Generation of Laguerre Functions and Associated Laguerre Functions
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This article explores methodologies for generating Laguerre functions and associated Laguerre functions, demonstrating how to visualize multiple Laguerre-Gaussian beam models using their analytical expressions. Through implementation examples, we examine the application of these functions in beam modeling and investigate their practical uses in optics and physics. The implementation typically involves recursive algorithms or special function libraries (such as SciPy's special.laguerre in Python) to compute polynomial values. We also discuss fundamental properties and mathematical derivations of these functions, including their orthogonal characteristics and recurrence relations, which are crucial for efficient numerical computation. The visualization component employs parametric plotting techniques where radial and angular components combine to create characteristic donut-shaped intensity profiles with orbital angular momentum properties.
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