In this contribution, we introduce a new practical approach to apply the fractional Fourier transform (FrFT) in the modeling of two optical systems: free space propagation (FSP) and a single lens processor (SLP). This formulation presents a simple way to stablish a direct relationship between physical parameters of the two optical systems and a real-valued fractional order. Furthermore, we employ and compare two numerical methods for evaluating the FrFT: the convolution and the Fresnel transform. Consequently, we apply this innovative approach to the digital decryption process in an opto-digital joint transform correlator cryptosystem, considering both the FSP and the SLP variants. We analyze both numerically and experimentally encrypted data to support our proposed method and to investigate the sensitivity of the decryption process with the fractional order. Notably, we obtain similar decryption results for both numerically and experimentally encrypted objects, demonstrating excellent agreement between the theoretical model, the numerical test, and the experiment.

中文翻译：

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分数阶傅里叶变换在实验光学密码系统中解密的应用

在这篇文章中，我们介绍了一种新的实用方法，将分数傅里叶变换（FrFT）应用于两个光学系统的建模：自由空间传播（FSP）和单透镜处理器（SLP）。该公式提供了一种简单的方法来建立两个光学系统的物理参数和实值分数阶之间的直接关系。此外，我们采用并比较了两种评估 FrFT 的数值方法：卷积和菲涅尔变换。因此，我们将这种创新方法应用于光数字联合变换相关器密码系统中的数字解密过程，同时考虑 FSP 和 SLP 变体。我们分析了数值和实验加密数据，以支持我们提出的方法，并研究分数阶解密过程的敏感性。值得注意的是，我们对数值和实验加密对象都获得了相似的解密结果，证明理论模型、数值测试和实验之间具有良好的一致性。