Three-dimensional (3D) imaging through a finite numerical aperture (NA) objective lens is fundamentally constrained by anisotropic spatial bandwidth, yielding high lateral but poor axial resolution. Forward scattering (FS) predominantly encodes volumetric information but suffers from the missing-cone problem, where high–axial-frequency components are inaccessible [1]. Backward scattering (BS), in contrast, carries high-frequency interfacial features but does not provide quantitative volumetric information [2], [3]. These channels are inherently complementary, yet are conventionally probed in separate transmission or reflection geometries [4], [5]. Extending the accessible 3D scattering spectrum is thus essential for accurate characterization, but existing solutions, such as mechanical sample rotation [6]–[8] or engineered reflectors [9], remain impractical.

In this Letter, we advance a substrate-enhanced diffraction tomography technique that simultaneously captures FS and BS using a multi-angle epi-illumination configuration [10]–[12], effectively forming a dual-view acquisition to expand the accessible 3D spatial frequency bandwidth threefold [Fig. 1 (a)]. In this arrangement, both channels interfere with the reflected incidence, providing a natural self-reference that enables quantitative phase reconstruction from intensity-only measurements. This substrate-enhanced acquisition strategy provides access to axially symmetric Fourier bands—one corresponding to FS and two to BS—whose complementary coverage enables phase–absorption separation in a non-Hermitian spatial frequency spectrum. In contrast, conventional reflection geometries capture only a single BS band, which is insufficient for such separation [3], [13].

A central challenge in recovery is that FS and BS contributions from different depths overlap in the recorded two-dimensional (2D) intensity patterns. To disentangle them, we employ diffraction tomography with angle-diverse measurements [Fig. 1 (b–c)], which reconstructs the 3D scattering potential by solving an inverse scattering problem. Our previous work employed the modified Born series (MBS) for this task [12], and while effective, such rigorous models, including the discrete dipole approximation [14], [15], require computationally intensive iterative reconstructions that obscure physical insight. To address this, we derive explicit 3D transfer functions under the first-Born approximation, which establish a linear relationship between the measured intensity and the underlying scattering potential. These transfer functions confine the reconstruction to the physical passbands, substantially reducing computational cost. We further exploit the spatial causality of scattering in the presence of a reflective substrate to establish an axial Kramers–Kronig (KK) relation, which constrains the inversion of the half-space scattering problem. Together, this transfer-function framework provides a clearer and more tractable alternative to iterative MBS model, enabling efficient separation and recovery of 3D FS and BS information, as validated in simulation and experiment.

Consider a 3D weakly-scattering object immersed in a medium permittivity \(\epsilon_{\mathrm{m}}\) above a reflective substrate, characterized by a permittivity perturbation \(\Delta\epsilon(\mathbf{r}) = \epsilon(\mathbf{r}) - \epsilon_{\mathrm{m}}\). The substrate surface is defined as the \(z = 0\) plane, with its optical response described by a reflection matrix \(R(\mathbf{k}_{\parallel})\). An obliquely incident plane wave \(\psi_{0, +} = e^{i(\mathbf{k}_{\parallel, \mathrm{in}}\cdot \mathbf{r}_{\parallel}+k_{z, \mathrm{in}} z)}\) with unit amplitude, propagating with \(k_{z, \mathrm{in}}\) along the positive \(z\)-direction, illuminates the object and substrate, as illustrated in Fig. 1 (c).

Under the single-scattering approximation, the reflective substrate first reflects the incidence back toward the object, generating a secondary incidence, \(\psi_{0, -} = R(\mathbf{k}_{\parallel, \mathrm{in}})e^{i(\mathbf{k}_{\parallel, \mathrm{in}}\cdot \mathbf{r}_{\parallel}-k_{z, \mathrm{in}} z)}\), forming a standing-wave illumination. Each incidence scatters at every depth, producing FS along the incidence (gray dashed line) and BS along the opposite direction (light blue dashed line). For \(\psi_{0,+}\) (purple arrows), FS (\(\psi_{\mathrm{FS},+}\)) propagates along \(+z\) and BS (\(\psi_{\mathrm{BS},+}\)) along \(-z\); For \(\psi_{0,-}\) (blue arrows), FS (\(\psi_{\mathrm{FS},-}\)) propagates along \(-z\) while BS (\(\psi_{\mathrm{BS},-}\)) propagates along \(+z\). Scattered fields propagating along \(-z\) are directly collected, while \(+z\)-propagating fields first reach the substrate, reflect at \(z=0\) to reverse the direction, and then enter the objective.

