## 2 Light Field Transmission Model  
The principle of the dual-wavelength laser heterodyne vibration measurement system based on a fiber optic Mach-Zehnder configuration, as proposed in this paper, is illustrated in Figure 1. Two lasers with closely matched wavelengths are coupled into the same optical fiber via a wavelength division multiplexing (WDM) device. The laser is split into reference and transmitted beams using a beam splitter. The transmitted light is focused onto the rough target using an optical fiber point diffraction method through a transmitter-receive telescope. The reflected light then returns through the telescope and optical fiber circulator, forming the measurement beam, which interferes with the reference light that has been frequency-shifted by a fiber Bragg grating. The heterodyne interference is generated at the coupler, and the interference signals of different wavelengths are separated by two WDM devices. Assume that \(P_{r}\) and \(P_{m}\) represent the powers of the reference and measurement beams incident on the detector, so that the interference intensity \(I(t)\) could be expressed as follows:  
\[I(t)=P_{r}+P_{m}+2\sqrt{P_{r}P_{m}}\cos(2\pi f_{0}t+\frac{4\pi}{\lambda}s(t)), \tag{1}\]  
In the equation, \(f_{0}\) is the modulation frequency of the fiber Bragg grating, \(s(t)\) represents the time domain displacement of the rough target, and \(\lambda\) is the laser wavelength. Finally, the photonic current \(i(t)\) is produced at two balanced photodetectors (BPDs) after photoelectric conversion, which can be expressed as below. Notice that the direct current (DC) component in Equation (1) can be effectively eliminated due to the principle of balanced detection, retaining only the alternating current (AC) component. \[i(t)=2K\sqrt{P_{r}P_{m}}\cos(2\pi f_{0}t+\frac{4\pi}{\lambda}s(t)), \tag{2}\]  
In the equation, \(K\) denotes the detector responsivity. In a laser heterodyne vibrometer system limited by speckle noise, the demodulated velocity noise, \(v_{n}\), is determined by the measurement of light power, as expressed by the following equation:  
\[v_{n}\propto\sqrt{\frac{1}{P_{m}}}, \tag{3}\]  
In the above equation, it is evident that higher measurement light power leads to reduced velocity noise. Therefore, the speckle coupling efficiency following the laser transmission and reception in the simulated vibrometer system is of significant importance. The laser transmission and reception process comprises four stages, as shown in Figure 1 and Table 1. These stages include the generation of the light field at the fiber's emission end face (F-face), the free-space propagation of the laser from the F-face to the rough target surface (T-face), the generation of a new emitted light field at the T-face due to phase accumulation, the free-space transmission from the T-face back to the F-face, and the light field coupling process. Figure 1: Schematic diagram of the dual-wavelength laser heterodyne vibrometric system. When a laser exits from a single-mode fiber, the expression for the stable light field distribution within a step-index fiber becomes complex. To simplify, the weakly guiding fiber assumption can be applied, where the refractive index of the fiber core is nearly identical to that of the cladding, thereby weakening the fiber's guiding effect on electromagnetic wave transmission.