We demonstrate that nonlinear computing can be achieved with a single linear diffractive surface under coherent illumination. We introduce a compact encoder-decoder co-localization (E+D) architecture in which an input-dependent dynamic encoder and a static optimized decoder are integrated within the same phase-only diffractive plane. Following free-space propagation, coherent interference between the encoder and decoder fields, combined with intensity detection, generates programmable nonlinear input-output mappings without requiring nonlinear optical materials or multiple diffractive layers. We prove that the proposed E+D optical processor is a universal approximator for arbitrary real-valued band-limited nonlinear functions and identify the physical factors governing its approximation fidelity, including the decoder degrees-of-freedom, detector aperture, and axial propagation distance. Crucially, we demonstrate that introducing a trained, frozen phase bias to the encoder region systematically enhances functional expressivity, providing robustness against coarse phase quantization on spatial light modulators. Using this framework, we accurately synthesize diverse nonlinear functions, including commonly used neural network activation functions and complex-valued nonlinear functions. Finally, we experimentally validate the proposed approach using a visible-light optical set-up trained through in situ learning, demonstrating the parallel approximation of 9 nonlinear functions in a single optical forward pass. By collapsing nonlinear optical computation into a single diffractive surface, the E+D architecture substantially reduces hardware and alignment complexity while preserving powerful function-approximation capabilities, providing a compact and scalable framework for analog information processing.

翻译：我们证明，在相干照明条件下，仅通过单个线性衍射表面即可实现非线性计算。我们提出了一种紧凑的编码器-解码器共定位（E+D）架构，其中输入相关的动态编码器与静态优化解码器集成在同一纯相位衍射平面上。经自由空间传播后，编码器与解码器场之间的相干干涉，结合强度探测，可在无需非线性光学材料或多层衍射层的情况下，产生可编程的非线性输入-输出映射。我们证明了所提出的E+D光学处理器是任意实值带限非线性函数的通用逼近器，并识别了影响其逼近保真度的物理因素，包括解码器自由度、探测器孔径以及轴向传播距离。关键在于，我们证明向编码器区域引入一个经过训练且冻结的相位偏置可以系统地增强功能表达性，从而提供对空间光调制器上粗糙相位量化的鲁棒性。利用这一框架，我们精确综合了多种非线性函数，包括常用神经网络激活函数和复值非线性函数。最后，我们通过使用原位学习训练的可见光光学装置实验验证了所提出的方法，在单次光学前向传播中并行逼近了9个非线性函数。通过将非线性光学计算压缩至单个衍射表面，E+D架构大幅降低了硬件和对准复杂度，同时保留了强大的函数逼近能力，为模拟信息处理提供了一种紧凑且可扩展的框架。