## 3 Numerical Simulation in Typical Application Scenarios  
In practical application scenarios, the LDV is employed to detect micro-vibration signals from rough targets located 10 m away. The selected single-mode fiber has a core radius of 5 \(\upmu\)m, a core refractive index of 1.45, and a cladding refractive index of 1.445, yielding an optical fiber numerical aperture (NA) of approximately 0.12. The lens features a focal length of 200 mm and an aperture diameter of 50 mm. The chosen laser wavelengths are 1550 nm, 1554 nm, 1558 nm, and 1562 nm. As discussed in Section 2, the stable light field transmission in the single-mode fiber results in the light field distribution at the fiber's emission end, as shown in Figure 2 for the various wavelengths. As shown in Figure 2, the light field transmitted stably within the fiber is approximately the same for different wavelengths under the simulation constraints and based on the light field transmission process discussed in Section 2. To ensure sufficient surface resolution and prevent \(FFT\{U_{T,RX}(x_{i},y_{i})\}\) spectral aliasing in the reconstruction process, we consider a rough target with a correlation length of 700 \(\upmu\)m and a root mean square (RMS) height deviation of 40 \(\upmu\)m. The resulting surface topography distribuThe laser beam waist (indicated by the red circle) at the target surface was set to 1 mm in diameter, matching the characteristic scale of the surface roughness features to enable proper optical interaction. The convolution operations in Equations (13) and (15) can be efficiently implemented using Fourier domain processing:  
1. Transform to frequency domain: Apply the Fourier transform to convert the spatial domain optical field into its spectral representation. 2. Point-wise multiplication: Instead of direct spatial convolution, perform a Hadamard (element-wise) product of the frequency domain fields. 3. Inverse transform: Reconstruct the convolved spatial result via the inverse Fourier transform. The light field distribution at the fiber end for different wavelengths reflected from the same rough target was obtained, as illustrated in Figure 4.  
Figure 3: Rough target surface topography. Figure 2: Optical field distribution of different wavelengths at the fiber end (transmitting light): (**a**) 1550 nm, (**b**) 1554 nm, (**c**) 1558 nm, and (**d**) 1562 nm. In Figure 4, it can be observed that the speckle patterns formed at the fiber end for different wavelengths exhibit similarities, but are not identical. As the wavelength difference increases, the discrepancies in the speckle patterns become more pronounced. The final calculated coupling efficiencies are 0.50, 0.30, 0.46, and 0.73, corresponding to different interference photocurrent amplitudes. It is important to note that these coupling efficiency values are specific to a particular rough target. Since the speckle characteristics vary across different targets, the coupling efficiency is not a constant. To better illustrate the differences in the light fields reflected from surfaces with varying roughness at the fiber end face, as well as the resulting variations in coupling efficiency, a new rough surface distribution was generated, and the reflected light field at the fiber end face was obtained, as shown in Figure 5.  
The final calculated coupling efficiencies are 0.33, 0.59, 0.70, and 0.65. In specific application scenarios, when the rough target moves and the reflected coupling efficiency  
Figure 4: Optical field distribution of different wavelengths at the fiber end (receiving light): (**a**) 1550 nm, (**b**) 1554 nm, (**c**) 1558 nm, and (**d**) 1562 nm. Figure 5: Optical field distribution of different wavelengths at the fiber end based on newly generated rough target(receiving light): (**a**) 1550 nm, (**b**) 1554 nm, (**c**) 1558 nm, and (**d**) 1562 nm. is low, the demodulated signal noise increases, potentially resulting in speckle noise.