the sensing signal. According to the interference theory [13], the intensity distribution between two light beams can be expressed by Eq. (1)

$$I(r,\varphi )={\left| {{E_1}(x,y)+{E_2}(x,y)} \right|^2},$$
(1)

where E1(x, y) and E2(x, y) are the electric field distributions of the OAM beam and plane wave. For the OAM beam with l = 1, the electric field on x–y plane can be expressed by Eq. (2):

$${E_1}\left( {r,\varphi } \right)={A_1}(r)\exp \left( {il\varphi } \right),$$
(2)

where A1 is the amplitude distribution. Due to the helical phase structure, it is convenient to describe the electric field of OAM beam in polar coordinate (r, φ). In this case, the center of the beam is located at r = 0. The intensity and phase distribution of the OAM beam with l = 1 is shown in Fig. 2a, b.

Fig. 2
figure 2

The intensity and phase distribution of OAM beam with l = 1 in a, b. The intensity and phase distribution of plane wave in c, d. The interference pattern by the OAM beam and plane wave with the phase difference of ϕ in e

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For plane wave with the propagation direction of (sinβ, 0, cosβ), the electric field on x–y plane can be expressed as

$${E_2}\left( {x,y} \right)={A_2}(x,y)\exp \left( {ikx\sin \beta +i\phi } \right),$$
(3)

where A2(x, y) is the amplitude distribution of the plane wave, and k refers to the plane wave numbers, β refers to the angle between the direction of plane wave propagation and z-axis. ϕ is the additional phase delay, which is equivalent to the phase difference between OAM beam and plane wave in the theoretical analysis of next section. The intensity and phase distribution of the plane wave is shown in Fig. 2c, d. In this work, the plane wave is generated by collimating the light from optical fiber, so the intensity distribution is Gauss type. When the two beams interfere with each other, the interference pattern of the light intensity is shown in Fig. 2e, which exhibits a fork shape at the OAM beam center.

2.3 Principle of phase interrogation

When the phase difference between the two beams is modified, the shape of fork pattern is changed. The interference pattern under the phase difference from ϕ = 0° to 315° is shown in Fig. 3a–h, respectively. The interval of phase difference between each two adjacent pattern is 45°.

Fig. 3
figure 3

The interference patterns by the OAM beam and plane wave under different phase differences. a ϕ = 0°, b ϕ = 45°, c ϕ = 90°, d ϕ = 135°, e ϕ = 180°, f ϕ = 225°, g ϕ = 270°, h ϕ = 315°

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The obvious change near the branch point of the fork pattern can be recognized in Fig. 3. In this part, the relationship between the interference pattern and phase difference is investigated. To speed the image processing, the method to quantitatively evaluate the feature of the pattern is proposed based on PCA in this work [17]. PCA is linear transformation method based on the statistical techniques, which can be applied in digital image processing (e.g., face recognition). In this work, it is employed to realize the pattern recognition. By reducing the data dimension, the speed of the digital image process can be largely increased. The obtained principal components can reflect the features of the original image. Each image is a two-dimensional array of 200 × 200 intensity values (8-bit), which can be transformed into a column vector with N2 × 1 dimension. In order to perform the recognition, 37 patterns with different ϕ from 0° to 720° calculated by Eq. (1) are chosen as the initial set of the patterns (i.e., the training set). The interval of the phase difference ϕ is 20°. The pattern set is represented by Φ = [Φ1, Φ2, Φ3, …Φ M ], where M = 37 is the total number of patterns. Each Φ i has been subtracted by the average vector of these patterns. Then, the eigenvalues and eigenvectors of covariance matrix are calculated. The covariance matrix is defined by C = ΦTΦ, which is one M × M matrix. Among these eigenvalues, one can find that only two values are dominant, which are λ1 = 1.57 × 104 and λ2 = 1.49 × 104, and other eigenvalues are near-zero values. The eigenvalues indicate their contributions to characterize the features of the training patterns are great. In this case, the two eigenvectors, V1 and V2, corresponding to the principal eigenvalues are considered in the following process. Therefore, using PCA method, the dimensionality of the training pattern set can be largely reduced. Similar with the concept of eigenfaces in face recognition process [14], we define the eigenpatterns of the training set in Eq. (4):

$$[\begin{array}{*{20}{c}} {{\mathbf{U}_1}}&{{\mathbf{U}_2}} \end{array}]=\varvec{\Phi} \times [\begin{array}{*{20}{c}} {{\mathbf{V}_1}}&{{\mathbf{V}_2}} \end{array}].$$
(4)

