Making neural networks work with nonlinear optics

Simulation of the nonlinear optical physics underlying ultrafast laser systems is computationally intensive and becomes a practical bottleneck in settings where rapid feedback is required. A new study by researchers at Stanford University, the University of California, Los Angeles (UCLA), and SLAC National Accelerator Laboratory introduces a deep learning surrogate that delivers orders of magnitude acceleration over traditional simulation methods while maintaining high fidelity over a difficult range of pulse shapes.

The research focuses on second-order nonlinear optics (χ² process), where light waves exchange energy within specially designed crystals, producing new frequencies and tailored pulse shapes. These processes play an important role in particle accelerator facilities. In SLAC’s Upgraded Linac Coherent Light Source (LCLS-II), the infrared laser pulse first becomes green light and then ultraviolet (UV) light. The UV pulse impinges on the cathode, releasing a flux of electrons that is then accelerated and modulated to produce an intense X-ray pulse. The temporal shape of a UV pulse directly affects its electron flux properties and ultimately the quality of the X-rays available for science. A surrogate model for the nonlinear χ² frequency conversion step that is central to this process is reported as follows. advanced photonics.

The standard simulation approach, which numerically solves the nonlinear Schrödinger equation using a split-step Fourier method (SSFM) with alternating time-domain and frequency-domain operations for each propagation step, is accurate but time-consuming, accounting for approximately 95% of the total run time of a complete laser simulation. Taking inspiration from previous work applying recurrent neural networks to optical fiber pulse propagation, the team developed an LSTM (long short-term memory) surrogate tailored to a more generalized multifield χ² setting. The noncollinear SFG involves three coupled optical fields evolving simultaneously over a wide range of pulse shapes and serves as a rigorous and broadly relevant testbed. An important design choice was to operate entirely on compressed frequency domain representations and avoid iterative domain transformations that increase the cost of SSFM.

(a) Schematic diagram of the non-collinear SFG process. Three coupled optical fields (A1, A2, A3) propagate through 100 discretized crystal slices, and at each step an LSTM surrogate is used instead of the traditional SSFM solver. (b) Architecture of the LSTM network showing the recursive layer and fully connected output layer. Credit: Hirschman et al., doi 10.1117/1.AP.8.3.036004

This surrogate accurately reconstructs temporal and spectral profiles over a wide range of pulse-shaping conditions, including cases with pronounced spectral holes and strong phase modulation. When run using batched GPU inference, the average simulation time per instance is reduced to a few milliseconds, orders of magnitude faster than traditional methods. In particular, this model appears to capture the global coupling dynamics of interactions. If the primary SFG output is well predicted, the secondary fields will also tend to match closely.

The broader goal of this research is to integrate such an alternative with experimental laser systems. The modular framework, where individual physical processes are each represented by trained surrogate blocks, points to a predictive model that is directly coupled to the running experiment.

Looking to the future, linking fast machine learning surrogates directly to experiments opens the door to applications such as complete digital twin development, adaptive control, and tight integration with downstream diagnostics across various laser-driven facilities.

For more information, see the original Gold Open Access article “Deep Learning-Assisted Modeling of χ” by Hirschman et al.⁽²⁾ “Nonlinear optics” Advanced photon. 8(3) 036004 (2026), doi: 10.1117/1.AP.8.3.036004