Without loss of generality [12], the objective’s focal plane is set at \(z=0\). The microscope effectively applies a Fourier filter to the field through its pupil function \(P(\mathbf{k}_{\parallel})\). The secondary incidence \(\psi_{0,-}\) interferes with all four scattered field components to form the measured intensity pattern \(I_{\mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{r}_{\parallel})\). Under the weak-scattering condition, the cross terms provide the main interference contrast, establishing a linear relation between the permittivity contrast and the intensity spectrum [16], \[\label{eq:slice-wiseTF} \tilde{I}_{\mathbf{k}_{\parallel, \mathrm{in}}}({\mathbf{k}_{\parallel}}) \approx \tilde{I}_{0, \mathbf{k}_{\parallel, \mathrm{in}}} +\int_{-\infty}^0 \mathbf{H}_{ \parallel,\mathbf{k}_{\parallel, \mathrm{in}}}({\mathbf{k}_{\parallel}},z)\widetilde{\Delta \boldsymbol{\epsilon} }_{\parallel}({\mathbf{k}_{\parallel}},z)\cdot\mathrm{d}z,\tag{1}\] where \(\mathbf{H}_{\parallel} = [H_{\mathrm{\parallel,Re}} ,H_{\parallel,\mathrm{Im}} ]\) denotes the slice-wise transfer function, \(\widetilde{\Delta \boldsymbol{\epsilon} }_{\parallel} = [\widetilde{\Delta \epsilon }_{\parallel,\mathrm{Re}}, \widetilde{\Delta \epsilon }_{\parallel,\mathrm{Im}}]^\top\) is the 2D spectrum of each slice, and \(\tilde{I}_{0, \mathbf{k}_{\parallel, \mathrm{in}}}\) is the DC component, which can be removed by background subtraction [16].

Unlike transmission geometry, where scattering occurs throughout the space unless previously bounded, the epi-configuration restricts the object to the upper half-space (\(z<0\)), leaving the region \(z>0\) free of scatterers. This asymmetry prevents a direct definition of a full 3D transfer function. However, the substrate-imposed cutoff naturally enforces a spatial causality constraint, where the scattering potential is confined to one half-space in real space, and its Fourier dual is analyticity in \(k_z\). This analyticity ensures that all scattered fields originate from the physical half-space and leads to an axial KK relation, linking the real and imaginary parts of \(\widetilde{\Delta \boldsymbol{\epsilon}}\) through a Hilbert transform \(\mathscr{H}_z\) along \(k_z\), \[\label{KK} \mathrm{Im}[\widetilde{\Delta \epsilon }({\mathbf{k}})] = \frac{1}{\pi} \mathrm{p.v.}\int_{-\infty}^\infty \frac{\mathrm{Re}[\widetilde{\Delta \epsilon }({\mathbf{k}}_{\parallel}, k'_z)]}{k_z - k'_z} \mathrm{d}k'_z,\tag{2}\] where p.v. denotes the Cauchy principal value, in direct analogy to the temporal KK relations that connect a material’s dispersive permittivity to causality.

We make this property explicit by expressing \(\Delta \epsilon({\mathbf{r}}) \to S(z)\Delta \epsilon({\mathbf{r}})\), where \(S(z)\) is a step function enforcing analyticity [Fig. 2 (a) and (b)]. This formalizes the causal nature of the scattering process and permits the integral in Eq. 1 to be extended over the entire space \((-\infty,\infty)\) without changing the physical content. Using the Parseval–Plancherel identity, scattering from an object above a reflective substrate can then be expressed linearly through a 3D transfer function, \[\label{eq:3DTF} \tilde{I}_{\mathbf{k}_{\parallel, \mathrm{in}}}({\mathbf{k}_{\parallel}}) \approx \tilde{I}_{0, \mathbf{k}_{\parallel, \mathrm{in}}} +\int_{-\infty}^\infty \mathbf{H}_{\mathbf{k}_{\parallel, \mathrm{in}}}({\mathbf{k}})\widetilde{\Delta \boldsymbol{\epsilon} }({\mathbf{k}})\cdot\mathrm{d}k_z,\tag{3}\] where \[\begin{align} \tag{4} H_{\mathrm{Re}, \mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{k}) &= \frac{ik_0^2}{2}\left[ H_{0, \mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{k}) - H^*_{0, \mathbf{k}_{\parallel, \mathrm{in}}}(-\mathbf{k}) \right],\\ \tag{5} H_{\mathrm{Im}, \mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{k}) &= -\frac{k_0^2}{2}\left[ H_{0, \mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{k}) + H^*_{0, \mathbf{k}_{\parallel, \mathrm{in}}}(-\mathbf{k}) \right]. \end{align}\] Here, \(H_{0, \mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{k})\) contains the essential contributions, consisting of two FS and two BS components, \[H_{0, \mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{k}) = \frac{R^*({\mathbf{k}_{\parallel, \mathrm{in}}})P^*({\mathbf{k}_{\parallel, \mathrm{in}}})P(\mathbf{k}_{\parallel} + \mathbf{k}_{\parallel, \mathrm{in}} ) }{k_{\bot}({\mathbf{k}_{\parallel}}+{\mathbf{k}_{\parallel, \mathrm{in}}})} (H_{\mathrm{FS}} + H_{\mathrm{BS}} ),\] with the FS and BS components \[\begin{align} \tag{6} H_{\mathrm{FS}} &= R(\mathbf{k}_{\parallel} + \mathbf{k}_{\parallel, \mathrm{in}} )\delta_{\mathrm{FS},+} + R(\mathbf{k}_{\parallel, \mathrm{in}})\delta_{\mathrm{FS},-}, \\ \tag{7} H_{\mathrm{BS}} &= \delta_{\mathrm{BS},+} + R(\mathbf{k}_{\parallel, \mathrm{in}}) R(\mathbf{k}_{\parallel} + \mathbf{k}_{\parallel, \mathrm{in}}) \delta_{\mathrm{BS}, -}, \end{align}\] where \(k_{\bot}({\mathbf{k}_{\parallel}}+{\mathbf{k}_{\parallel, \mathrm{in}}})\equiv [ \epsilon_{\mathrm{m}} k_0^2- ({\mathbf{k}_{\parallel}}+{\mathbf{k}_{\parallel, \mathrm{in}}})^2]^{1/2}\), \(\delta_{\mathrm{FS}+/-} \equiv \delta[k_z \mp k_{\bot}({\mathbf{k}_{\parallel}}+{\mathbf{k}_{\parallel, \mathrm{in}}}) \pm k_{z, \mathrm{in}} ]\) and \(\delta_{\mathrm{BS},+/-} \equiv \delta[k_z \pm k_{\bot}({\mathbf{k}_{\parallel}}+{\mathbf{k}_{\parallel, \mathrm{in}}}) \pm k_{z, \mathrm{in}} ]\). Details in Supplemental Material [17].

