[Paper Review] Controlling Light Transmission Through Highly Scattering Media Using Semi-Definite Programming as a Phase Retrieval Computation Method

Moussa N’Gom, Miao-Bin Lien+4 more|PubMed|Dec 24, 2016

Random lasers and scattering media19 references3 citations

TL;DR

This paper proposes a non-iterative wavefront shaping method that uses semi-definite programming (SDP) to reconstruct the complex transmission matrix of a highly scattering medium from intensity-only measurements, enabling precise control of light focusing without a reference beam. The method achieves a 48-fold intensity enhancement at the focus and allows steering of multiple foci using a single set of measurements, demonstrating fast, stable, and high-precision control in quasi-static media.

ABSTRACT

Complex Semi-Definite Programming (SDP) is introduced as a novel approach to phase retrieval enabled control of monochromatic light transmission through highly scattering media. In a simple optical setup, a spatial light modulator is used to generate a random sequence of phase-modulated wavefronts, and the resulting intensity speckle patterns in the transmitted light are acquired on a camera. The SDP algorithm allows computation of the complex transmission matrix of the system from this sequence of intensity-only measurements, without need for a reference beam. Once the transmission matrix is determined, optimal wavefronts are computed that focus the incident beam to any position or sequence of positions on the far side of the scattering medium, without the need for any subsequent measurements or wavefront shaping iterations. The number of measurements required and the degree of enhancement of the intensity at focus is determined by the number of pixels controlled by the spatial light modulator.

Motivation & Objective

• To overcome the limitations of iterative wavefront shaping methods that are slow and prone to local minima in highly scattering media.
• To eliminate the need for a reference beam in transmission matrix (TM) measurement, which is typically required in holographic or interferometric methods.
• To develop a non-iterative, closed-form method for computing optimal wavefronts that focus light to any desired position beyond a scattering medium.
• To demonstrate that SDP-based phase retrieval enables high-intensity, stable, and multi-point focusing using a single set of intensity measurements.
• To validate the method in quasi-static scattering media such as ground glass and fresh yogurt, showing practical relevance for biomedical and imaging applications.

Proposed method

• The method uses semi-definite programming (SDP) to retrieve the phase of the transmission matrix from intensity-only measurements of transmitted speckle patterns generated by random phase-modulated wavefronts.
• A spatial light modulator (SLM) generates a sequence of random phase patterns on the input aperture, and the resulting intensity distributions are recorded by a camera.
• The SDP algorithm reconstructs the complex transmission matrix by solving a convex optimization problem based on the intensity measurements, leveraging the algorithm of Waldspurger et al. for phase retrieval.
• Once the transmission matrix is reconstructed, the optimal input wavefront is computed in closed-form using Eq. (19) to maximize intensity at a target output pixel.
• The number of required measurements scales as $\lceil M \log M \rceil$, where $M$ is the number of SLM pixels, ensuring efficient and scalable reconstruction.
• The method supports multi-focus generation by extending the transmission matrix to include multiple target output pixels, enabling simultaneous focusing at multiple locations.

Experimental results

Research questions

• RQ1 Can semi-definite programming be effectively used to retrieve the phase of a transmission matrix from intensity-only measurements in optical wavefront shaping?
• RQ2 Does the SDP-based method achieve high-intensity focusing without requiring a reference beam or iterative optimization?
• RQ3 To what extent does the intensity enhancement at focus scale with the number of SLM modes (M)?
• RQ4 Can the method generate multiple foci simultaneously using a single set of measurements?
• RQ5 How stable and practical is the method in quasi-static scattering media such as fresh yogurt or ground glass?

Key findings

• The SDP-based method achieved a 48-fold intensity enhancement at the focal spot compared to the average speckle intensity, demonstrating high focusing efficiency.
• The method successfully reconstructed the transmission matrix using only intensity measurements, eliminating the need for a reference beam or interferometric setup.
• With $M = 100$ SLM pixels, the method required approximately 461 random measurements to achieve high-fidelity focusing, consistent with the $M \log M$ scaling law.
• The method enabled the generation of a multi-focus pattern spelling 'MICHIGAN' (157 foci) using the same initial measurement set, proving scalability to multiple targets.
• The intensity focus was confined to a single detector pixel when $M = 100$, demonstrating sub-pixel resolution control.
• The method was validated on a quasi-static yogurt sample with a speckle persistence time of minutes, showing practical feasibility for real-world applications.