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Surface-Encoded Partial Coherence Transformation: Modeling Source Coherence Effects in Wave Optics

Netzer Moriya

TL;DR

This work introduces the Surface-Encoded Coherence Transformation (SECT) framework to model partial coherence in wave optics by decoupling surface interactions from propagation. It defines two deterministic operators, SECT$_S$ for surface coherence encoding and SECT$_P$ for propagation, and shows how their combination reproduces traditional VCZ coherence under appropriate conditions, while reducing computational complexity for multi-surface systems. The authors establish convergence to ensemble averages, energy conservation, and linear, stable operator properties, and provide a discrete, FFT-friendly formulation with clear guidance on regime-specific dominance of surface vs propagation effects. The framework offers practical pathways for efficient simulations in complex optical systems such as multi-surface interferometers and astronomical instruments, with explicit guidelines for near-field versus far-field regimes and spectral generalizations. Overall, SECT provides a physically transparent, computationally advantageous alternative to ensemble averaging and full 4D coherence propagation, while maintaining fidelity to established coherence theory.

Abstract

We present a new mathematical framework for incorporating partial coherence effects into wave optics simulations through a comprehensive surface-to-detector approach. Unlike traditional ensemble averaging methods, our dual-component framework models partial coherence through: (1) a surface-encoded transformation implemented via a linear integral operator with a spatially-dependent kernel that modifies coherence properties at the reflection interface, followed by (2) a propagation component that evolves these coherence properties to the detection plane. This approach differs fundamentally from conventional models by explicitly separating surface interactions from propagation effects, while maintaining a unified mathematical structure. We derive the mathematical foundation based on the coherence function formalism, establish the connection to the Van Cittert-Zernike theorem, and prove the equivalence of our framework to conventional partial coherence theory. The method reduces the dimensional complexity of coherence calculations and offers potential computational advantages, particularly for systems involving multiple surfaces and propagation steps. Applications include optical testing and astronomical instrumentation. We provide rigorous mathematical proofs, demonstrate the convergence properties, and analyze the relative importance of surface and propagation effects across different optical scenarios.

Surface-Encoded Partial Coherence Transformation: Modeling Source Coherence Effects in Wave Optics

TL;DR

This work introduces the Surface-Encoded Coherence Transformation (SECT) framework to model partial coherence in wave optics by decoupling surface interactions from propagation. It defines two deterministic operators, SECT for surface coherence encoding and SECT for propagation, and shows how their combination reproduces traditional VCZ coherence under appropriate conditions, while reducing computational complexity for multi-surface systems. The authors establish convergence to ensemble averages, energy conservation, and linear, stable operator properties, and provide a discrete, FFT-friendly formulation with clear guidance on regime-specific dominance of surface vs propagation effects. The framework offers practical pathways for efficient simulations in complex optical systems such as multi-surface interferometers and astronomical instruments, with explicit guidelines for near-field versus far-field regimes and spectral generalizations. Overall, SECT provides a physically transparent, computationally advantageous alternative to ensemble averaging and full 4D coherence propagation, while maintaining fidelity to established coherence theory.

Abstract

We present a new mathematical framework for incorporating partial coherence effects into wave optics simulations through a comprehensive surface-to-detector approach. Unlike traditional ensemble averaging methods, our dual-component framework models partial coherence through: (1) a surface-encoded transformation implemented via a linear integral operator with a spatially-dependent kernel that modifies coherence properties at the reflection interface, followed by (2) a propagation component that evolves these coherence properties to the detection plane. This approach differs fundamentally from conventional models by explicitly separating surface interactions from propagation effects, while maintaining a unified mathematical structure. We derive the mathematical foundation based on the coherence function formalism, establish the connection to the Van Cittert-Zernike theorem, and prove the equivalence of our framework to conventional partial coherence theory. The method reduces the dimensional complexity of coherence calculations and offers potential computational advantages, particularly for systems involving multiple surfaces and propagation steps. Applications include optical testing and astronomical instrumentation. We provide rigorous mathematical proofs, demonstrate the convergence properties, and analyze the relative importance of surface and propagation effects across different optical scenarios.

Paper Structure

This paper contains 42 sections, 9 theorems, 119 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Coherence Function Formalism
  4. Van Cittert-Zernike Theorem Extension
  5. Equivalence to Van Cittert-Zernike Theorem
  6. Two-Component Formulation
  7. Component 1: Surface-Encoded Coherence Transformation (SECT$_S$)
  8. Component 2: Propagation Component (SECT$_P$)
  9. Generalization to Curved Surfaces and Arbitrary Detection Geometries
  10. Complete Two-Component Framework
  11. Surface-Encoded Coherence Operator
  12. Propagation Operator
  13. Combined Formulation
  14. Relation to Spatial Filtering Theory
  15. Coherence Kernel Construction
  16. ...and 27 more sections

Key Result

Theorem 2.1

Given a fully coherent incident field $U_i(\bm{r}) = A_0$ (uniform plane wave illumination) impinging on a surface with coherence operator $\mathcal{C}_S$ characterized by kernel $K_S(\bm{r}, \bm{r}', \lambda)$, followed by propagation operator $\mathcal{P}_z$, the mutual coherence function at the d where $\mathcal{F}$ denotes the Fourier transform operator.

Figures (4)

  • Figure 1: Geometric configuration of the SECT framework showing a curved reflecting surface and detection plane. The framework accounts for the actual three-dimensional coordinates of each point on both surfaces connected by black line to illustrate the point-to-point propagation distance.
  • Figure 2: Comparison between exact path calculation (required for curved surfaces and large angles) and the paraxial approximation (valid when lateral dimensions are much smaller than propagation distance).
  • Figure 3: Validity regions for the paraxial approximation. The approximation is valid when the ratio of lateral dimensions $L$ to propagation distance $z$ is small, typically requiring $L/z < 0.2$ for phase errors below $\lambda/20$. Curved surfaces further restrict the valid region.
  • Figure 4: Application of the dual-component SECT framework to a curved optical system. The SECT$_S$ component encodes coherence properties at the surface, while the SECT$_P$ component evolves these properties during propagation to the detector.

Theorems & Definitions (17)

  • Theorem 2.1: SECT-VCZ Equivalence
  • proof
  • Corollary 2.2: Circular Source Equivalence
  • proof
  • Corollary 2.3: Gaussian Source Equivalence
  • proof
  • Theorem 2.4: Surface-Encoded Coherence Transformation
  • proof
  • Definition 2.5: Admissible Coherence Kernel
  • Theorem 4.1: Convergence to Ensemble Average
  • ...and 7 more