Fiber lasers have been widely used in industry, national defense, and other fields, with the advantages of compact structure, high efficiency, and flexible energy transmission. Beam quality is one of the most important parameters of the fiber laser and directly determines the performance and propagation effects of the laser. Various beam quality parameters have been proposed so far, such as factor M2, Strehl ratio, barrel power, and diffraction limit multiples. In all these parameters, factor M2 is a relatively perfect evaluation parameter of the laser beam quality and can reflect both the near-field and far-field characteristics of the laser. With the increase in the laser power, thermal accumulation and nonlinear effects in the laser can cause dynamic changes in the beam quality. Dynamic measurement of factor M2 is beneficial to reveal the physical mechanism of mode field changes of the laser and control the real-time distribution of the laser mode field. Therefore, it has great significance to the design, manufacture, and application of lasers.

In this study, we propose a dynamic measurement method of factor M2 of a fiber laser using coaxial interferometry. The far-field complex amplitude of the laser under test is determined by the spatially phase-shifted coaxial interferograms. The intensity distributions of the laser at different places in the vicinity of the laser near-field are then obtained through angular spectrum transmission and lens transformation. Factor M2 of the laser is finally determined by fitting the beam diameters at different places. Our new method has a compact structure that avoids manufacturing and assembly errors caused by optical lenses. Compared with off-axis holographic technology, the method described in this study can obtain fast, high-precision, and complex amplitude information with high resolution and realize the fast and accurate measurement of laser beam quality factor M2 factor.

In order to verify the feasibility of the factor M2 measurement method, the measured results of lasers from different fibers are compared with those measured by the commercial factor M2 measurement system (BEAM SQUARED). Two experimental setups are shown in Fig. 7. The fiber laser under test is collimated by an infinitely conjugated microscope objective lens and then reflected into the commercial factor M2 measurement system through two aluminized mirrors, as shown in the dashed box of Fig. 7. Through comparison, our coaxial interferometer in the solid line frame of the figure is more compact. It should be noted that to ensure the consistency of the laser under test in the two measurement systems, we only measure the beam quality of the P-light in the dashed box.

Figure 8 shows the factor M2 results of the output from 630-HP fiber (Nufern), 1060-XP fiber (Nufern), and SMF-28e fiber (Corning) with two different systems shown in Fig. 7. The three fibers can transmit 1, 6, and 10 modes in a single polarization direction, respectively. For each fiber laser, we repeat the measurement three times and use their average values as the results. Figure 7 also compares the light intensity at the waist position with different methods. For 630-HP fiber, the measurement result of the proposed method is Mx_c2=1.09 and My_c2=1.03, and that of the commercial BEAM SQUARED is Mx_m2=1.04 and My_m2=1.04. The average measurement error of M² is 0.028. For 1060-XP fiber, the measurement result of the proposed method is Mx_c2=1.16 and My_c2=1.46, and that of the commercial BEAM SQUARED is Mx_m2=1.10 and My_m2=1.58. The average measurement error of M² is 0.065. For SMF-28e fiber, the measurement result of the proposed method is Mx_c2=1.95 and My_c2=2.53, and that of the commercial BEAM SQUARED is Mx_m2=2.06 and My_m2=2.45. The average measurement error of M2^{is 0.043. In the verification experiment of the beam quality, in order to avoid the system error introduced by building the measurement system repeatedly, we only move the fiber to be tested to switch the system. However, when the measurement system is switched, the attitude of the optical fiber will be slightly changed, which will change the output mode field of the laser and thus cause the measurement error of M2 of the beam to be measured. The error is more obvious when the number of modes is large, and excessive modes will lead to mode coupling, which is also the reason for the poor repeatability of multiple measurements of SMF-28e in the above experimental results.}

We propose a new method for the dynamic measurement of beam quality factor M2 of a fiber laser by using far-field coaxial interferometry. The far-field complex amplitude of the laser under test is determined by the spatially phase-shifted coaxial interferograms. The intensity distributions of the laser at different places of free space are then obtained through angular spectrum transmission and lens transformation. factor M2 of the laser is finally determined by fitting the beam diameters at different places. In the experiment, we have measured factor M2 of the laser output from fibers with different core diameters at 633 nm. The results are consistent with those determined by the commercial beam quality instrument. In addition, it takes 0.02 s to complete the factor M2 calculation of lasers by the proposed method, which is more than two orders of magnitude faster than that of the commercial instrument. Our new method has a compact structure that avoids manufacturing and assembly errors caused by optical lenses. The method provides a technical means for the quality detection and state monitoring of few-mode fibers and their devices and is conducive to the design, manufacture, and development of fiber lasers and their fiber devices.