Figure 4a, b shows the two eigenpatterns, which are obtained by reshaping U1 and U2 to be matrixes with the size of 200 × 200, respectively. The two eigenpattern vectors can be considered as the base vectors to define a new two-dimensional space. One pattern (Φ i ) with arbitrary phase difference ϕ i can be projected into this defined space by the following operation in Eq. (5):

Fig. 4
figure 4

The two principal eigenpatterns for the training set of 37 patterns with different ϕ values

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$${\Omega _i}=\left[ {\begin{array}{*{20}{c}} {{\omega _1}} \\ {{\omega _2}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {\mathbf{U}_{1}^{T}} \\ {\mathbf{U}_{2}^{T}} \end{array}} \right]{\varvec{\Phi}_i}.$$
(5)

From Fig. 4, one can recognize that, near the branch point of the fork patterns U1 has the anti-symmetric distribution, and U2 has the symmetric distribution. So the vector of Ω can be used to evaluate the degree of symmetry of an interference pattern.

When ϕ i equals to a series of value from 0° to 340°, the corresponding points of Ω i are plotted in the ω1–ω2 plane as shown in Fig. 5a. From the calculation results, the position of Ω i point rotates around the original point of ω1–ω2 plane in the clockwise direction. Therefore, the argument of Ω i is employed to represent the main feature of the interference pattern between the OAM beam and plane wave. The relationship between the light phase difference and the argument of Ω i is shown in Fig. 5b. In Fig. 5b, all the 37 patterns in the training set are processed by Eq. (5). The phase difference ϕ ranges from 0° to 720°. Due to the wrapped phase effect, the argument value is restricted in [− π, π], which also limits the dynamic measuring range of this phase interrogation method. Within the range of [0°, 360°], there was a good linear relationship between phase difference ϕ i and the argument of Ω i . Specifically, the relationship is Ω i  = 180° − ϕ i . This interrogation method can be applied in the optical fiber sensing technology. When the variation of the experimental parameter can influence the propagation phase delay in optical fiber, this phase delay can be monitored by analyzing the interference pattern. The image analysis process only requires the operation of Eq. (5), when the two eigenpattern vectors are already determined by the training set. The efficient method is helpful to measure the phase difference between the measuring path and the reference path in real time.

Fig. 5
figure 5

a The position of Ω i in the ω1–ω2 plane, when ϕ = 0°, 20°, 40°, …, 340°. b The relationship between the phase difference ϕ and the argument of Ω i

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2.4 The principle of strain measurements

In this work, the strain measurement is carried out to verify the phase interrogation method. When the stress is applied on optical fiber, the phase delay of the propagating light can be generated. By considering the length variation and the elasto-optic coefficient of fiber, the phase delay can be expressed by Eq. (6):

$$\Delta \phi =n\frac{{2\pi }}{\lambda }L\left\{ {1 - 0.5{n^2}\left[ {\left( {1 - \nu } \right){p_{12}} - \nu {p_{11}}} \right]} \right\}\varepsilon$$
(6)

Here λ is the wavelength of the light source, L is the length of the fiber under strain, n is the effective refractive index of the fiber mode, p11 and p12 are the elasto-optic coefficient of the fiber, ν is the Poisson’s ratio, and ε is the strain. Therefore, there is a linear relationship between phase delay and applied strain on optical fiber. In the following experiments, the strain on optical fiber is converted to light phase difference by fiber MZI. Finally, the phase difference can be determined by image processing method based on PCA.

