Fig. 1. Virtual imaging experiment and scattering matrix tomography (SMT). (a) The scattering matrix S(kout,kin,ω) relates any incident field E˜in(kin,ω) to the resulting scattered field E˜out(kout,ω). (b) After one-time measurements of each spectrally-resolved scattering matrix S(kout,kin,ω), the data are processed computationally to reconstruct the image. (c) SMT imaging that digitally performs triple gating, spatiotemporal wavefront corrections, volumetric scanning, and optimization via Eqs. (2) and (3). The reconstructed image acts as virtual guidestars intrinsic to the sample, providing feedback noninvasively. (d) The scattering matrix can synthesize input spatial gating, output spatial gating, and time gating by summing over the incident momentum kin, outgoing momentum kout, and frequency ω, respectively. (e)–(g) Dispersion, refractive index mismatch at interfaces, and wavefront distortions (from both the optical system and the sample, including both aberrations and multiple scattering) degrade the gates. SMT corrects all of them digitally through a frequency-dependent phase θ(ω) that acts as a virtual pulse shaper, appropriate momenta and phase coefficients for the medium that acts as a virtual index-corrected objective lens, and angle-dependent phases ϕin(kin) and ϕout(kout) that act as two virtual spatial light modulators (SLMs).

Fig. 2. SMT digital dispersion compensation and wavefront corrections. SMT finds the digital corrections θ(ω), ϕin(kin), and ϕout(kout) by optimizing an image quality metric M, Eq. (3). Different spatial zones of the image use different ϕin/out. Each panel shows the SMT image ISMT(r) of the USAF-target-under-tissue sample of Fig. 4 in this process, at the depth where M is maximized. (b)–(f) A direct optimization (red dashed arrows) leads to overfitting and gets trapped in poor local optima. (g)–(k) Through regularization and progression strategies described in the text (green solid arrows), SMT finds digital corrections that restore the spatiotemporal focus and enable high-resolution imaging. Note that the image before corrections in panel (a) already incorporates triple (temporal, input spatial, and output spatial) gating, index-mismatch correction, and removal of the dispersion and aberrations in the input path of the optical system. Scale bar: 10μm.

Fig. 3. Measurement of the hyperspectral reflection matrix. (a) We use off-axis holography to measure the phase and amplitude of fields scattered by the sample. BS, beam splitter; BE, beam expander; TL, tube lens. See Fig. S1 in the Supplementary Material for a detailed schematic. (b) Construction of the data cube by mapping the output angles with the camera, scanning the input angle with the galvo, and scanning the frequency with the tunable laser.

Fig. 4. Noninvasive imaging through thick tissue. (a) Schematic of the sample—a USAF resolution target underneath 0.98 mm of mouse brain tissue—and a scanning electron microscope image of the USAF target before covered by the tissue. (b) A standard bright-field microscope image of the sample (with white-light illumination). (c)–(f) Reflectance confocal microscopy (RCM), optical coherence tomography (OCT), optical coherence microscopy (OCM), and volumetric reflection-matrix microscopy (VRM) images at the USAF target plane, synthesized from the measured hyperspectral reflection matrix. (g) SMT image, ISMT(r), from Eqs. (2) and (3). Each pair of full-view and zoom-in uses the same colorbar, and all images share the same normalization, with scales indicated on the colorbars. Scale bar in panels (b) and (c): 10μm. (h) The wavefront correction phase maps for the 8×8 zones in SMT. (i)–(n) Corresponding point spread function PSF(r) of the sample (i)–(m) and of a mirror in air (n), centered at (xin,yin)=(23.4,45.0)μm.

Fig. 5. Role of triple gating and double-path wavefront correction. (a)–(c) Reconstructed image of the USAF target under tissue following the same procedure as SMT but without input spatial gating. (d)–(f) Reconstructed image with triple gating but with digital aberration correction only in the reciprocal space q of the image. (g) SMT image with triple gating and double-path wavefront correction. Scale bar: 10μm.

Fig. 6. Volumetric imaging inside a dense colloid. The sample consists of 500-nm-diameter TiO2 nanoparticles dispersed in PDMS, with an estimated transport mean free path of 0.47 mm. (a)–(e) SMT, VRM, OCM, OCT, and RCM images built from the measured hyperspectral reflection matrix. (f)–(j) A longitudinal slice of the images at y=23.2μm and close-up views of three particles at different depths in the SMT image. (k) Cross sections of the three particles; Δr=r−rpeak. (l) Transverse slices at the depths of the three particles. (m), (n) SMT images of particles separated horizontally (m) and vertically (n), with center-to-center cross sections. rP11=(46,39,31.29,1487.31)μm, rP12=(46.90,31.96,1487.74)μm, rP21=(44.04,39.12,1569.56)μm, and rP22=(44.18,38.92,1571.02)μm. Scale bars in panels (f) and (l): 10μm. All images share the same normalization. Volumetric images and 2D slices use the same color.