非傍轴光束级数修正解的有效性

当光束具有较大的发散角或光束束腰可与波长相比拟时，傍轴近似不再成立。需要发展更为严格的非傍轴处理方法。迄今已提出许多方法．例如微扰法、级数展开法、算子法、角谱分析法和波动方程积分解等方法，用以研究光束的非傍轴传输行为。不同方法具有各自的优缺点和适用范围。分析了非傍轴光束级数解的有效性。对波动方程积分解和级数解作了详细的数值计算和比较，指出当ω0≤0．2251λ时(ω0为束腰宽度，λ为波长)，级数解对傍轴解有修正作用。但是，当ω0≤0．2251λ时，级数解会给出无意义的结果，并且会发散。级数解必须在其有效范围内使用。级数解的适用范围与光束的束腰宽度、传输距离以及所使用的级数解的阶次有关。

Validity of Series Expansion Corrections to the Paraxial Approximation of Non-Paraxial Beams

It is well known that the paraxial approximation is no longer valid for the beams with large divergence angle and/or small spot size comparable with the wavelength. Thus, a rigorous nonparaxial treatment becomes necessary. As yet, a variety of approaches, such as the perturbation method, power-series expansion, angular spectrum representation and integral solution of the wave equation etc. have been proposed to treat the beam propagation behavior beyond the paraxial regime. In this paper the validity of series expansion corrections to the paraxial approximation of non-paraxial beams is studied. Detailed numerical calculations for the integral expression and series expansion of the wave equation are performed and compared. It is shown that the series expansion provides a valid correction to the paraxial solution as w0>0.2251λ, with w0 and λ being the waist width and wavelength, respectively. However, the series expansion solution delivers unrealistic results and may become divergent if w0≤0.2251λ. The applicable range of the series expansion approach depends on the waist width, propagation distance and order of the series expansion. The results obtained in this paper are useful for studying the propagation of nonparaxial beams.