3 Experimental results

3.1 The OAM mode conversion

The OAM mode converter is an important part of the proposed system. There have been many attempts to generate and manipulate OAM beam. In free space, there are some optical devices to realize the OAM beam conversion, such as phase plates, cylindrical lenses, and spatial light modulators. Meanwhile, there is increasing interest in the generation of OAM beams in optical fibers [15, 16], because the OAM modes are intrinsic high order modes in optical fiber. To be compatible with the fiber-sensing structure, a kind of OAM mode converter based on the all fiber structure is proposed in this work as shown in Fig. 6a. Under the weak-guidance approximation condition, linearly polarized (LP) modes can be obtained by solving the scalar wave equation for optical fiber. There are two indices to describe LP lm mode, where l is the azimuthal index and m is the radial index. Each LP lm with l > 0 consists of twofold degenerate modes labeled by a and b, respectively. The OAM modes can be obtained by combining the two degenerate LP lm modes with π/2 or 3π/2 phase difference (i.e., OAM = LP lma  ± i × LP lmb ). For example, the OAM mode with l = 1 can be generated by the combination of LP11a and LP11b modes in TMF, which can support LP01 and LP11 modes [17]. Therefore, the LP01 mode, which cannot contribute to the OAM mode, should be suppressed in TMF. In the previous work, long-period grating is inscribed in TMF to convert the LP01 mode to LP11 mode at the resonance wavelength of the long-period grating [17]. This conversion process is efficient, but the operating wavelength is difficult to adjust. In this work, FMF is spliced to lead-in SMF with the core-offset of 15 µm. In this structure, when the LP01-mode incidents from SMF, the power ratio between LP11 and LP01 is above 20 dB in C-band (λ is from 1520 to 1570 nm) according the theoretical simulation. Therefore, the LP11 mode can be excited in FMF for a broadband wavelength range by the highly asymmetric structure in Fig. 6b. However, the phase difference between LP11a and LP11b modes is unknown. In order to obtain OAM mode, the rotator is employed in the system to clamp the TMF and apply a twist on the fiber, which can modify the phase difference between LP11a and LP11b modes. During rotating the fiber, the near-field field profile is monitored by CCD camera. In the experiment, the length of the twisted TMF is 25 cm. When the rotating angle is 45° approximately, the light beam with the annular intensity profile is obtained in Fig. 6c.

Fig. 6
figure 6

a The schematic diagram of OAM mode converter. b The beam profile of LP11 mode measured by CCD camera before rotating. c The beam profile of OAM mode measured by CCD camera after rotating

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3.2 The experimental setup

To verify the feasibility of the phase interrogation method, the sensing section in the measuring path is applied by stress. Both the measuring path and reference path are placed in the same environment simultaneously to eliminate the influence of temperature variation. As mentioned above, the strain causes the phase delay between the two paths of the Mach–Zehnder interferometer. In this case, the strain signal is converted to shape variation of the interference pattern, which is real-time monitored by the CCD camera. The information of the recorded digital image is extracted by the image processing method based on PCA as discussed in Sect. 2.3.

The photograph of the optical fiber strain sensing experiment system is shown in Fig. 7. In the system, distributed feedback (DFB) laser with linewidth of 3 MHz with a central wavelength of 1550 nm is used as the light source. The optical coupler is a 2 × 2 single-mode fiber coupler with splitting ratio of 1:1. The core diameters of SMF and TMF are 8.2 and 15 µm. The refractive index differences between the core and cladding of the two kinds of fiber are both (n1 − n2)/n1 = 0.336%. FMF is spliced to SMF with the core-offset of 15 µm. The splicing point is protected by epoxy resin and fixed by a holder. The length between the holder and the rotator is 25 cm. the output beams of SMF and TMF are collimated by 20 × objective lens. The non-polarized beam splitter (NPBS) is employed to generate the inference pattern on the CCD camera, whose maximum resolution is 1296 × 964. The fork interference pattern is obtained when the incident light power is 10 µW, which indicates that the sensing system is low power consumption.