Eqs. 4 and 5 show that \(\widetilde{\Delta \epsilon }_{\mathrm{Re}}({\mathbf{k}})\) and \(\widetilde{\Delta \epsilon }_{\mathrm{Im}}({\mathbf{k}})\) are encoded asymmetrically in Fourier domain: \(H_{\mathrm{Re}, \mathbf{k}_{\parallel, \mathrm{in}}}\) is imaginary and anti-symmetric, while \(H_{\mathrm{Im}, \mathbf{k}_{\parallel, \mathrm{in}}}\) is real and symmetric. The substrate effectively doubles the accessible Fourier domain along \(k_z\) by folding \(+z\)-propagating FS and BS components into the pupil, effectively forming a dual-view acquisition. This yields four \(\delta_{\mathrm{FS}/\mathrm{BS}+/-}\) terms that define Ewald spherical caps centered at \(\mathbf{k}_{\parallel,\mathrm{in}}\), truncated by the NA, and separated symmetrically along \(k_z\), forming Fourier pairs that enable separation of phase and absorption in both FS and BS from any non-Hermitian spectrum. [Eqs. 6 and 7 ; Fig. 1 (e)].

This asymmetry and separation imply that phase and absorption, as well as FS and BS contributions, are inherently decoupled and thus separable during retrieval. As the illumination angle varies, FS and BS caps sweep through Fourier domain and fill their respective supports [Fig. 1 (a)], yielding three isolated bands with lateral bandwidths \(B_x = B_y = 4\mathrm{NA}\cdot k_0\) and axial bandwidth \(B_z = 2\epsilon^{1/2}_mk_0(1 - \sqrt{1 - \mathrm{NA}^2/\epsilon_{\mathrm{m}}})\). Two BS bands (green) and the FS band (gray) are highlighted in Fig. 1 (e).

The inverse scattering problem seeks to recover the unknown \(\Delta \epsilon ({\mathbf{r}})\) of the object from a series of angle-diverse measurements. The forward model, defined in Eq. 3 , is linear and decouples across in-plane spatial frequencies \({\mathbf{k}_{\parallel}}\). In the discrete formulation, we assume sampling in all three dimensions meets the Nyquist criterion imposed by the respective bandwidths.

At a given \({\mathbf{k}_{\parallel}}\), the key challenge is to disentangle FS and BS components, which are superimposed in the forward model. To address this, we collect the full stack of \(M\) angle-diverse measurements, leading to a discrete linear system [Fig. 1 (e)], \[\mathbf{A}({\mathbf{k}_{\parallel}}) \widetilde{\Delta \boldsymbol{\epsilon} }({\mathbf{k}_{\parallel}}) = \tilde{\mathbf{I}}({\mathbf{k}_{\parallel}}),\] where \(\tilde{\mathbf{I}}({\mathbf{k}_{\parallel}})\in \mathbb{R}^{M\times 1}\) is the column vector of measured intensity spectra at \({\mathbf{k}_{\parallel}}\), \(\widetilde{\Delta \boldsymbol{\epsilon} }({\mathbf{k}_{\parallel}}) \in \mathbb{C} ^ {2N \times 1}\), given by \([\widetilde{\Delta \boldsymbol{\epsilon} }^{\top}_{\mathrm{Re}}({\mathbf{k}_{\parallel}}) , \widetilde{\Delta \boldsymbol{\epsilon} }^{\top}_{\mathrm{Im}}({\mathbf{k}_{\parallel}}) ]^{\top}\), contains the unknown Fourier components of the permittivity contrast within the passband (for \(N\) axial frequency components), and forward operator \(\mathbf{A}\in\mathbb{C}^{M\times 2N}\) mapping \(\widetilde{\Delta \boldsymbol{\epsilon}}\) to the measurement spectra across all illuminations, \[\label{eq:ForwardMatrix} \mathbf{A} \equiv \left(\mathbf{H}_{\textrm{Re}},\mathbf{H}_{\textrm{Im}}\right) \begin{pmatrix} (1+i\mathscr{H}_z)/2&0\\0 & (1+i\mathscr{H}_z)/2 \end{pmatrix},\tag{8}\] where the axial KK relation in Eq. 2 is imposed by invoking the identity \((1+i\mathscr{H}_z)\widetilde{\Delta \boldsymbol{\epsilon} }({\mathbf{k}_{\parallel}})/2 = \widetilde{\Delta \boldsymbol{\epsilon} }({\mathbf{k}_{\parallel}})\). In practice, we apply discrete Fourier transforms (DFTs) to the discretized slice-wise transfer functions [Eq. 1 ] to compute the 3D transfer function. This approach is numerically more stable than direct discretization of Eqs. 4 and 5 , as it naturally enforces the Nyquist bandlimit and properly handles cases where delta locations fall off the Fourier sampling grid. Example 3D transfer functions are shown in Fig. 2 (c), and details are shown in Supplemental Material [17].

