The textbook description of multiple slit interference employs a standard path difference formula that was originally calculated for the purpose of analyzing Young’s double slit experiment. It was arrived at on the basis of a pair of assumptions that help simplify the geometry of the slit-barrier-screen arrangement and the ensuing formalism needed to estimate the positions of interference fringes on the detection screen. According to the conventional approach, convergent rays of light that emanate from different slits can be treated as effectively parallel in the far field limit. Such a parallel ray approximation-based analysis was shown by Thomas (2019; 2020) to be redundant, paradoxical and limited in accuracy. In this paper, the task of reformulation that began previously with the elementary two-slit experiment is now extended to encompass the N-slit scenario, diffraction at a single-slit and the formation of circular fringes. In order to derive one of the key results (the generalized hyperbola theorem for N-point sources), a novel computational technique called analytical induction is introduced. The procedure involves a synthesis of concepts from Cartesian geometry and discrete mathematics, namely coordinate transformation by Euclidean translation and a method of proof called the principle of mathematical induction, respectively. The final goal attained are two distribution functions that succinctly and precisely describe the variation of intensity of interference fringes on the distant screen, when it is oriented parallel and perpendicular relative to the line joining the sources. A comparison is drawn between the predictions of the conventional and the new analyses by means of numerical-graphical simulation with MATLAB. The theoretical methods and results presented herein may have a significant bearing in areas of applied optics like interferometry and diffraction crystallography. On the pedagogic front, it is suggested that the current geometrical program being visually more intuitive, be incorporated into the standard curriculum of an advanced undergraduate/graduate level course in physical optics, to complement the conventional approach.

多缝干涉的教科书描述采用了标准路径差公式，该公式最初是为了分析杨氏双缝实验而计算的。它是在两个假设的基础上得出的，这些假设有助于简化狭缝屏障屏幕布置的几何形状，以及随后估计检测屏幕上干涉条纹位置所需的形式主义。根据传统方法，从不同狭缝发出的会聚光线可以被视为在远场极限内有效平行。Thomas (2019; 2020) 表明，这种基于平行射线近似的分析是多余的、自相矛盾的并且精度有限。在本文中，之前从基本的双缝实验开始的重构任务现在扩展到包括N缝场景、单缝衍射和圆形条纹的形成。为了推导出一个关键结果（N的广义双曲线定理）-点源），引入了一种称为分析归纳的新型计算技术。该过程涉及笛卡尔几何和离散数学概念的综合，即分别通过欧几里得平移的坐标变换和称为数学归纳原理的证明方法。达到的最终目标是两个分布函数，当它相对于连接源的线平行和垂直时，它们简洁而准确地描述了远处屏幕上干涉条纹强度的变化。通过使用 MATLAB 进行数值图形模拟，对传统分析和新分析的预测进行了比较。本文提出的理论方法和结果可能对干涉测量和衍射晶体学等应用光学领域具有重要意义。在教学方面，建议将当前的几何程序在视觉上更加直观，并纳入物理光学高级本科/研究生课程的标准课程，以补充传统方法。