With the increasing demand for high-precision flow velocity measurement in various biomedical fields, such as blood flow rate monitoring [1,2], quantitative drug delivery [3], and biochemical analysis [4], various methods have been explored to measure flow accurately [5,6]. The available flow velocity measurement instruments can be classified into two categories: contact method and non-contact method. The former means that the fluid has contact with the measurement device, including electrochemical measurement [7] and mechanical structure measurement [8]. By contrast, non-contact measurement can avoid corrosion and contamination of the fluid, such as ultrasonic flow meters [9] and a laser Doppler velocimeter (LDV) [10–12], which has been widely reported in velocity measurement owing to the advantages of high accuracy and non-invasive measurement. However, the reference optical path and accurate optical path alignment is required in traditional LDV, making the construction of the sensing system relatively complicated and difficult, which limits the application scope of LDV [13]. In recent years, laser self-mixing interferometry (SMI) has drawn great attention due to the merits of a single optical path, auto-alignment, and high sensitivity [14]. By monitoring the power changes caused by the mixing of the feedback light from an external object and the intracavity light, SMI has been extensively used to measure basic physical quantities such as displacement [15,16], vibration [17,18], velocity [19,20], distance [21,22], refractivity index, and other physical quantities [23–27].

In particular, combining Doppler effect with SMI, researchers proposed a self-mixing LDV, which can obtain the flow velocity by recording the beat frequency signal of the original laser field and the scattered light field [28,29]. Subsequently, various lasers such as laser diodes [30,31] and microchip lasers [32] are employed as the sensing light source to measure the flow velocity. These self-mixing LDVs mainly focus on the spatial optical configurations that are unable to be obstructed by the measured target and are difficult to meet the real-time measurement needs of space-restricted or extreme environments [33,34]. To solve these problems, some researchers proposed a flow measurement system with an all-fiber structure [35,36], which has the advantages of optical path flexibility, counteracting electromagnetic interference. However, the aforementioned self-mixing LDVs rely on a one-dimensional measurement system to obtain the velocity signal of flow, which usually needs to know the angle between the laser beam and the velocity. In addition, the rates and direction of the flow are usually difficult to be simultaneously measured in the one-dimension measurement system. Importantly, the flow vector information is essential for health monitoring in the human blood flow vector field [37,38]. Multi-dimensional measurements may be considered an effective method to solve the above-mentioned problems, with a simpler structure compared to traditional heterodyne interference systems. Furthermore, it is difficult to identify each dimensional signal and to solve the crosstalk of multi-dimensional signals. To the best of our knowledge, the acquisition of flow vector information based on the SMI systems has not been reported. Therefore, there is an urgent need for a method to solve the above problems encountered in practical flow velocity measurement.

In this study, a 2D flow vector measurement is realized based on the laser feedback frequency-shifted multiplexing technology. The impact of the incident angle on actual fluid measurement can be eliminated by using an orthogonal beam structure. Optical feedback frequency-shifted multiplexing technology is introduced into this system to overcome the challenges posed by low-frequency noise, which not only can decompose 2D vector velocity signals into 1D scalar signal, but also distinguish velocity signals for two dimensions in the frequency domain. Meanwhile, the distributed measurement of flow velocity is naturally realized based on a platform-like frequency spectrum. Furthermore, comprehensive experimental investigations and numerical simulations were conducted to validate that the rate and direction of the fluid can be easily obtained in this 2D system, indicating the effectiveness, applicability, and practicability of our proposed system with an unknown incidence angle. This work may offer a reference for the acquisition and processing of high-dimensional vector signals, which has promising applications in the fields of biomedical monitoring and atmospheric turbulence.