To validate the forward model, we simulated scattering from a synthetic object placed above a mirror under oblique illumination [Fig. 2 (a)]. A mirror is used as the substrate to maximize collection efficiency, with the reflection matrix \(R(\mathbf{k}_{\parallel})=-1\). The object consists of randomly distributed beads confined to a thin spherical shell (radius 3 \(\mu\)m, thickness 100 nm) with volume fraction \(\rho=10\)–40% and permittivity contrast \(\Delta \epsilon=0\)–0.2 on a background of \(\epsilon_{\mathrm{m}}=1.80\), illuminated by a 632 nm plane wave with \(\mathbf{k}_{\parallel, \mathrm{in}}=(0.6k_0,0)\). The normalized root-mean-square error (NRMSE) between the computed patterns using the transfer function approach and the rigorous MBS model [Fig. 2 (d)] remains below 0.3 across all tested parameters, confirming that Eq. 3 accurately captures both forward- and backward-scattered fields. Representative patterns at \(\Delta \epsilon=0.05,~0.1,\) and \(0.15\) (inset) show that Eq. 3 reproduces the interference features of the rigorous MBS model with consistent contrast variations as \(\Delta \epsilon\) increases. The error grows nearly linearly with \(\Delta \epsilon\), reflecting stronger multiple scattering beyond the single-scattering regime, yet remains quantitatively reliable for typical biomedical imaging conditions.

For recovering 3D FS and BS information, the linear forward model enables efficient non-iterative inversion, while the axial KK relation enforces spatial causality as a physical prior, confining the solution to the upper half-space. For practical implementation, rather than discretizing the entire real space at the Nyquist limit as required by our previous MBS method [12], the axial separation of FS and BS components in the 3D Fourier space allows a band-limited reconstruction confined to the three Fourier-domain supports illustrated in Fig. 1 (e). This restriction reduces the dimensionality of the forward operator \(\mathbf{A}\) to the axial passband, substantially lowering the memory cost of inversion. The inverse is computed using truncated singular value decomposition, with \(\mathbf{A}^{+} \approx \mathbf{V}\mathbf{\Sigma}_{\alpha}^{-1}\mathbf{U}^\dagger\) constructed by discarding singular values below a tunable threshold \(\alpha\), thereby suppressing low signal-to-noise modes while preserving the dominant scattering components.

We first validate the method in simulation with three beads (radius 1 \(\mu\)m, \(|\Delta\epsilon_b|=8\times10^{-3}\)) embedded in a thin spherical shell (radius 6.5 \(\mu\)m, thickness 100 nm, \(|\Delta\epsilon_s|=2.7\times10^{-2}\)) on a background \(\epsilon_{\mathrm{m}}=1.80\) (full figures in Supplemental Material [17]). The top bead and upper half-shell are purely absorptive (\(\Delta\epsilon_{\mathrm{Im}} \neq 0\), \(\Delta\epsilon_{\mathrm{Re}} = 0\)), whereas the bottom beads and lower half-shell are purely phase objects (\(\Delta\epsilon_{\mathrm{Re}} \neq 0\), \(\Delta\epsilon_{\mathrm{Im}} = 0\)). Angle-diverse intensity measurements are generated using our rigorous MBS model under 138 illuminations at 632 nm, arranged in seven concentric rings with uniform NA spacing up to 0.95 at the outermost ring.

After inversion, three recovered bands are assembled into the 3D Fourier spectrum. Figure 3 (a) shows the \(k_x\)–\(k_z\) cross-section, while the \(k_x=0.5k_m\) slice on the right confirms the axial KK relation of the recovered spectrum. Because the spectrum is non-Hermitian, regions symmetric about the axial frequency \(k_z\) must be combined to decouple phase and absorption. For the FS band, centered at zero frequency with support inherently symmetric about \(k_z=0\), this symmetry alone suffices for decoupling. By contrast, BS bands occupy two distinct high-frequency regions symmetric about the \(k_z\) axis; both must be combined to achieve decoupling, with their real and imaginary parts in real space encoding high-\(k_z\) phase and absorption, respectively. The spectral gap between two BS bands introduces oscillations in \(z\)-profiles. To obtain smooth BS \(z\)-profiles, a single BS band is isolated, and its magnitude gives the oscillation envelope. Accordingly, a single BS band is used for \(z\)-profiles, whereas both bands are combined for \(x\)–\(y\) cross-sections. Figure 3 (b) shows the reconstructed \(x\)–\(z\) cross-sections from the FS band and the top BS band. Since the spectra obey the axial KK relation along \(k_z\), both \(\Delta\epsilon_{\mathrm{FS}}\) and \(\Delta\epsilon_{\mathrm{BS}}\) are confined to the upper half-space (\(z<0\)); for clarity, the \(z>0\) regions are truncated in the figure (dashed lines).

