Coherent Mode Decoupling: A Versatile Framework for High-Throughput Partially Coherent Light Transport

Accurate and efficient wave-optics simulation of partially coherent light transport systems is critical for the design of advanced optical systems, ranging from computational lithography to diffraction-limited storage rings (DLSR). However, traditional approaches based on Coherent Mode Decomposition suffer from high computational costs due to the propagating massive sets of two-dimensional modes. In this paper, we propose the Coherent Mode Decoupling (CMDC) algorithm, a high-throughput computational framework designed to accelerate these simulations by orders of magnitude without compromising physical fidelity. The method factorizes 2D modes into efficient one-dimensional (1D) components, while crucially incorporating a subspace compression strategy to capture non-separable coupling effects. We demonstrated the generality and robustness of this framework in applications ranging from computational lithography to coherent beamlines of DLSR.

💡 Research Summary

The paper introduces Coherent Mode Decoupling (CMDC), a high‑throughput computational framework that dramatically reduces the cost of partially coherent wave‑optics simulations. Traditional Coherent Mode Decomposition (CMD) expands the cross‑spectral density (CSD) into a large set of orthogonal two‑dimensional (2‑D) eigenmodes, each of which must be propagated individually through the optical system. For realistic sources, the number of required modes can reach the thousands, leading to computational complexities on the order of O(N·Nx·Ny) when Fourier‑based propagators are used, where N is the number of modes and Nx, Ny are the transverse pixel counts.

CMDC tackles this bottleneck by exploiting the fact that many 2‑D modes are nearly separable. Each mode is treated as a complex matrix and subjected to a Singular Value Decomposition (SVD). The leading singular value and its associated left and right singular vectors provide a rank‑1 approximation, i.e., a product of two one‑dimensional (1‑D) functions u(x)·v(y). The proportion of a mode’s energy captured by this separable part is quantified by the Separable Energy Ratio (η). A global separability ratio, obtained by weighting η with the eigenvalues of the CMD expansion, indicates how much of the total field can be handled by 1‑D propagation alone.

The remaining higher‑order SVD terms constitute a non‑separable residual. Propagating each residual term directly would defeat the purpose of the method, so CMDC aggregates all residuals into a global residual matrix C and performs a second SVD. By retaining only the top M residual modes that satisfy a cumulative energy threshold (e.g., 95 %), the algorithm captures the essential non‑separable physics (astigmatism, rotation, irregular apertures) with a dramatically reduced set of 2‑D modes. This “subspace compression” balances accuracy and speed.

CMDC also applies the same SVD‑based decoupling to optical elements. Any 2‑D transmission function T(x, y) is expressed as a sum of separable 1‑D profiles multiplied by singular values. Consequently, the interaction of a 1‑D field with a complex element becomes a series of 1‑D multiplications and summations, preserving the computational advantage throughout the entire simulation pipeline.

The authors validate CMDC on three representative problems:

1. Computational Lithography – Using a 1024×1024 pixel binary mask and 61 Transmission Cross‑Coefficient (TCC) kernels, the method achieves a separable energy ratio of 0.58. With only the 1‑D decoupled modes, aerial images reach R² = 95.4 % compared to the full 2‑D SOCS reference. Adding 2 and 5 residual 2‑D kernels raises R² to 98.7 % and 99.9 % respectively, while the total convolution time drops from 3.4 s (full 2‑D) to 0.02 s (pure 1‑D) and up to 0.41 s when residuals are included—a speed‑up factor of 8–167.

2. Lens Aberration Analysis – A Gaussian‑Schell source illuminates a refractive lens with measured surface‑height errors. The phase error is decomposed into 1‑D components; the first 15 modes capture 99 % of the error energy. CMDC accurately reproduces the asymmetric intensity distribution caused by the non‑separable phase modulation, demonstrating its capability for tolerance analysis of realistic optics.

3. HEPS Hard‑X‑ray Coherent Scattering Beamline – Full wave‑optics propagation of the beamline, including all source, optics, and apertures, is performed with CMDC. The simulation runs roughly 10³ times faster than a conventional 2‑D CMD approach while maintaining excellent agreement with experimental ptychography measurements.

Key advantages of CMDC include: (i) reduction of computational complexity from O(N·Nx·Ny) to roughly O(N·max(Nx,Ny)) plus a modest cost for the residual SVD; (ii) a tunable accuracy‑speed trade‑off via the number of retained residual modes; (iii) a unified framework that can be applied to both sources and optical elements, making it suitable for a wide range of wavelengths and system sizes.

Limitations are acknowledged. When the global separability ratio is low (strongly non‑separable fields), the number of residual modes required may become large, eroding the speed benefit. Additionally, the initial SVD of large matrices can be memory‑intensive; the authors suggest GPU‑accelerated or randomized low‑rank approximations as future improvements.

In conclusion, CMDC provides a powerful, flexible tool for high‑throughput partially coherent light transport simulations. It enables rapid design iterations, tolerance analyses, and inverse‑design workflows across micro‑fabrication, microscopy, and synchrotron beamline engineering, while preserving the physical fidelity needed for quantitative predictions. Future work may extend the method to nonlinear media, time‑dependent sources, and integration with machine‑learning‑based mode prediction.