## 3 Critical Technical Approach  
### Large-Aperture, Far-Field Uniformity Modeling and Design  
Conventional wavelength-specific homogenization techniques--such as diffractive optics, lens arrays, and spatial light modulators (SLMs)--are fundamentally constrained by chromatic aberrations, limited scalability, and power-handling inefficiencies, making them unsuitable for applications requiring simultaneous large-aperture and far-field uniformity [20; 21; 22; 23; 24; 25; 26]. In contrast, our fiber-apodizer architecture overcomes these limitations by exploiting controlled multimodal propagation in a tailored optical fiber, synergized with precision apodization for optimized beam shaping. This composite design achieves exceptional far-field intensity uniformity propagation while maintaining wavelength selectivity--a critical capability for advanced laser systems demanding joint spectral and spatial control. The shaping principle is based on the Multimode Interference-Mode Interference (MIMI) mechanism. The homogenizing fiber supports multiple guided modes, and when the input is a Gaussian beam, each guided mode undergoes phase evolution along distinct propagation constants within the fiber. The output optical field distribution \(E(x,y)\) is given by Equation (1):  
\[E(x,y)=\sum\limits_{m=1}^{N}c_{m}\psi_{m}(x,y)e^{i\beta_{m}L} \tag{1}\]  
where \(\psi_{m}\) denotes the modal eigenfunction, \(\beta_{m}\) represents the propagation constant, and L is the fiber length. Due to phase differences between the supported modes, the output field forms a spatially varying multimode coherent superposition. After temporal or statistical averaging, the output intensity distribution can be approximated by Equation (2), which averaging effectively mitigates central bias and power fluctuations in the input beam. However, residual high-order modes can still induce localized modulation near the beam edge, resulting in a non-uniform energy distribution in the peripheral region.  
\[I\approx\frac{1}{A}\iint\limits_{\text{core}}I_{in}(x,y)dxdy \tag{2}\]  
where \(I_{in}(x,y)\) denotes optical intensity distribution, and \(A\) denotes cross-sectional area of the fiber core.  
To further suppress high spatial frequency components, an apodizing diaphragm with a specific aperture diameter \(D\) is introduced at the output end of the homogenizing fiber. This diaphragm physically truncates the edge components of the beam, functioning as a spatial frequency filter. The relationship between the homogenization performance \(U_{E}(D)\) in the diaphragm diameter D is given by Equation (3):  
\[U_{E}(D)=1-\frac{\int_{r>D/2}I(r)dr}{\int\limits_{0}^{R}I(r)dr} \tag{3}\]  
where \(I(r)\) denotes optical intensity distribution within radius \(r\), and \(r\) denotes the radius within which the uniformity requirement is satisfied.  
To verify the theoretical effectiveness of the beam homogenization scheme, the energy distribution at the equivalent focal plane of the homogenizing fiber was measured and analyzed. Theoretical calculations indicate that within the central 30% region of the output beam diameter, the irradiance uniformity can reach up to 90%. This result provides design constraints for determining the optimal diaphragm size.  
During the theoretical design phase, to ensure that the irradiance uniformity within the central region exceeds 90%, the optimal diameter range of the apodizing diaphragm Dwas determined to be between 15 mm and 40 mm. Within this range, high-frequency modal structures at the beam edges can be effectively filtered out, significantly improving spatial energy uniformity at the cost of a slight reduction in total transmitted energy. The specific dimensions of the apodizer used in the experimental setup are detailed in the following experimental section. The working principle of the homogenization structure combining the fiber and apodizer is illustrated in Figure 2.