The FS reconstruction reveals smooth volumetric features similar to transmission, offering high lateral resolution but suffering from axial elongation and blurred boundaries due to the missing-cone in the low-frequency region [white shading in Fig. 3 (a)], with real and imaginary parts corresponding to phase (left) and absorption (right), respectively. The absence of high-\(k_z\) components causes the top and bottom shell interfaces to vanish in \(\Delta\epsilon_{\mathrm{FS}}\). In contrast, the recovered BS envelope shows enhanced sensitivity to lateral interfaces, effectively recovering the missing boundaries of the shell and thereby complementing the FS information. Phase–absorption separation in BS is given in Supplemental Material [17].

The technique was further validated experimentally on a fixed C. elegans placed on a silver mirror and immersed in water (\(\epsilon_{\mathrm{m}}=1.80\)). Detailed information and additional breast cancer cell results are shown in Supplemental Material [17]. The experiment used a reflection-mode LED microscope with a 10\(\times\)/0.28NA objective [Fig. 4 (a)]. Illumination was provided by a 25-LED array, relayed by a 4\(f\) system to the objective’s back focal plane, generating plane waves with NAs of 0.14 and 0.28. Each LED emitted at 632 nm (20 nm bandwidth), and the resulting intensities were recorded on a camera (2.74 \(\mu\)m pixel size), as shown in the inset of Fig. 4 (a).

The transfer function–based reconstruction was applied to recover \(\Delta\epsilon_{\mathrm{FS}}\) and \(\Delta\epsilon_{\mathrm{BS}}\). Representative \(x\)–\(y\) cross-sections are shown in Fig. 4 (b), with depth-dependent contrast and a zoom-in \(z\)-stack in Fig. 4 (c). For the BS reconstructions, both BS bands were used to reconstruct \(x\)–\(y\) slices, while only the top BS band was used to generate the contrast profile. This profile, defined as \(\sum_{\mathbf{r}_{\parallel}}|\Delta \epsilon|^2\), rapidly vanishes beyond \(z=0\), confirming that the axial KK relation confines the reconstruction to the upper half-space. At \(z=-23.0~\mu\)m near the worm’s top surface, the reconstructed \(\Delta\epsilon_{\mathrm{FS}}\) suffers from blurred, low-contrast interfaces due to the missing cone problem, whereas \(\Delta\epsilon_{\mathrm{BS}}\), arising from interfacial reflections, captures high-frequency details and reveals pronounced, continuously varying features that complement the FS reconstruction. Deeper inside (\(z=-3.5~\mu\)m), \(\Delta\epsilon_{\mathrm{FS}}\) delineates internal anatomy, including the pharynx, intestinal lumen, and nuclei, while BS contrast diminishes but remains spatially correlated with FS. The contrast profile and \(z\)-stack highlight this complementarity: FS contrast peaks near the mirror plane, where internal structures are best resolved but lateral interfaces are blurred, whereas BS contrast is maximal at the surface and decays with depth, recovering the high-\(k_z\) interfacial information.

In conclusion, we have introduced a substrate-enhanced diffraction tomography method that simultaneously retrieves FS and BS in epi-configuration, mitigating the missing cone of transmission and tripling the accessible 3D Fourier bandwidth. FS resolves internal volumetric structures, while BS captures high-frequency surface interfaces, providing complementary contrast confirmed in simulations and experiments. We derive explicit 3D transfer functions for FS and BS and establish an axial KK relation that constrains inversion to the physical half-space. These transfer functions enable band-limited reconstruction restricted to the physical passbands, substantially reducing computational cost. The approach achieves high-resolution, label-free volumetric imaging, disentangles FS–BS mixing in epi-geometry, and extends the effective bandwidth of conventional configurations.

Acknowledgment—The work was supported by National Science Foundation (1846784) and in part by grant number 2023-321163 from the Chan Zuckerberg Initiative DAF, an advised fund of Silicon Valley Community Foundation.

