This supplementary text provides a detailed explanation of the high-precision two-dimensional displacement metrology based on matrix metasurface. The principle of the 2D displacement sensor is based on matrix Fourier optics, where the spatially-varying Jones matrix of the metasurface is used to achieve diffraction and polarization analysis. The metasurface is designed to have specific polarization states for different diffraction orders, enabling the sensor to measure displacement in two dimensions. The sensor uses a 4F optical system to recombine the light beam after the metasurface, and the output optical field is calculated based on the Jones matrix and the incident polarization state. The power of the output field is analyzed to determine the displacement parameters. The Pancharatnam-Berry phase is used to analyze the polarization evolution of the light beams contributing to the output optical interference, enabling the measurement of displacement with a phase shift of π/2 between the horizontal and vertical polarization components. The metasurface is optimized to maximize the power of the three diffraction orders and ensure equal power distribution among them. The optimization process involves gradient descent to find the optimal parameters for the metasurface. The metasurface is fabricated using a diagrammatic process involving lithography, etching, and resist removal. The data acquisition and processing involve measuring the modulation powers and extracting the phase information using Lissajous figures and ellipse fitting. The calibration process involves determining the linear relationship between the measured phase and the displacement parameters. The power contrast of the output light is optimized to enhance the contrast and ensure accurate displacement measurement. The results show that the metasurface can achieve high-precision displacement measurement with a sensitivity of up to 1.4 times higher for the (1,0) diffraction order compared to the (0,1) and (0,-1) orders. The study demonstrates the effectiveness of the matrix metasurface in achieving high-precision two-dimensional displacement metrology.This supplementary text provides a detailed explanation of the high-precision two-dimensional displacement metrology based on matrix metasurface. The principle of the 2D displacement sensor is based on matrix Fourier optics, where the spatially-varying Jones matrix of the metasurface is used to achieve diffraction and polarization analysis. The metasurface is designed to have specific polarization states for different diffraction orders, enabling the sensor to measure displacement in two dimensions. The sensor uses a 4F optical system to recombine the light beam after the metasurface, and the output optical field is calculated based on the Jones matrix and the incident polarization state. The power of the output field is analyzed to determine the displacement parameters. The Pancharatnam-Berry phase is used to analyze the polarization evolution of the light beams contributing to the output optical interference, enabling the measurement of displacement with a phase shift of π/2 between the horizontal and vertical polarization components. The metasurface is optimized to maximize the power of the three diffraction orders and ensure equal power distribution among them. The optimization process involves gradient descent to find the optimal parameters for the metasurface. The metasurface is fabricated using a diagrammatic process involving lithography, etching, and resist removal. The data acquisition and processing involve measuring the modulation powers and extracting the phase information using Lissajous figures and ellipse fitting. The calibration process involves determining the linear relationship between the measured phase and the displacement parameters. The power contrast of the output light is optimized to enhance the contrast and ensure accurate displacement measurement. The results show that the metasurface can achieve high-precision displacement measurement with a sensitivity of up to 1.4 times higher for the (1,0) diffraction order compared to the (0,1) and (0,-1) orders. The study demonstrates the effectiveness of the matrix metasurface in achieving high-precision two-dimensional displacement metrology.