Phase and coherence retrieval from near- and far-field intensities

Technical Summary: Phase and Coherence Retrieval from Near- and Far-Field Intensities

Problem Statement
Quantifying the coherence properties of complex optical fields is critical for applications ranging from high-power laser arrays to quantum coherent systems. While iterative phase retrieval algorithms, such as the Gerchberg–Saxton (GS) framework, are standard for reconstructing complex fields from intensity measurements, they typically assume fully coherent fields. Many important physical systems, including large laser arrays, X-ray/electron beams, and structured quantum states, exhibit only partial spatial coherence. In these cases, the field cannot be described by a simple complex amplitude but requires a four-dimensional mutual intensity function, $J(r, r') = \langle E^*(r)E(r') \rangle$, representing the first-order spatial coherence. Existing reconstruction methods often rely on restrictive assumptions (e.g., homogeneous coherence, known modal bases, or low-rank density matrices) or fail to recover the full mutual intensity from intensity measurements in only two planes (near-field and far-field).

Methodology
The authors propose a new paradigm for reconstructing the full mutual intensity of partially coherent fields from intensity measurements in two Fourier-conjugate planes. The approach adapts the GS algorithm to iteratively propagate and project the mutual intensity between the near-field and far-field domains. Two complementary implementations are developed:

1. Tensor GS Algorithm: This method directly reconstructs the mutual intensity by applying alternating projections to the four-dimensional mutual intensity tensor. The process involves:

   • Propagating the mutual intensity between planes using four-dimensional Fourier transforms.
   • Enforcing target intensity constraints by projecting the diagonal elements of the mutual intensity onto the measured near-field and far-field intensities.
   • Updating off-diagonal elements to preserve the underlying coherence structure while enforcing Hermitian symmetry and physical consistency (ensuring $|\tilde{g}_1(k, k')| \leq 1$).
   • Iterating until convergence.

2. Monte Carlo (MC) GS Variant: To reduce the computational cost associated with four-dimensional Fourier transforms, this method represents the partially coherent field as an ensemble of $P$ independent random complex realizations.

   • Fields are propagated between planes using standard two-dimensional Fourier transforms.
   • Field amplitudes are multiplied by a common function at each step to ensure the ensemble-averaged intensity matches the target intensity.
   • The mutual intensity is estimated by ensemble averaging over the $P$ realizations.
   • The accuracy scales as $1/\sqrt{P}$, allowing for controlled approximation. The computational complexity scales as $PM \log(M)$ (where $M$ is the number of spatial modes), compared to $M^2 \log(M)$ for the Tensor GS method.

Key Contributions

• General Reconstruction Framework: The paper presents a general method for reconstructing the full spatial mutual intensity from intensity measurements in only two planes, without requiring predefined modal bases, low-rank parameterization, or multiple detection planes.
• Dual Algorithmic Approaches: The introduction of both a high-accuracy Tensor GS method and a computationally efficient MC GS variant provides flexibility for different system sizes and accuracy requirements.
• Handling Inhomogeneity: The methods are specifically designed to handle spatially inhomogeneous partially coherent fields, a regime where traditional inverse Fourier transform (IFT) approaches fail.

Results
The authors validated both methods through numerical simulations and experimental demonstration:

• Numerical Simulations:
  • Linear Arrays: Simulations on linear arrays of up to $N=600$ Gaussian beams with prescribed Gaussian-decaying coherence showed that while IFT methods accurately recover coherence only for short coherence lengths ($\xi \ll Nd$), both Tensor GS and MC GS methods maintain accuracy even when the coherence length exceeds the array size ($\xi > Nd$), where mutual intensity becomes strongly inhomogeneous.
  • Ring Arrays: Simulations on ring arrays ($N=50$) demonstrated that the IFT method fails qualitatively due to the breakdown of translational invariance assumptions. In contrast, both proposed methods accurately recovered the correct structure of the mutual intensity, with Tensor GS providing high-fidelity reconstruction and MC GS offering a computationally efficient approximation.
• Experimental Demonstration:
  • The Tensor GS method was applied to a triangular array of 130 coupled lasers with controlled coupling and staggered chirality.
  • The system generated states with non-monotonous mutual coherence decay and non-trivial phases (staggered anti-vortex and equal superposition of vortex/anti-vortex states).
  • The reconstruction achieved excellent agreement with theory, reproducing expected phase windings (e.g., $2\pi/3$) and Gaussian decay of coherence magnitude.
  • The root-mean-square (RMS) phase error was measured at $2\pi/250$ for the staggered anti-vortex state and $2\pi/65$ for the equal superposition state.

Significance and Claims
The paper claims that these results indicate the approach is practical for characterizing large-scale partially coherent optical systems. The methods complement existing tomographic approaches by removing the need for prior knowledge of the coherence structure or multiple detection planes. The authors note that while the theoretical uniqueness of mutual intensity reconstruction from two Fourier-conjugate planes remains an open question, their numerical and experimental results demonstrate stable convergence toward physically meaningful solutions for strongly inhomogeneous coherence structures and robustness to experimental noise.

The work suggests potential future extensions, such as incorporating free-space propagation to reconstruct fields from non-Fourier-conjugate planes and applying the framework to quantum tomography, given the formal analogy between mutual intensity reconstruction and density matrix reconstruction. However, the paper maintains a modest stance, focusing on the demonstrated ability to reconstruct phase and coherence in specific, challenging inhomogeneous systems without overextending claims to arbitrary optical fields.