1 Derivation Details of the Transfer Function for Substrate-Enhanced Diffraction Tomography↩︎

1.1 First-Born single scattering approximation↩︎

In this section, we derive the transfer function for the substrate-epi configuration. We start with the scalar inhomogeneous Helmholtz equation, expressed as \[\label{Scal95Helm} \nabla^2\psi(\mathbf{r})+\epsilon(\mathbf{r})k_0^2\psi(\mathbf{r}) = 0,\tag{9}\] where \(\psi(\mathbf{r})\) is the field at position \(\mathbf{r}\equiv [\mathbf{r}_\parallel, z]\), \(\epsilon(\mathbf{r})\) is the permittivity distribution. Here, \(\lambda\) is the wavelength of the incidence, and \(k_0=2\pi/\lambda\) is the wavenumber in vacuum. The solution to Eq. 9 is referred to the total field, \(\psi_\text{tot}(\mathbf{r})\), as it includes both the original incident field, \(\psi_{0}(\mathbf{r})\), and the scattered field, \(\psi_{\text{s}}(\mathbf{r})\), \[\label{TotalFieldDef} \psi_\text{tot}(\mathbf{r}) = \psi_{0}(\mathbf{r}) + \psi_{\text{s}}(\mathbf{r}).\tag{10}\] By separating the permittivity into a background component and a scattering component, the scattering potential can be defined as \(V(\mathbf{r}) = \epsilon(\mathbf{r})k_0^2 -\epsilon_{\text{m}}k_0^2 = \Delta \epsilon(\mathbf{r}) k_0^2\), where \(\epsilon_{\text{m}}\) is the background permittivity, and the permittivity contrast distribution is \(\Delta \epsilon(\mathbf{r}) = \Delta \epsilon_{\text{Re}}(\mathbf{r}) + i \Delta \epsilon_{\text{Im}}(\mathbf{r})\). Substituting this into Eq. 9 gives, \[\label{Scal95Helm95scatpot} \nabla^2\psi(\mathbf{r})+\epsilon_{\text{m}}k_{0}^2(\mathbf{r})\psi(\mathbf{r}) = -V(\mathbf{r})\psi(\mathbf{r}).\tag{11}\] Using Green’s function theorem, the solution of Eq. 11 can be written as, \[\label{Scal95Helm95solution} \psi(\mathbf{r})=g_{\text{m}}(\mathbf{r})*[V(\mathbf{r})\psi(\mathbf{r})],\tag{12}\] where \(*\) refers to the convolution operation, i.e., \(A(\mathbf{r})*B(\mathbf{r})=\int A(\mathbf{r}-\mathbf{r}')B(\mathbf{r}')\mathrm{d}\mathbf{r'}\). Here, \(g_{\text{m}}(\mathbf{r})\) is the Green’s function in the background medium, which satisfies \[\nabla^2g_{\text{m}}(\mathbf{r})+\epsilon_{\text{m}}k_{0}^2g_{\text{m}}(\mathbf{r}) = -\delta(\mathbf{r}),\] where \(\delta\) is the delta function. The Fourier spectrum of the scalar Green’s function in free space is \(\tilde{g}_{\text{m}}(\mathbf{k}) = 1/(|\mathbf{k}|^2-\epsilon_{\text{m}}k_{0}^2)\), where \(\mathbf{k} \equiv [\mathbf{k}_\parallel, k_z]\) is the coordinate of Fourier space. Under the first Born approximation, which assumes that the incident field is scattered only once, the solution of Eq. 12 simplifies to \[\label{Scal95Helm95FB} \psi_\text{tot}(\mathbf{r})\approx\psi_{0}(\mathbf{r}) +g_{\text{m}}(\mathbf{r})*[V(\mathbf{r})\psi_{0}(\mathbf{r})],\tag{13}\] where the scattered field under the first Born approximation is \(\psi_{\text{s}}(\mathbf{r}) \approx g_{\text{m}}(\mathbf{r})*[V(\mathbf{r})\psi_{0}(\mathbf{r})]\).

1.2 Slice-wise transfer function for substrate-epi configuration↩︎

In substrate-epi configuration, assuming a reflective substrate is set at \(z=0\) plane with its optical response described by a reflection matrix \(R(\mathbf{k}_{\parallel})\), the source and sample are put on the \(z<0\) side of the substrate, occupying the region of \((z_1, 0)\) with \(z_1< 0\), and the \(k_{z,\text{in}}\) of the incidence is along the \(+z\) direction. An obliquely incident plane wave \(\psi_{0, +} = e^{i(\mathbf{k}_{\parallel, \mathrm{in}}\cdot \mathbf{r}_{\parallel}+k_{z, \mathrm{in}} z)}\) with unit amplitude, propagating with \(k_{z,\mathrm{in}}\) of unit amplitude illuminates the object and substrate. With the presence of the reflective substrate, the primary incidence is reflected back, serving as a secondary incidence, \(\psi_{0, -} = R(\mathbf{k}_{\parallel, \mathrm{in}})e^{i(\mathbf{k}_{\parallel, \mathrm{in}}\cdot \mathbf{r}_{\parallel}-k_z z)}\), together forming a standing-wave illumination distribution, \[\psi_{\mathrm{0}}(\mathbf{r}) = \psi_{\mathrm{0},+}(\mathbf{r}) + \psi_{\mathrm{0},-}(\mathbf{r}).\] As mention in the manuscript, each incidence is scattered by the object at every depth to produce forward scattering (FS) along the incidence direction and backward scattering (BS) along the opposite direction. For the primary incidence \(\psi_{0,+}\), FS (\(\psi_{\mathrm{FS},+}\)) propagates along \(+z\) and BS (\(\psi_{\mathrm{BS},+}\)) along \(-z\); For the secondary incidence \(\psi_{0,-}\), FS (\(\psi_{\mathrm{FS},-}\)) propagates along \(-z\) while BS (\(\psi_{\mathrm{BS},-}\)) propagates along \(+z\). The \(-z\)-propagating scattered fields are directly collected by the objective, whereas the \(+z\)-propagating scattered fields first reach the substrate, reflect at \(z=0\) to reverse the direction, and then enter the objective.

