Reflected-Probe Imaging: Principles and Applications

• RPIs are optical intensity measurements obtained from controlled probe illumination, capturing quantitative information for reflection-mode imaging.
• They enable 3D refractive index mapping, multilayer structure characterization, and aberration sensing through adaptive optics and ptychographic techniques.
• RPIs benefit from both physics-based inversion and deep learning methods, achieving high-resolution reconstructions in biological and semiconductor applications.

Reflected-probe images (RPIs) are optical intensity measurements formed by detecting the light reflected or backscattered from a sample under carefully controlled probe illumination. These images play a central role in a variety of quantitative reflection-mode microscopy and tomography frameworks, particularly for characterizing multilayer structures, extracting three-dimensional refractive index maps, and sensing aberrations in scattering samples. RPIs form the raw data backbone in fields including adaptive optics with deep neural networks, ptychographic reflection microscopy, and reflection-mode diffraction tomography. Their quantitative information content and reconstruction methodologies are governed by a combination of physical optics forward models specific to the reflective geometry, substrate–interface boundary conditions, spectral and angular scanning protocols, and—when applicable—data-driven inference architectures.

1. Physical Principles and Mathematical Modeling

RPIs originate from the coherent or partially coherent interaction of probe illumination with a sample–substrate system, encompassing both the excitation and detection wavefronts and potentially multiple scattering and interface effects. In adaptive optics configurations, the sample imparts spatially varying phase aberrations characterized by $\varphi_{\rm exc}(\rho,\theta)$ (excitation path) and $\varphi_{\rm det}(\rho,\theta)$ (detection path), each expanded in a Zernike basis:

$$\varphi_{\rm exc}(\rho, \theta) = \sum_{n=1}^{N} a_{n}^{\rm exc} Z_{n}(\rho, \theta), \quad \varphi_{\rm det}(\rho, \theta) = \sum_{n=1}^{N} a_{n}^{\rm det} Z_{n}(\rho, \theta)$$

for $N$ Zernike terms, with the reflected-probe image in epi-detection given by

$$I_{\rm RPI}(u,v) \propto \left| \mathcal{F}\big\{ E_0(x,y) e^{i\varphi_{\rm exc}(x,y)} \big\} \right|^2 \otimes \left| \mathcal{F}\big\{ E_0(x,y) e^{i\varphi_{\rm det}(x,y)} \big\} \right|^2$$

where $E_0(x,y)$ is the nominal excitation field, $\mathcal{F}$ denotes the Fourier transform, and $\otimes$ is convolution (Vishniakou et al., 2020).

In multilayer ptychographic reflection, the probe at position $\mathbf{R}_j$ and wavelength $\lambda$ interacts with each interface. The spatial exit wave is

$$\Psi_{\rm exit}(\mathbf{r},\lambda) = P(\mathbf{r} - \mathbf{R}_j,\lambda)\, O_r(\mathbf{r}, \lambda) + \left[ H \! \left\langle P(\mathbf{r} - \mathbf{R}_j,\lambda)\, O_{t1}(\mathbf{r},\lambda) \right\rangle \right] O_{t2}(\mathbf{r}, \lambda)$$

yielding a measured intensity $$I(\mathbf{k},\lambda) = |\mathcal{F}\{\Psi_{\rm exit}(\mathbf{r},\lambda)\}|^2$$ (Gao et al., 2024).

In reflection-mode diffraction tomography, RPIs are captured under multiple oblique illuminations. The forward model employs the modified Born series with Green’s functions and enforces Bloch (lateral) and perfect-electric-conductor (PEC, substrate) boundary conditions. The camera images the intensity at the detection plane:

$$I_m(\mathbf{x}) = \bigl| \hat{D}\{\psi(\mathbf{r})|_{z=z_0}\} \bigr|^2$$

(Li et al., 2024).

2. Experimental Generation and Acquisition of RPIs

Experimental modalities for producing RPIs are highly context-dependent. In adaptive optics with deep learning, a two-photon laser-scanning microscope is equipped with spatial light modulators (SLMs) in both excitation and detection arms. The sample is illuminated, and the reflected signal is relayed to multiple cameras (at several z-planes), producing a stack of 192 × 192 × 3 images for each RPI (Vishniakou et al., 2020). Guide-star configurations utilize metallic spheres or fluorescent beads as embedded reflectors.

In ptychographic reflection microscopy, a nanofabricated pinhole (200 nm) admits a spatially localized probe at a designated angle, with the detector capturing far-field reflection intensities on a high-resolution CCD. The scan grid is designed for high overlap with random jitter to avoid periodic artifacts. Dual- or multi-wavelength illumination enables spectrum multiplexing (Gao et al., 2024).

For reflection-mode diffraction tomography, a programmable LED array in the objective back focal plane produces oblique plane-wave illuminations. The sample, mounted directly on a highly reflective mirror substrate, is imaged under each illumination angle, forming a series of RPIs with sample-dependent intensity modulation (Li et al., 2024).

