多奇异涡旋高斯光束的解析解：散射模的数学理论
Analytical solution for multisingular vortex Gaussian beams: The mathematical theory of scattering modes
摘要：
本文提出了一种新的方法来求解薛定谔方程，该方程在光学中是傍轴波动方程。这个初始条件有一个平面中的一些奇点横向传播嵌入在高斯光束。我们使用的散射模式，这是解决方案的傍轴波动方程，可以直接组合来表达的初始条件，因此允许解决问题。为了构造散射模式，需要获得一组特定的多项式，其作用类似于拉盖尔-高斯模式的拉盖尔多项式。我们在这里演示的递归关系需要确定这些多项式。为了强调该方法的实用性和强度，我们首先解决了具有两个正奇点和一个负奇点的初始高斯光束的问题。我们表明，该解决方案允许一个获得解析表达式。这些可以用来获得有意义的量的封闭表达式，例如正负奇点合并的距离，闭合涡线的环。此外，我们提出了一个具体的离散高斯态的计算的例子，这是一个拉盖尔-高斯态的衍射的解决方案，显示一定的角动量（即，高电荷的漩涡）薄衍射元件显示一定的离散对称性。我们表明，从而这个问题是解决在一个更简单的方式比使用以前的程序的基础上的积分菲涅耳衍射方法。
英文摘要：
We present a novel procedure to solve the Schrödinger equation, which in optics is the paraxial wave equation, with an initial multisingular vortex Gaussian beam. This initial condition has a number of singularities in a plane transversal to propagation embedded in a Gaussian beam. We use the scattering modes, which are solutions of the paraxial wave equation that can be combined straightforwardly to express the initial condition and therefore permit to solve the problem. To construct the scattering modes one needs to obtain a particular set of polynomials, which play an analogous role than Laguerre polynomials for Laguerre-Gaussian modes. We demonstrate here the recurrence relations needed to determine these polynomials. To stress the utility and strength of the method we solve first the problem of an initial Gaussian beam with two positive singularities and a negative one embedded in. We show that the solution permits one to obtain analytical expressions. These can used to obtain closed expressions for meaningful quantities, like the distance at which the positive and negative singularities merge, closing the loop of a vortex line. Furthermore, we present an example of calculation of an specific discrete-Gauss state, which is the solution of the diffraction of a Laguerre-Gauss state showing definite angular momentum (that is, a highly charged vortex) by a thin diffractive element showing certain discrete symmetry. We show that thereby this problem is solved in a much simpler way than using the previous procedure based in the integral Fresnel diffraction method.
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