Determination of the optimal truncation ratio for focused flattened Gaussian beams based on encircled power analysis

In atmospheric propagation of high-power lasers, far-field energy concentration plays a critical role in determining system performance, and the encircled power radius serves as an important evaluation metric. Although the influence of truncation ratio on Gaussian beams has been extensively investigated, an effective truncation criterion for flattened Gaussian (FG) beams remains unavailable. In this work, an analytical expression for the focal-plane optical field of truncated FG beams in free-space propagation is derived. Based on the encircled power distribution, a quantitative criterion termed the lossless truncation ratio is proposed to determine the optimal transmitting aperture. The relationships among encircled power radius, truncation ratio, and beam order are systematically established. The results show that increasing the truncation ratio improves far-field energy concentration, although the improvement gradually saturates. The lossless truncation ratio decreases with beam order and approaches approximately 2.8 when the beam order reaches 9. To extend the analysis to realistic atmospheric environments, numerical simulations incorporating turbulence and thermal blooming effects are performed. The results reveal that stronger thermal blooming induced by higher transmitted power and stronger turbulence both reduce the optimal truncation ratio, while the influence of beam order gradually weakens at higher orders. Furthermore, comparisons with super-Gaussian beams demonstrate that the proposed criterion remains consistent for practical flat-top beam profiles and can be extended to non-integer beam orders. The proposed criterion provides a practical guideline for transmitting-aperture design and far-field energy optimization of high-power flattened Gaussian beam systems operating in atmospheric environments.