Fig. 1: Schematics of imaging geometry and time-reversal process.
From: High-throughput volumetric adaptive optical imaging using compressed time-reversal matrix
a Description of the reflection matrix R in point illumination basis. \({\boldsymbol{R}}[{{\bf{r}}}_{\rm{o}};{{\bf{r}}}_{\rm{i}}^{{\prime}}]=[{E}_{\rm{o}}({{\bf{r}}}_{\rm{o}};{{\bf{r}}}_{\rm{i}}^{{\prime}})]\) is a set of E-field responses \({{E}}_{\rm{o}}({{\bf{r}}}_{\rm{o}};{{\bf{r}}}_{\rm{i}}^{\prime })\) to impulse input fields \({E}_{\rm{i}}({{\bf{r}}}_{\rm{i}})=\delta ({{\bf{r}}}_{\rm{i}}-{{\bf{r}}}_{\rm{i}}^{{\prime} })\). For clarity, the reflection pathway is unfolded to the transmission side. O is the object’s reflection coefficient matrix, and \({{\boldsymbol{P}}}_{\rm{i(o)}}=[{P}_{\rm{i(o)}}({\bf{r}};{{\bf{r}}}_{\rm{i(o)}})]\) is the transmission matrix of the scattering medium for wave propagation from position r to \({{\bf{r}}}_{\rm{i(o)}}\). The red and blue curves represent incident and reflected waves, respectively. b Reflection matrix RS in the case of speckle illumination. \({{\boldsymbol{R}}}_{\rm{S}}[{{\bf{r}}}_{\rm{o}};j]=[{E}_{\rm{o}}({{\bf{r}}}_{\rm{o}};j)]\) is a set of E-field responses \({E}_{\rm{o}}({{\bf{r}}}_{\rm{o}};j)\) for speckle input channels \(S({{\bf{r}}}_{\rm{i}};j)\). RS can be written as RS = RS, where \({\boldsymbol{S}}[{{\bf{r}}}_{\rm{i}};j]=[S({{\bf{r}}}_{\rm{i}};j)]\) is the input illumination matrix. c Geometric interpretation of the time reversal matrix. The time-reversal matrix \({\boldsymbol{W}}[{{\bf{r}}}_{\rm{o}};{{\bf{r}}}_{\rm{o}}^{{\prime} }]={{\boldsymbol{R}}}_{\rm{S}}{{\boldsymbol{R}}}_{\rm{S}}^{\dagger }\) can be interpreted as a reflection matrix describing a roundtrip process for a wave propagating from the output plane ro to the object plane r and reflecting back to the output plane by the target object |O|2