The setup of the two-dimensional laser frequency-shifted feedback flow velocity measurement system is depicted in Fig. 1. A distributed feedback Bragg (DFB) fiber laser operating at 1550.10 nm is employed as the light source of the entire sensing system, represented by the purple dotted box. In Fig. 1, it can be seen that this DFB fiber laser has two output ports. In particular, the port with higher output power is designated for optical measurement, whereas the other port with lower output power is allocated for signal detection. Then the detection port is connected to a photodetector (PD), and subsequently its output is transferred to a spectrum analyzer. The output laser is coupled into an erbium-doped fiber amplifier 1 (EDFA1) through an optical coupler 1 (OC1). Afterwards, the amplified laser is then fed into port 1 of circulator 1 (CIR1). The laser output from port 2 of CIR1 is guided toward a collimating lens and then focused on the capillary by a convex lens. The dashed box in the upper right corner of Fig. 1 shows the actual schematic diagram of fluid velocity measurement based on an orthogonal beam (Appendix A). The inner diameter of the glass capillary is 0.5 mm, which is placed on a three-dimensional movable platform for accurate position adjustment. The milk inside the pipeline is pushed into the capillary by the syringe pump at a set rate. The scattered particles in the milk will flow together and follow the fluid motion. Due to the laminar distribution of fluid in the pipeline, the flow rates are related to the position inside the pipeline, and the flow rates vary at different positions (Appendix B), which provides different frequency shifts.

As scattered particles inside the pipeline coincide with the incident laser beam, a fraction of the back-scattered light will return to the DFB laser cavity along the original optical path, which causes the change of laser output power due to the laser SMI effect. The feedback light emerging from port 3 of CIR1 enters OC2, then traverses through acoustic-optic modulator (AOM) pairs, OC1, and eventually injects into the laser cavity. In this case, the feedback light passes through the AOM pairs only once, resulting in a single frequency shift in the measurement channel. Note that frequency shifting is generated by a pair of AOMs, whose frequency shifts are set to −100MHz and 101 MHz, respectively. When the light passes through the AOMs one time, it will cause a frequency shift of 1 MHz. After the frequency-shifted light returns into the laser cavity, SMI will happen and then the X-dimension measurement of flow velocity can be implemented by recording the frequency spectrum; whereas another beam of laser output from OC1 passes through the AOM pairs first and is subsequently connected to the common port of OC2 with a splitting ratio of 50:50. The light output from OC2 enters port 2 of CIR2 and is amplified by EDFA2. Then, the light from port 2 of CIR3 passes through the collimating lens and is focused on the capillary by a convex lens. Similarly, a fraction of the back-scattered light will return to the DFB fiber laser cavity along the original optical path, which causes the change of laser output power and frequency spectrum due to the laser SMI effect. In particular, the light in this optical path crosses AOMs twice, resulting in a double frequency shift (2 MHz) in the measurement channel; therefore, the Y-dimension information of flow velocity is also achieved in the overall measurement system. Due to the particles scattering, the laser beam emitted from the x-axis can either return from the x-axis direction or along the y-axis direction, which may result in a mutual coupling. However, the flow vector measurement will not be affected by it due to negligible crosstalk Doppler frequency shift. The detailed theoretical explanation of 2D laser feedback frequency-shifted multiplexing is provided in Appendix C.

Note that when the laser feedback direction forms an acute angle with the velocity direction, a positive Doppler frequency shift occurs and then the corresponding signal is located to the right of the AOM peak. Conversely, when the laser feedback direction forms an obtuse angle with the flow velocity, a negative Doppler frequency shift occurs, and the signal is located to the left of the AOM peak. Figure 2 illustrates the simulated spectral distributions for different directions of flow rate measurements when the angle between the pipeline and the beam is 45°. The frequency shifts of different positions are continuously distributed, presenting a platform-like signal, owing to a laminar state in the pipeline. The numerical simulation results consist of two parts, namely the feedback signal of the laser after the AOM frequency shift and the Doppler frequency shift signal after fluid scattering. When an AOM is introduced into the system, the feedback light is directly modulated by the AOM to generate a beat frequency signal fΩ. The process of generating flow velocity signals inside the pipeline can be regarded as the superposition of multiple Doppler frequency shift signals.

