In the general case, upon free-space propagation of a paraxial optical vortex (OV) its topological charge (TC) is not conserved, unlike the orbital angular momentum (OAM), which remains unchanged. In this work, we discuss a Gaussian beam with fractional TC in the source plane, showing theoretically and numerically in which way the TC is changing (but remains integer) upon free-space propagation. There are four evolution scenarios for an original OV with fractional TC that depend on how close the original TC is to an integer even or odd number. If the TC in the initial plane has an arbitrary fractional value, it becomes an odd integer in the Fourier plane. If the initial TC is close to an odd integer number, then, on propagation, an OV with TC +1 is born in the Fresnel diffraction zone, while in the far field an OV with TC -1 appears. For simple OVs (like Laguerre-Gaussian or Bessel-Gaussian modes), TC is conserved both upon propagation and following weak phase distortions. When scattered from a random phase screen, the integer TC of an OV is experimentally shown to conserve until random path-difference distortion reaches half wavelength. Because of this, under weak-turbulence conditions, it makes sense to measure a discretely changing TC, rather than measuring the continuously varying OAM.

中文翻译：

---

具有初始分数拓扑电荷的光学涡旋的演化

在一般情况下，旁轴光学涡旋（OV）在自由空间传播时，其拓扑电荷（TC）不守恒，这与轨道角动量（OAM）保持不变相同。在这项工作中，我们讨论了在源平面中具有分数TC的高斯光束，从理论和数值上显示了自由空间传播时TC发生变化（但仍为整数）的方式。具有分数TC的原始OV有四种演变方案，具体取决于原始TC与整数偶数或奇数的接近程度。如果初始平面中的TC具有任意小数值，则它在傅立叶平面中成为奇数整数。如果初始TC接近奇数整数，则在传播时，菲涅耳衍射区将生成TC +1的OV，而在远场中将出现TC -1的OV。对于简单的OV（如Laguerre-Gaussian或Bessel-Gaussian模式），在传播时以及跟随弱相位失真后，TC都是守恒的。当从随机相位屏幕散射时，实验表明OV的整数TC可以保存，直到随机的路径差失真达到一半波长为止。因此，在弱湍流条件下，测量离散变化的TC而不是测量连续变化的OAM是有意义的。