Understanding Multi-layered Transmission Matrices

Wavefront shaping systems attempt to correct aberrations caused by tissue scattering, by placing a spatial light modulator (SLM) in the optical path and using it to reshape the light wave emitted from a target of interest deep inside the tissue. However, the field-of-view we can correct with one SLM pattern is extremely small, because inherently the aberration is caused by the 3D tissue structure while the SLM is planar. To overcome this, multi-conjugate correction systems built with multiple layered SLMs have been introduced, which attempt to approximate the 3D tissue structure with multiple planar aberrations. However, multi-conjugate systems with a large number of SLMs are not easy to construct. Therefore, it is important to understand how many SLM layers are actually needed for a good correction, and whether there are practical benefits from correction systems with a relatively small number of layers. To help in the design of future systems, this paper analyzes multi-layer corrections. We show that the well-known missing cone problem which fundamentally limits the axial resolution of microscopes, turns into an advantage when designing 3D correction systems. Due to this property less information should be captured by the correction system, facilitating sparser approximations. We also show that even if the number of available layers is insufficient for a reasonable approximation of the full 3D tissue volume, it can significantly expand the field-of-view when compared to single-layer correction systems, and therefore can largely accelerate their operation.