光学奇点的短程和长程屏蔽
Short and Long Range Screening of Optical Singularities
摘要：
讨论了具有短程关联和长程关联的近轴光场拓扑荷（奇点）的屏蔽。对于短程屏蔽，半径为$R$的圆形区域中的电荷方差随$R$线性增长，而不是如在没有屏蔽的情况下所预期的那样随$R^{2}$线性增长;对于长程屏蔽，电荷方差比$R$增长更快：对于自相关函数为零阶贝塞尔函数J_{0}的场，电荷方差增长为R ln R$。一个J_{0}关联函数是无法实现的，在实践中，但我们展示了如何产生一个光场的关联函数密切接近这种形式。电荷方差可以通过计算区域A内的正负奇点来测量，或者更容易地通过计算A周边上的带符号的零交叉来测量。在\For中，第一种方法通过对电荷关联函数C（r）积分来计算电荷方差，第二种方法通过对过零关联函数Gamma（r）积分来计算电荷方差。使用C（r）和Gamma（r）的显式形式，我们表明两种计算方法产生相同的结果。我们表明，对于短距离筛选的零交叉可以计数沿直线的长度等于P，但对于长距离筛选这种简化不再成立。我们还证明了，对于可实现的光场，当R足够小时，电荷方差为R^2，而当R足够大时，电荷方差为R。这些普遍规律适用于短程和伪长程关联函数。
英文摘要：
Screening of topological charges (singularities) is discussed for paraxial optical fields with short and with long range correlations. For short range screening the charge variance in a circular region with radius $R$ grows linearly with $R$, instead of with $R^{2}$ as expected in the absence of screening; for long range screening it grows faster than $R$: for a field whose autocorrelation function is the zero order Bessel function J_{0}, the charge variance grows as R ln R$. A J_{0} correlation function is not attainable in practice, but we show how to generate an optical field whose correlation function closely approximates this form. The charge variance can be measured by counting positive and negative singularities inside the region A, or more easily by counting signed zero crossings on the perimeter of A. \For the first method the charge variance is calculated by integration over the charge correlation function C(r), for the second by integration over the zero crossing correlation function Gamma(r). Using the explicit forms of C(r) and of Gamma(r) we show that both methods of calculation yield the same result. We show that for short range screening the zero crossings can be counted along a straight line whose length equals P, but that for long range screening this simplification no longer holds. We also show that for realizable optical fields, for sufficiently small R, the charge variance goes as R^2, whereas for sufficiently large R, it grows as R. These universal laws are applicable to both short and pseudo-long range correlation functions.
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