As shown in Fig. 2(a), it is evident that there is a plateau-like signal of flow velocity on the left side of the fΩ peak, which reveals that the X-dimension of the laser feedback direction forms an obtuse angle with the flow direction. This is caused by the superposition of Doppler frequency shifts arising from the velocity distribution inside the pipeline. Meanwhile, we can observe that the signal locates on the left side of the 2fΩ peak, showing the Y-dimension of the laser feedback direction also forms an obtuse angle with the flow direction. Similarly, Figs. 2(b)–2(d) illustrate the spectral distributions of velocity signals in other situations, and the inset image in each figure illustrates the projection of flow velocity on two axes. Here the X- and Y-axes represent the feedback light beam direction of the corresponding coordinate axis, respectively. Additionally, the (+) sign means the flow causes a positive frequency shift on this axis, while the (−) sign means the flow causes a negative frequency shift on this axis. By this way, the direction of flow can be determined by observing the frequency spectrum characteristics of the SMI signal. Meanwhile, the actual flow velocity and incident angle can be calculated from the scalar information of the two signals. Hence the flow vector signal can be obtained in the case of an unknown incident angle in the orthogonal dual-beam measurement system, which has the potential to be applied to blood flow monitoring and fluid vector field monitoring of the ocean and atmosphere.

Note that the output power of the laser is 85.9 μW at the microwatt level, and the insertion loss of the AOM is −3dB/cycle unit, which causes the incident light intensity to be attenuated to 1/4 of its original value. Therefore, the EDFA is utilized for amplifying the laser in the sensing system to ensure sufficient optical feedback level. When the syringe pump speed is set to 5000 μL/min and the incident angle is 45°, a typical result of two-dimensional flow rate measurement is obtained by a spectrum analyzer, as presented in Fig. 3.

In Fig. 3, two signal peaks appear in the spectrum corresponding to the SMI velocity signals caused by the feedback light passing through the AOMs once and twice, respectively. The inset image with a red background in the top left corner and the inset image with a blue background in the top right corner illustrate the zoomed-in picture of the two flow velocity components, respectively. It is clear that there is a plateau-like signal on the left side of the 1 MHz peak (X-dimension), which indicates that the laser feedback direction forms an obtuse angle with the flow direction. Similarly, the plateau-like signal is on the right side of the 2 MHz peak (Y-dimension), showing that the laser feedback direction forms an acute angle with the flow direction. These experimental results are in good agreement with the theoretical simulation provided in Fig. 2. In addition, there is also a peak located at 234 kHz to be observed, which corresponds to the relaxation oscillation noise of the DFB fiber laser.

To verify the feasibility and reliability of the measurement system under different flow velocity states, a syringe pump is utilized to control the flow velocity. The pump rate is adjusted within a range of 200–2000 μL/min at an interval of 200 μL/min when the incident angle is 30°. Convert the velocity at the set syringe pushing speed based on the volume relationship of the flow, obtaining the corresponding linear velocity of 372–3720 μm/min at an interval of 372 μm/min. The obtained Doppler frequency shift varies with the variation of the flow rates, and the corresponding SMI signals behaviors are also different in the frequency domain, as shown in Fig. 4. The spectrograms of the two dimensions (X- and Y-dimensions) with different syringe pump speeds are given, as shown in Figs. 4(a) and 4(b), separately. Notably, the central shift frequencies of 1 MHz and 2 MHz remain constant. As the flow rate increases, the bright areas gradually widen. The broadening of the spectrum is mainly attributed to the superposition of Doppler signals generated by continuously distributed particles in the pipeline. The width of the frequency signal represents the Doppler frequency shift caused by the maximum flow velocity. That is to say, the higher the flow velocity, the wider the Doppler signal. In addition, the linewidth of the laser source, spot size, and the vibrations of the glass capillary can also affect the broadening of the frequency signals. Figures 4(c) and 4(d) present the Doppler shift caused by maximum flow velocity. The data points represent the average value of the Doppler frequency shift, and the error bars indicate the standard deviation of the measurement data. Figure 4(c) shows the experimental results for the X-dimension; the frequency shift linearly increases from 2.1 to 20.7 kHz with the increasing syringe pump speed. Similarly, the frequency shift linearly increases from 1.3 to 14.3 kHz in the Y-dimension, as presented in Fig. 4(d). The red lines in the graph represent the linear fitting of the experimental data (Pearson coefficient above 99%). In the experiment, the measurement error mainly originates from the disturbances caused by the instability of the pipeline as well as the discrepancies between the beam diameter and the internal diameter of the pipeline.