To model the scattering process, the Weyl expansion of Green’s function is used to calculate the scattered field at a specified depth, \[\label{WeylExpan} g_{\text{m}}(\mathbf{r}) =\frac{i}{2} \mathscr{F}^{-1}_{\parallel}\left[\frac{e^{ik_{\bot}(\mathbf{k}_\parallel)|\Delta z|}}{k_{\bot}(\mathbf{k}_\parallel)}\right],\\\tag{14}\] where \(\mathscr{F}_{\parallel}\) and \(\mathscr{F}^{-1}_{\parallel}\) denote in-plane 2D Fourier and inverse Fourier transform, respectively, and \(k_{\bot}({\mathbf{k}_{\parallel}}+{\mathbf{k}_{\parallel, \mathrm{in}}})\equiv [ \epsilon_m k_0^2- ({\mathbf{k}_{\parallel}}+{\mathbf{k}_{\parallel, \mathrm{in}}})^2]^{1/2}\). The convention of the Fourier basis is defined as \(e^{-i\mathbf{k}\cdot\mathbf{r}}\) in this work. We first formulate the \(-z\)-propagating scattered fields, \(\psi_{\mathrm{FS},-} + \psi_{\mathrm{BS},+}\), at \(z_\infty\ll z_1<0\) plane, \[\begin{align} \psi_{\mathrm{FS},-}(\mathbf{r}_{\parallel},z_\infty) + \psi_{\mathrm{BS},+}(\mathbf{r}_{\parallel},z_\infty)&=\frac{i}{2}\int^{0}_{-\infty}\mathrm{d}z \cdot \mathscr{F}^{-1}_{\parallel}\left\{\frac{e^{ik_{\bot}(\mathbf{k}_\parallel)|z-z_\infty|}}{k_{\bot}(\mathbf{k}_\parallel)}\cdot\mathscr{F}_{\parallel}[V(\mathbf{r})\psi_{0}(\mathbf{r})]\right\}\\ &=\frac{i k_0^2 }{2}\int^{0}_{-\infty} \mathscr{F}^{-1}_{\parallel}\left\{\widetilde{\Delta\epsilon}_{\parallel}(\mathbf{k_\parallel}-\mathbf{k}_{\parallel,\text{in}},z) \frac{e^{ik_{\bot}(\mathbf{k}_\parallel)(z-z_\infty)}}{k_{\bot}(\mathbf{k}_\parallel)}\left[e^{ik_{z,\text{in}}z}+R(\mathbf{k}_{\parallel,\text{in}})e^{-ik_{z,\text{in}}z}\right]\right\}\mathrm{d}z,\\ \end{align}\] where \(\widetilde{\Delta\epsilon}_{\parallel} \equiv \mathscr{F}_\parallel\Delta\epsilon\) is the 2D spectra of the sliced permittivity contrast. For the \(+z\)-propagating scattered fields, \(\psi_{\mathrm{FS},+} + \psi_{\mathrm{BS},-}\), first propagates to the substrate at \(z = 0\), \[\begin{align} \psi_{\mathrm{FS},+}(\mathbf{r}_{\parallel},0) + \psi_{\mathrm{BS},-}(\mathbf{r}_{\parallel},0)&=\frac{i}{2}\int^{0}_{-\infty}\mathrm{d}z\cdot \mathscr{F}^{-1}_{\parallel}\left\{\frac{e^{ik_{\bot}(\mathbf{k}_\parallel)|z-0|}}{q_{\bot}(\mathbf{k}_\parallel)}\cdot\mathscr{F}_{\parallel}[V(\mathbf{r})\psi_{0}(\mathbf{r})]\right\}\\ &=\frac{i k_0^2 }{2}\int^{0}_{-\infty} \mathscr{F}^{-1}_{\parallel}\left\{\widetilde{\Delta\epsilon}_{\parallel}(\mathbf{k_\parallel}-\mathbf{k}_{\parallel,\text{in}},z) \frac{e^{-ik_{\bot}(\mathbf{k}_\parallel)z}}{k_{\bot}(\mathbf{k}_\parallel)}\left[e^{ik_{z,\text{in}}z}+R(\mathbf{k}_{\parallel,\text{in}})e^{-ik_{z,\text{in}}z}\right]\right\}\mathrm{d}z,\\ \end{align}\] then it is reflected back by the substrate and propagates to the \(z_\infty\) plane, \[\psi'_{\mathrm{FS},+}(\mathbf{r}_{\parallel},z_\infty) + \psi'_{\mathrm{BS},-}(\mathbf{r}_{\parallel},z_\infty) =\frac{i k_0^2 }{2}\int^{0}_{-\infty} \mathscr{F}^{-1}_{\parallel}\left\{\widetilde{\Delta\epsilon}_{\parallel}(\mathbf{k_\parallel}-\mathbf{k}_{\parallel,\text{in}},z) \frac{R(\mathbf{k}_{\parallel})e^{-ik_{\bot}(\mathbf{k}_\parallel)(z+z_\infty)}}{k_{\bot}(\mathbf{k}_\parallel)}\left[e^{ik_{z,\text{in}}z}+R(\mathbf{k}_{\parallel,\text{in}})e^{-ik_{z,\text{in}}z}\right]\right\}\mathrm{d}z.\\\] These four terms yield the total scattered field \(\psi_{\mathrm{s}}(\mathbf{r}_{\parallel},z_\infty) = \psi_{\mathrm{FS},-}(\mathbf{r}_{\parallel},z_\infty) + \psi_{\mathrm{BS},+}(\mathbf{r}_{\parallel},z_\infty) + \psi'_{\mathrm{FS},+}(\mathbf{r}_{\parallel},z_\infty) + \psi'_{\mathrm{BS},-}(\mathbf{r}_{\parallel},z_\infty)\) at \(z_\infty\) plane, \[\psi_{\mathrm{s}}(\mathbf{r}_{\parallel},z_\infty) =\frac{ik_0^2}{2}\int^{0}_{-\infty} \mathscr{F}^{-1}_{\parallel}\left\{\frac{\widetilde{\Delta\epsilon}_{\parallel}(\mathbf{k_\parallel}-\mathbf{k}_{\parallel,\text{in}},z) }{k_{\bot}(\mathbf{k}_\parallel)} \left[R(\mathbf{k}_{\parallel})e^{-ik_{\bot}(\mathbf{k}_\parallel)(z+z_\infty)}+e^{ik_{\bot}(\mathbf{k}_\parallel)(z-z_\infty)}\right]\left[e^{ik_{z,\text{in}}z}+R(\mathbf{k}_{\parallel,\text{in}})e^{-ik_{z,\text{in}}z}\right]\right\}\mathrm{d}z.\]