Configuration	Illumination Scheme	Detector Arrangement
Adaptive AO w/DNN	Focused laser (920 nm)	CMOS cameras, axial stack
Ptychographic Reflection	Monochromatic/multi-λ pinhole	2048 × 2048 CCD, far-field
Diffraction Tomography	Multi-angle (LED array)	Focused image via microscope objective

3. RPI Data Processing and Preprocessing

Post-acquisition preprocessing prepares RPIs for either direct analysis, machine learning, or inversion algorithms. In adaptive optics applications, each camera frame is normalized to [0,1], cropped, and stacked to form a multi-channel tensor (192 × 192 × 3) ready for neural network ingestion. No explicit background subtraction is applied, as normalization is generally sufficient (Vishniakou et al., 2020).

Ptychography requires additional auto-correlation filtering of the raw intensity images owing to the coherent superposition of multiple reflection modes. After taking the Fourier transform of the measured diffraction pattern, cross-terms arising from interference between surface and buried interface reflections appear as off-center peaks in autocorrelation space. These components are masked out; inverse transforming yields the filtered RPI containing only the (incoherent) sum of the individual reflection channels. This step is essential for accurate phase retrieval in multilayer samples (Gao et al., 2024).

In RMDT, RPI data are demultiplexed by illumination angle and referenced to the detection plane. Filtering the spectral content in the $k_z$-domain suppresses high-frequency artifacts while preserving the desired spatial information (Li et al., 2024).

4. Reconstruction and Inference Methodologies

Two broad classes of RPI utilization dominate: data-driven inference and physics-based inversion.

In deep learning–based adaptive optics, the normalized RPI tensor is input to a convolutional neural network. The network's output vector comprises the Zernike coefficients describing the sample-induced aberrations for either the excitation, detection path, or both. The learned mapping enables real-time calculation of phase correction patterns for SLM actuation (Vishniakou et al., 2020). The architecture includes interleaved convolutional, pooling, and dense layers, with mean-absolute error loss relative to ground-truth coefficients engineered via SLM patterns.

Ptychographic reflection reconstruction in multilayer samples is performed by iteratively updating probe and object functions through a Douglas–Rachford solver (sDR). After auto-correlation filtering, the states to reconstruct are (per wavelength): probe $P^k$, surface reflection $O_r^k$, transmission through the top and bottom layers $O_{t1}^k$, $O_{t2}^k$. The forward model propagates these through off-axis angular spectrum operators, and the reconstruction process minimizes a composite loss over all scan positions and wavelengths (Gao et al., 2024). Regularization strategies (e.g., total variation, positivity constraints) are optional but often employed.

Reflection-mode diffraction tomography exploits the modified Born series, embedding Bloch boundary conditions for lateral periodicity and a PEC at the substrate. The forward-model solution is iteratively refined; the gradient with respect to the refractive index is computed via the adjoint method, yielding efficient updates even in the presence of multiply scattered fields. Low-pass filtering in axial $k_z$ is routinely applied to mitigate back-scattered stripe artifacts (Li et al., 2024).

5. Applications and Domain-Specific Impact

RPIs constitute the fundamental observables for aberration correction in biological microscopy, nanoscale multilayer characterization in semiconductor metrology, and volumetric refractive index imaging in complex media.

In adaptive optics, RPI-driven neural correction achieves 2–3× improvements in two-photon fluorescence intensity, lateral PSF narrowing from ~1.2 μm → ~0.6 μm, and substantial gains in image symmetry and focus (Vishniakou et al., 2020). The approach is robust in both planar-mirror calibration and guide-star–embedded biological contexts, with limitations associated with Zernike order truncation and residual misregistrations.

Ptychographic RPI analysis enables resolution of sub-10 nm features, quantitative topography with nanometer-scale accuracy (as verified by AFM), and element-specific contrast via spectrum multiplexing. This capability is particularly emergent in the PISM framework for nanofabrication and semiconductor inspections (Gao et al., 2024).

In RMDT, RPIs enable high-contrast 3D RI reconstructions for dual-layer micro-patterns, scattering objects, and biological samples, preserving fine axial structure and interfacial detail while maintaining computational tractability (Li et al., 2024).

6. Limitations, Error Sources, and Prospects for Improvement

Constraints inherent to RPI-based methodologies include the representational limits of Zernike mode expansions for strongly aberrated fields, network generalization in DNN-driven AO, and the challenge of coherent noise or stripe artifacts in multilayer phase retrieval. Shot noise, high-order uncorrected modes, axial sampling granularity, and imperfect experimental alignment also contribute to residual reconstruction errors (Vishniakou et al., 2020, Gao et al., 2024, Li et al., 2024).

Proposed advancements include higher-order modal decompositions, expansion of simulated and real RPI datasets, introduction of more sophisticated network architectures (e.g., ResNet, Inception variants), and algorithmic innovations in spectral filtering and multi-slice propagation modeling. Future PISM implementations may leverage tabletop extreme ultraviolet sources for further increased specificity and resolution (Gao et al., 2024).

A plausible implication is that continuing development of RPI acquisition and computational analysis will broaden the applicability of reflective imaging across both high-scatter biological samples and the layered nanostructures of advanced manufacturing.