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Forward and backward scattering provide complementary volumetric and interfacial information, yet conventional three-dimensional (3D) imaging typically accesses only one. We present a substrate-enhanced diffraction tomography approach that simultaneously recovers both channels under multi-angle epi-illumination. This geometry captures one forward- and two backward-scattering bands in axially symmetric Fourier regions, where their complementary coverage enables phase–absorption separation in a non-Hermitian spectrum. Explicit 3D transfer functions are derived for both channels, and an axial Kramers–Kronig relation is established to incorporate substrate-induced boundary conditions in a unified framework. Our results establish a label-free, high-resolution 3D imaging modality that surpasses the limits of existing methods. 

Abstract This paper presents fully-discrete error analysis of the dynamically regularized Lagrange multiplier method (DRLM), a novel stabilization approach for the incompressible Navier–Stokes equations that incorporates kinetic energy evolution with a squared Lagrange multiplier regularization term into the system. The first-order implicit-explicit temporal discretization is employed for the DRLM formulation, in combination with the Taylor–Hood finite elements for spatial approximation. We establish optimalH¹-norm error estimate for the velocity andL²-norm error estimate for the pressure through a mathematical induction process. Specifically, we first derive suboptimal error estimates, which serves as a basis for bounding the numerical solutions via the inverse inequalities, and optimal convergence can then be obtained by repeating the similar arguments provided with the uniformly bounded numerical solutions. Furthermore, we prove optimalL²-norm error estimate for the velocity by using the negative norm techniques. Finally, numerical experiments are given to verify the theoretical convergence rates. 

We introduce a reflection-mode diffraction tomography technique that enables the simultaneous recovery of forward- and backward-scattering information for high-resolution 3D refractive index reconstruction. Our technique works by imaging a sample on a highly reflective substrate and employing a multiple-scattering model and a reconstruction algorithm. It combines the modified Born series as the forward model, Bloch and perfect electric conductor boundary conditions to handle oblique incidence and substrate reflections, and the adjoint method for efficient gradient computation in solving the inverse-scattering problem. We validate the technique through simulations and experiments, achieving accurate reconstructions in samples with high refractive index contrasts and complex geometries. Forward scattering captures smooth axial features, while backward scattering reveals complementary interfacial details. Experimental results on dual-layer resolution targets, 3D randomly distributed beads, phase structures obscured by highly scattering fibers, fixed breast cancer cells, and fixedC. elegansdemonstrate its robustness and versatility. This technique holds promise for applications in semiconductor metrology and biomedical imaging.