ABSTRACT Time-gated reflection matrix (RM) has been successfully used for optical imaging deep inside scattering media. Recently, this method was extended to enhance the spatiotemporal

focusing of light ultra-deep inside scattering media. This is achieved by calibrating the decomposition of the RM with the Tikhonov regularization parameter to convert multiply scattered

photons that share the same time of flight with the singly scattered photons into singly scattered photons. Such a capability suggests a reshaping to the interaction mechanism between light

and scattering media, which may benefit or inspire wide optical applications that desire enhanced spatiotemporal focusing of light at depths inside scattering media. Light is playing an

photons for imaging in some techniques, such as diffuse optical tomography1, ultrasound-modulated optical tomography2, and photoacoustic tomography3, albeit with compromised resolution. To

achieve a high resolution, singly scattered (SS) photons (also known as ballistic photons) that only probe the in situ target need to be screened out from the predominant MS photons4.

Endeavors to this purpose can be divided into two categories. In the first category, SS photons are spatially filtered out from MS photons, and the representative technology is confocal

microscopy where a small pinhole aperture only allows fluorescence emission from the desired focal plane to pass through5. In the second category, SS photons are temporally filtered out from

MS photons, as demonstrated in optical coherence tomography (OCT) based on time-gating technique6. For both categories, the imaging depth are still restricted to ~1 mm beneath tissue

surface because the number of SS photons decreases exponentially with optical thickness7. Inspired by the transmission matrix approach8 and random matrix model in ultrasound imaging through

strongly scattering media9, a time-gated reflection matrix (RM)-based method called “Smart OCT” was proposed to enhance the imaging depth10,11. In this implementation, a singular value

decomposition of the RM was used to screen out most of the MS photons. Despite of that, MS photons with the same time of flight as the SS photons were still dominant for targets located

deeper than a few scattering mean free paths (SMFP). The common wisdom is that the residual MS photons have impeded the imaging quality and hence in order to yield high resolution, they need

to be removed or suppressed through methods such as iteration12 or a spatial input–output correlation11. Most recently, in _Light: Science & Applications_, Cao et al. suggest that the

part of MS photons that share the same time of flight as the SS photons can be utilized and converted into SS photons to enhance in situ optical energy delivery spatially and temporally13.

In this work, the main goal is to retrieve singular values of the part of MS photons with the same time of flight as the SS photons. First, a coherent gating is created inside the scattering

medium using an ultra-short pulse beam. The back-reflected photons from the scattering medium can be divided into three types: SS photons (ξSS), MS2 photons (ξMS2) sharing the same time of

flight as the SS photons, and other remaining MS photons (ξMS1). After the construction of RM, a singular value decomposition (also termed as “time reversal operator”) is applied to the RM

(R): \(R = U\Sigma V^{\rm T}\), where _T_ represents transpose, Σ is a diagonal matrix containing the real positive singular values in a descending order \(\sigma _1 \,>\, \sigma _2

\,>\, \cdots \,>\, \sigma _N\) (_N_ is the number of the singular values in Σ), _U_ and _V_ are two unitary matrices whose columns correspond to the input and output mode,

respectively. Singular values in the diagonal matrix corresponding to the SS, MS2, and MS1 photons are also in a descending order. Practically, Σ is not actually a standard diagonal matrix

and it cannot be used to filter out the MS2 photons. Thus, matrix\(R^{\dagger} R = USV^T\)(\({\dagger}\) represents conjugate transpose) that has a more standard diagonal matrix will be

taken into consideration in the inversion process. As a result, _S_ is a diagonal matrix containing the square of the singular values (\(\sigma _i^2\)) of the diagonal matrix Σ. Note that

however, this operator is very labile in ultra-deep position due to the noise and there are a lot of non-zero elements at adjacent positions of diagonal line of the diagonal matrix. To

reduce the influence of noise, the Tikhonov regularization parameter14 is introduced to create a calibration matrix to optimize the reversal results in the inversion process. The calibrated

matrix is \(C = UFSV^T\), where _F_ is a diagonal matrix with diagonal elements \(\alpha _i = \sigma _i^2{{{\mathrm{/}}}}\left( {\sigma _i^2 + \lambda ^2} \right)\) (_λ_ is a variance

ranging from 10−8 to 108 for different penetration depths). During the selection of _λ_, there is a tradeoff between the retrieval number of eigenstates and the retrieval accuracy rate of

each eigenstate. In the optimization process, the target is to make the output field from calibrated matrix close to the measured output field. After successfully retrieving singular values

of the SS and MS2 photons, the desired wavefront can be acquired, and the corresponding phase pattern will be loaded on the spatial light modulator. At last, the optical energy delivery can

be enhanced by a magnitude at an ultra-deep (~14.4 SMFP) position. As demonstrated in this study, the optical energy delivery can be enhanced by shaping some part of the MS photons into SS

photons, it can be potentially used to increase the signal-to-noise ratio (SNR) or the imaging depth of “Smart OCT”. Fundamentally, optical scattering arises from the interaction between

photons and matter. Therefore, shaping MS photons into SS photons suggests that the light-matter interaction for the particular photons is changed and furtherly, the underlying physical

mechanism can be reshaped from the conventional realm. This method may also benefit or inspire other optical applications that desire enhanced spatiotemporal focusing of light at depths

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Opin. Otolaryngol. Head Neck Surg._ 29, 79–85 (2021). Article  Google Scholar  Download references AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Department of Biomedical Engineering, The

Hong Kong Polytechnic University, Hong Kong SAR, China Zhipeng Yu, Huanhao Li, Tianting Zhong & Puxiang Lai * Shenzhen Research Institute, The Hong Kong Polytechnic University, Shenzhen,

China Zhipeng Yu, Huanhao Li, Tianting Zhong & Puxiang Lai * Photonics Research Institute, The Hong Kong Polytechnic University, Hong Kong SAR, China Puxiang Lai Authors * Zhipeng Yu

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and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Yu, Z., Li, H., Zhong, T. _et al._ Enhancing spatiotemporal focusing of light deep inside scattering media with Time-Gated Reflection

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