The frequency shifts of different positions are continuously distributed, presenting a platform-like signal. That is to say, distributed measurement of flow rates can be implemented by recording the spectrum information. Correspondingly, the velocity distribution can be obtained, as shown in Fig. 5.

Two parabolas located in the coordinate axis plane illustrate the distribution of flow velocity in the X- and Y-dimensions. The flow rates at the center of the pipeline are maximum and gradually decrease in all directions until the position of the pipe wall is 0. The maximum rates in the X- and Y-dimensions are 1.59 mm/s and 1.02 mm/s, respectively. The resultant velocity distribution of fluid by synthesizing the X-dimension and Y-dimension is displayed with a maximum flow rate of 1.89 mm/s, which also presents a parabolic distribution. Meanwhile, the angle θ between the projection of the synthesized velocity in the X–Y plane and the X-axis is calculated to be 32.8° [see Eq. (A4) in Appendix A]. Therefore, with the unique feature of SMI, the flow rate at each position where the beam passes by in the pipeline can be obtained, achieving distributed measurement of flow velocity and incident angle acquirement.

In the experiment, the syringe pump speed maintains a constant rate at 800 μL/min when adjusting the angle between the capillary and the output laser beam. Figure 6(a) illustrates that the Doppler frequency shift of the X-dimension decreases with an increase of angle, which is attributed to the cosine relationship between the Doppler frequency shift and the angle. Similarly, the Doppler frequency shift of the Y-dimension increases with the increasing angle, as presented in Fig. 6(b), which follows a sinusoidal function.

The actual flow rates are obtained by the composition of velocities in the X- and Y-dimensions as shown in Fig. 6(c). In our 2D laser feedback frequency-shifted system, the flow velocity is independent of the angle. Thus, when the syringe pump speed maintains a constant rate, the velocity should be the same at different angles. The velocity basically maintains at 0.0114 m/s, as shown in Fig. 6(c). It is pointed out that the error mainly comes from the variation of the spot position with the rotation of the pipeline and the weak feedback light intensity at small angles. This also leads to an experimental measurement angle θ of 10°–80°. In addition, by calculating the Doppler shift of two dimensions, the angle between the flow direction and the laser beam can be determined. Thus, the direction of the flow in the plane can also be obtained, as shown in Fig. 6(d). The purple data points represent the calculated angle, and corresponding linear fitting of the measured results is denoted in the red line, which shows a good linear relationship with the Pearson coefficient above 0.999. Furthermore, we calculated the corresponding standard deviation for each angle, as represented by the blue bar column, and the maximum standard deviation is less than 1.5° in the angle range of 10° to 80°. Results indicate that the proposed all-fiber frequency-shifted SMI system can measure any flow vector in the plane under different angles.

In summary, an orthogonal measurement system based on laser feedback frequency-shifted multiplexing technology is proposed and experimentally demonstrated. By employing frequency-shifted multiplexing technology, the signals of two dimensions are distinctly distinguished in the frequency domain. 2D vector information of fluid motion is decomposed into 1D scalar signal, and the rate and direction of fluid can be easily obtained based on the strong correlation between the fundamental frequency signal and doubling frequency signal. In addition, the velocity distribution of laminar fluid inside the pipeline is also characterized. The experimental results are in good agreement with the numerical simulation of the SMI in this 2D sensing system. When the syringe pump speed is within the range of 200–2000 μL/min, the flow velocity is measured with a high level of linearity. With the incidence angle changing, the frequency shift is different in the X- and Y-dimensions when keeping the syringe pump speed constant, whereas the flow rate remains basically unchanged, which further verifies that the proposed flow vector measurement system is independent of the incidence angle. Meanwhile, the incidence angle can also be obtained with the maximum standard deviation below 1.5°. These results highlight the effectiveness of SMI sensors in detecting flow velocity in a glass capillary and determining the angle between the velocity and the laser beam. By overcoming the limitations of 1D measurements and avoiding the influence of unknown angles, the proposed sensor allows for the measurement of the in-plane vector velocity and its distribution, which may pave the way for multi-dimensional fluid flow distributed measurement in the field of biomedical and atmospheric turbulence.