Self-mixing interferometry (SMI), also known as laser feedback interferometry (LFI), refers to the reflection or scattering of light from a laser by an external target surface, returning to the laser cavity along the original optical path, and interfering with the optical field in the cavity, leading to alterations in the laser's power, phase, frequency, polarization state, and other physical properties. This phenomenon was discovered by King and Steward in 1963 and has since garnered the attention of numerous academics [[1], [2], [3]].

In comparison to conventional two-beam interferometers, such as the Michelson interferometer, SMI has the advantages of high sensitivity, simple structure, and ease of collimation. Furthermore, it is capable of completely non-contact measurement, which enables online monitoring of precision processing equipment and processes in real-time. Consequently, SMI has a promising future in the field of precision measurement and sensing technology and has been applied to the measurement of displacement [[4], [5], [6]], vibration [[7], [8], [9]], distance [[10], [11], [12]], velocity [[13], [14], [15]], angle [16,17], biological signal [18,19], trace detection [20,21], refractive index [22,23], acoustical signal [24,25], etc.

Currently, researchers have constructed various SMI systems, and SMI based on microchip laser has garnered considerable interest for a long time due to its ultrahigh sensitivity [[26], [27], [28], [29], [30], [31]]. Nevertheless, since the entire optical path is interferometric, the output light in propagation is susceptible to environmental change and thermal creep of optical devices, leading to an increase in error. In a self-mixing interferometer, the system error is mainly composed of the error inside the interferometer and the external optical path error. Existing error compensation schemes can compensate for errors inside the interferometer, Wan et al. proposed the quasi-common-path scheme based on frequency shifting [32], which requires the placement of a glass plate at a specific angle to reflect the reference light because the reference light cannot return to the laser cavity along the original optical path. Not only is it difficult to adjust the optical path, but in practice, the glass plate can only be placed closer to the laser and therefore only compensates for errors inside the interferometer. When the measurement target is far away, it is not possible to compensate for errors in the external optical path. Zhang et al. proposed the common-path scheme by combining polarization and frequency shifting [33], which compensates for errors in the external path due to environmental changes. However, the method uses two light sources and requires beam splitting, resulting in a complex system structure and difficult adjustment of the optical path. In addition to this, it has more optical elements, which will produce weak reflections on the surface of the optical elements, reducing the signal-to-noise ratio of the self-mixing signal.

To resolve the limitations of the aforementioned measurement systems, in this paper, a self-mixing interferometry scheme based on microchip laser and heterodyne frequency shifting technology is proposed, in which the diffracted light of different orders of the acousto-optic modulator is utilized as the measurement and reference light to compensate the error, which not only compensates for errors inside the interferometer but also eliminates disturbances in the external optical path due to environmental changes. Furthermore, it has a simple system structure, high robustness, and easy optical path adjustment, which improve stability while ensuring high precision.