1) in which the SLM plane is conjugated to the scattering layer has been adopted by many studies 13, 14, 15, 16, 17, however, others have implemented a different approach in which the SLM plane is conjugated to the pupil plane of an objective lens placed before the scattering medium 2, 6. The configuration we wish to analyze (Fig. To maximize intensity at point P a phase map of linear dimension D is projected by an SLM onto the scattering layer and is optimized using a continuous sequential algorithm 29 (Fig. Let’s consider one typical scenario to achieve an enhanced focal point P at a plane located a distance z from a scattering layer. ( 1) represents the upper limit of intensity enhancement that can be reached. Here we find that the intensity enhancement is not generally constant when focusing light behind a thin scattering medium and that Eq. via an SLM) of N independent sub-sources within the beam so that they constructively interfere at a desired location the pre-factor γ depends on several experimental parameters such as the operation mode of the SLM, the sensitivity of the camera to small intensity changes, the noise level throughout the enhancement process and the stability of the scattering medium 26, 27, 28. ( 1) can be understood intuitively as the result of adjusting the phases (e.g. Where N is the number of controllable degrees of freedom on the phase profile of the illuminating beam and γ is an experimental scaling factor (For polychromatic light sources the enhancement is reduced by the number of independent frequency components 21, 22, 23, 24 and for focusing into a general field distribution other factors should be considered 25). scattering originating from a random interface between two materials 20, the generalization to volume scattering corresponding to thick materials is much more complex and beyond the scope of this study. In this work we focus on the regime of surface scattering, i.e. We show that this intrinsic limit imposes practical constraints on focusing protocols, as it effectively limits the size of a focal point enhanced through a thin scattering layer, and/or sets an upper-bound to the intensity flux delivered to a given location behind a thin scattering medium. We present a theoretical derivation and experimental demonstration that as the focal plane gets closer to the thin scattering material leading to smaller speckle sizes, the intensity enhancement of the focal point within this plane is compromised. To this end, here we report on a tradeoff between the size of the smallest speckle (serving as focal point) that can be obtained behind a thin scattering medium and the intensity of such focal point achieved via wave-front shaping. However, although, in relative terms, large intensity enhancements can be obtained at the focal point compared to the average intensity of the speckle pattern, the intensity at the focal point is still typically only several percentages of the incident intensity. Several enabling features of such “scattering lens” systems have been described such as super-resolution focusing 17, versatile focal length and structural compactness 18, 19. The underlying concept shared by these works is that the combination of a scattering medium with spatially-resolved control of the beam phase profile can effectively work as a lens. The various methods designed to enhance a focal point through a scattering layer have attracted a great deal of interest for diverse applications such as deep-tissue focusing 10, optogenetic modulations 11, imaging of hidden objects 12 and high resolution focusing/microscopy 13, 14, 15, 16. Following this work, in the past decade, several methods have been developed to obtain such intensity enhancement at a focal point either by iteratively modifying the incident beam phase profile with a spatial light modulator (SLM) 2, 3, 4, 5, by directly measuring the optical transmission matrix of the scattering medium 6, 7 or by recording the field fluctuations induced by the medium 8, 9. They achieved this by tailoring the relative phases of light in the scattering medium to constructively interfere at a point of interest. In their pioneering work, Vellekoop and Mosk 2 obtained a “focal point” from a typical random speckle pattern generated behind a highly scattering layer. Light-based imaging and focusing methods have been historically limited to transparent materials or shallow depths due to multiple light scattering in complex media 1.
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