Advances Simulations of Partially Coherent Light Transport by Orders of Magnitude

Scientists are tackling the computationally intensive problem of simulating how partially coherent light travels through complex systems, a crucial capability for technologies like advanced lithography and diffraction-limited storage rings. Han Xu, Ming Li, and Shuo Wang, from their respective institutions, alongside Zhe Ren, Peng Liu, Yi Zhang et al, present a novel algorithm called Coherent Mode Decoupling (CMDC) that dramatically accelerates these simulations without sacrificing accuracy. This research is significant because CMDC factorises complex two-dimensional modes into simpler, one-dimensional components, coupled with a clever compression strategy, achieving speed-ups of several orders of magnitude , paving the way for more efficient design and optimisation of light-based technologies.

This innovative approach allows for simulations that were previously impractical, opening new avenues for optical system design and optimization. The study reveals a high-throughput framework that circumvents the limitations of Coherent Mode Decomposition (CMD), a standard technique which often requires propagating massive sets of two-dimensional modes, a process demanding significant computational resources. CMDC cleverly decouples these 2D modes, transforming the problem into a series of efficient 1D operations.

Unlike simplistic approximations, the CMDC algorithm includes a robust residual correction mechanism, extracting and propagating “coupled energy” using a small set of 2D residual modes, ensuring both speed and physical fidelity. This hybrid approach provides a flexible and powerful tool for applications requiring rigorous simulations, such as those found in advanced lithography and synchrotron beamlines. Experiments demonstrate the versatility of CMDC across diverse applications, yielding substantial performance gains. For computational lithography, the convolution between kernels and masks was accelerated by factors ranging from 8 to 167, all while maintaining an impressive R2 value of 99.9%, 95.4%. The core innovation of this work lies in its ability to determine whether a complex 2D wave function can be factorized into the product of two independent 1D functions, effectively reducing computational complexity. Researchers extracted the “coupled energy” responsible for these effects and propagated it using a small set of 2D residual modes, creating a flexible balance between simulation speed and fidelity.
Experiments employing computational lithography demonstrated acceleration factors ranging from 8 to 167, while maintaining an impressive R2 value of 99.9%, 95.4%. For partially coherent beamline design, the CMDC method achieved an approximate 103-fold speedup, with accuracy rigorously verified through comparison with ptychography experiments. The team harnessed the Cross-Spectral Density (CSD) function to expand the optical field into a sum of uncorrelated orthogonal modes, as defined by the equation W(r1, r2, ω) = ∑λn n φn(r1, ω)φn ∗(r2, ω). The CMDC algorithm represents a substantial methodological innovation, offering a practical solution to the longstanding challenge of computationally intensive partially coherent light calculations and paving the way for more efficient and accurate optical system modelling.

CMDC accelerates coherent light transport simulations

This new framework was successfully applied to diverse applications, including lithography and coherent beamlines of diffraction-limited storage rings, showcasing its generality and robustness. Experiments revealed a substantial improvement in computational efficiency when applying CMDC to a standard lithography configuration with a wavelength of 193nm and a numerical aperture of 0.3. The illumination source, defined as an annular pupil with a coherence factor of 0.69, typically requires a large number of coherent modes for accurate representation; however, CMDC efficiently handled this complexity. Simulations utilized a 1024×1024 pixel grid to discretize a binary mask containing complex circuit patterns, and the Transmission Cross Coefficient was decomposed into 61 orthogonal eigenfunctions, capturing 98.3% of the signal.

Results demonstrate that the 1D decoupled modes propagation maintained 58% of the total source energy, with the remaining 42% represented by 2D coupled modes for accuracy adjustment. Measurements confirm a speedup factor of approximately 167 when comparing the CMDC method with traditional 2D Sum of Coherent Systems (SOCS) methods, achieving a calculation time of just 0.02 seconds for the 1D decoupled convolution. Incorporating two and five 2D coupled kernel modes increased the computational time to 0.20 and 0.41 seconds respectively, while simultaneously boosting the R2 value, a measure of accuracy, from 95.4% to 98.7% and ultimately 99.9%. The aerial images calculated using CMDC closely matched those produced by the SOCS method, with a difference map showing an initial R2 of 95.4% and improving to 99.9% with the addition of coupled modes.

Researchers then evaluated the algorithm’s performance in the presence of lens aberrations, employing measured surface height error data from a fabricated refractive lens to simulate a strongly aberrated system with a Strehl Ratio of 0.5. The light source was modelled using Gaussian-Schell Modes, and the team incorporated the measured error profile to accurately reflect realistic manufacturing defects. This rigorous computational wave-optics simulation allowed for detailed prediction of the intensity distribution, including asymmetric side-lobes induced by the aberrations, demonstrating the CMDC framework’s capability for tolerance analysis and high-fidelity modelling of complex optical systems.

CMDC accelerates coherent light transport simulations

This method addresses a key computational bottleneck in traditional Coherent Mode Decomposition (CMD) techniques, which struggle with the propagation of numerous two-dimensional modes. Crucially, experimental validation at the HEPS HXCS beamline confirmed the algorithm’s ability to accurately model the complex modal structure of X-ray sources, reducing simulation times from over an hour to just 3.4 seconds. The authors acknowledge a limitation in the current availability of the underlying data, stating it may be obtained upon reasonable request. Future work could focus on expanding data accessibility and exploring the algorithm’s potential for real-time tolerance analysis, large-scale inverse design, and optimization of advanced partially coherent imaging systems. This research establishes CMDC as a versatile and scalable tool for optical engineers, offering a tunable balance between computational speed and precision for a range of applications.