Nelly: A User-Friendly and Open-Source Implementation of Tree-Based Complex Refractive Index Analysis for Terahertz Spectroscopy

Terahertz (THz) spectroscopy is a powerful tool for unambiguously extracting complex-valued material properties (e.g., refractive index, conductivity, etc.) from a wide range of samples, with applications ranging from materials science to biology. However, extracting complex refractive indices from THz time-domain spectroscopy data can prove challenging, especially for multilayer samples. These challenges arise from the large number of transmission-reflection paths the THz pulse can take through the sample layers, leading to unwieldy strings of Fresnel coefficients. This issue has often been addressed using various approximations. However, these approximations are only applicable to specific classes of samples and can give erroneous results when misapplied. An alternative to this approach is to programmatically model all possible paths through the sample. The many paths through the sample layers can be modeled as a tree that branches at every point where the paths diverge, i.e., whenever the pulse can either be transmitted or reflected. This tree can then be used to generate expressions relating the unknown refractive index to the observed time domain data. Here, we provide a freely available open-source package implementing this method as both a MATLAB library and a corresponding graphical user interface, which can also be run without a MATLAB license (https://github.com/YaleTHz/nelly). We have tested this method for a range of samples and compared the results to commonly used approximations to demonstrate its accuracy and wide applicability. Our method consistently gives better agreement than common approximations.

太赫兹（Terahertz, THz）光谱技术是一种能够无歧义地从各类样品中提取复数值材料特性（如折射率、电导率等）的强大工具，其应用场景覆盖材料科学乃至生物学领域。然而，从太赫兹时域光谱数据中提取复折射率往往极具挑战，对于多层样品而言更是如此。此类挑战源于太赫兹脉冲在样品层间传播时存在大量透射-反射路径，进而生成繁杂冗长的菲涅耳（Fresnel）系数序列。此前，学界常通过各类近似方法解决该问题，但这类近似仅适用于特定类别的样品，误用则可能产生错误结果。

另一种解决方案是对样品内所有可能的传播路径进行程序化建模。样品层间的多条传播路径可被建模为一棵在每一个路径分叉点（即脉冲可发生透射或反射的节点）处产生分支的树状结构。借助该树结构，可生成将未知折射率与观测到的时域数据相关联的表达式。

本文提供了一款免费开源软件包，该工具以MATLAB库及配套图形用户界面（Graphical User Interface, GUI）的形式实现了上述方法，且无需MATLAB许可证即可运行（项目地址：https://github.com/YaleTHz/nelly）。我们已针对多种样品测试了该方法，并将所得结果与常用近似方法进行对比，以验证其准确性与广泛适用性。本方法的拟合一致性始终优于常规近似方案。