Fig. 7
figure 7

Interference pattern for different strains applied on the sensing fiber. a 0 µε, b 20 µε, c 40 µε, d 60 µε, e 80 µε, f 100 µε, g 120 µε, h 140 µε

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During the measurement, a segment of SMF on the measuring path is clamped by two fiber holders, which are fixed on two movable stages, respectively. The stress is applied on the fiber by stretching the fiber along the axial direction. At the same time, the shape variation of the fork interference pattern is recorded on the CCD. Before the feature extraction, preprocessing is required to enhance the image quality. The median filter and mean filter are both used to process the fork interference patterns obtained in the experiment. The filter template size is 3 × 3. In the experiments, the length between the two fiber holders is 50 cm, and the minimum moving distance of the stages is 10 µm. In this experimental system, the minimum strain of 20 µε can be provided. When the applied strain on optical fiber is changed from 0 to 240 µε, the corresponding 13 interference patterns are recorded. The first 8 patterns are shown in Fig. 8. From the shape of the patterns, the branch point can be recognized in each pattern. It can be proved that the OAM beam with l = 1 is generated in the mode conversion process. The similar shape evolution process as the simulated results in Fig. 3 can be recognized, which indicates that the applied stress can modify the phase difference between two paths. It should be mentioned that, due to the small area of the interference patterns, the region of interest with the size of 430 × 430 is picked up to accomplish the following analysis.

Fig. 8
figure 8

The contribution rates of the eigenpatterns of the experimental images. Insets are the two eigenpatterns of λ1 and λ2

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3.3 Sensing performance

The PCA in Sect. 2.3 is used to analyze the fork interference pattern measured above. The eigenvalues representing the contributions of all the eigenpattern are shown in Fig. 8. The sum of contribution rates by the first two eigenpattern (i.e., principal components) is 89.9%, which is lower than the theoretical results (nearly 100%). This may be caused by the image distortion and the serious noise of the near infrared CCD camera. The two eigenpatterns of λ1 and λ2 (as shown in the insets of Fig. 8) are obtained for further analysis of the fork interference patterns. Then each image is projected into the (ω1, ω2) space by the linear transformation of Eq. (5). The results of PCA when the fiber is under different strain are shown in Fig. 9a, b. In ω1–ω2 plane, Ω i point experience the circular motion around the original point as predicted in Fig. 5a, when the strain on fiber is increased. Therefore, the argument of Ω i is still selected to characterize the shape of each pattern. When the strain continues to increase, the argument of Ω i may be outside that range from − π to π. However, the calculated value by the arctangent function will be still wrapped within the range of [− π, π]. When the sampling rate is sufficiently high and the noise level is low, the algorithm of one-dimensional phase unwrapping can be employed to increase the dynamic sensing range [18].

Fig. 9
figure 9

a The position of Ω i in the ω1–ω2 plane, when the applied strain ranges from 0 to 240 µε. b The relationship between the applied strain on optical fiber and the argument of Ω i

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To evaluate the sensitivity of this interrogation method, the linear fitting is used as shown in Fig. 10. The fitting curve can be expressed as Eq. (7), which indicates that the sensitivity is 0.04688 rad/µε:

Fig. 10
figure 10

The linear fitting results of the measured results processed by PCA

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$$Y=0.04688X+2.5457.$$
(7)

When the strain is further increased (i.e., larger than 240 µε), the interference pattern recorded by CCD becomes unstable over time. This may be explained by the viscoelastic nature of the polymer coating layer of optical fiber under large strain. The theoretical design of the phase interrogation method is verified by the strain-sensing experiments. The resolution of the experimental system is mainly affected by parameters of the imaging system and CCD camera. It should be noted that the bare optical fiber structure is employed as the sensing section without any process to increase sensitivities. If the more sensitive materials and fiber structure is introduced, the sensitivity and the resolution would be further improved.

4 Conclusion

In this work, the phase interrogation method based on OAM beam for optical fiber sensor is proposed and discussed. The feasibility of the proposed strain sensor has been verified by the simulation and experiments. PCA method is developed to extract the main features of the fork interference patterns. The patterns can be projected into a two-dimensional space by a simple linear transformation. The theoretical design is verified by the experiments of strain sensing. The sensitivity of the strain measurement is 0.04688 rad/µε. It is possible to increase the sensitivity and the resolution of the proposed strain sensor by improving the sensitive material and the parameters of the structure. With the advantage of high-resolution, real-time measurement and low power consumption, the proposed method has a great application potential in industrial fields required high-resolution strain sensing.