The objective focal plane is set at \(z=0\), so that the microscope effectively filters the fields through its pupil \(P(\mathbf{k}_{\parallel})\), corresponding to a convolution with its coherent point-spread function \(h(\mathbf{r})\), \[\mathscr{F}_{\parallel}[h(\mathbf{r})]= e^{ik_{\bot}(\mathbf{k}_\parallel)z}P(\mathbf{k}_{\parallel}).\] Finally, the secondary incidence \(\psi_{0,-}\) interferes with all four scattered components to form the measured intensity patterns on the camera, \[I_{\mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{r}_{\parallel}) = \left| [\psi_{0,-}(\mathbf{r}_{\parallel},z_\infty) + \psi_{\mathrm{s}}(\mathbf{r}_{\parallel},z_\infty)]* h(\mathbf{r_{\parallel}, z_\infty})\right|^2.\] In its expansion, \(|\psi_{0,-}* h|^2\) gives a constant background, with a 2D spectrum of \(\tilde{I}_{0, \mathbf{k}_{\parallel, \mathrm{in}}} = |P(\mathbf{k}_{\parallel,\text{in}})R(\mathbf{k}_{\parallel,\text{in}})|^2\delta(\mathbf{k}_{\parallel})\), which can be removed by background subtraction. \(|\psi_{s}* h|^2\) is negligible for weak scatterers, and the cross terms \([\psi_{0,-}* h]^*[\psi_{s}* h] + \mathbf{c}.\mathbf{c}.\) dominate the interference contrast, whose 2D spectrum can be formulated as, \[\begin{align} \mathscr{F}_{\parallel}\big\{[\psi_{0,-}* h]^*[\psi_{s}* h] \big\} &= \frac{ik_0^2R^*(\mathbf{k}_{\parallel,\text{in}})}{2}\int_{-\infty}^0\mathrm{d}z\cdot \frac{\widetilde{\Delta\epsilon}_{\parallel}(\mathbf{k_\parallel},z) }{k_{\bot}(\mathbf{k}_\parallel+\mathbf{k}_{\parallel,\text{in}})} P^*(\mathbf{k}_{\parallel,\text{in}})P(\mathbf{k}_{\parallel}+\mathbf{k}_{\parallel,\text{in}})\\ &\cdot\left[R(\mathbf{k}_{\parallel}+\mathbf{k}_{\parallel,\text{in}})e^{-ik_{\bot}(\mathbf{k}_\parallel+\mathbf{k}_{\parallel,\text{in}})z}+e^{ik_{\bot}(\mathbf{k}_\parallel+\mathbf{k}_{\parallel,\text{in}})z}\right]\left[e^{ik_{z,\text{in}}z}+R(\mathbf{k}_{\parallel,\text{in}})e^{-ik_{z,\text{in}}z}\right].\\ \end{align}\] It yields a linear relation between the 2D spectrum of the sliced permittivity contrast and the 2D measurement spectrum: \[\label{eq:slice-wiseLinear} \tilde{I}_{\mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{k_\parallel}) \approx \tilde{I}_{0, \mathbf{k}_{\parallel, \mathrm{in}}}+\int_{-\infty}^0[H_{\parallel,\text{Re}}(\mathbf{k_\parallel},z)\widetilde{\Delta\epsilon}_{\parallel,\text{Re}}(\mathbf{k_\parallel},z)+H_{\parallel,\text{Im}}(\mathbf{k_\parallel},z)\widetilde{\Delta\epsilon}_{\parallel,\text{Im}}(\mathbf{k_\parallel},z)]\cdot\mathrm{d}z,\tag{15}\] where \(\widetilde{\Delta\epsilon}_{\parallel,\text{Re}/\text{Im}} \equiv \mathscr{F}_\parallel (\Delta\epsilon_{\text{Re}/\text{Im}})\), and the slice-wise transfer function of phase and absorption can be formulated as, \[\begin{align} H_{\parallel,\text{Re}}(\mathbf{k_\parallel},z) &= \frac{ik_0^2}{2} \left[ H_{\parallel, 0}(\mathbf{k_\parallel},z) - H^*_{\parallel, 0}(-\mathbf{k_\parallel},z) \right],\\ H_{\parallel,\text{Im}}(\mathbf{k_\parallel},z) &= -\frac{k_0^2}{2} \left[ H_{\parallel, 0}(\mathbf{k_\parallel},z) + H^*_{\parallel, 0}(-\mathbf{k_\parallel},z) \right], \end{align}\] with, \[\begin{align} H_{\parallel, 0}(\mathbf{k_\parallel},z) &= \frac{R^*(\mathbf{k}_{\parallel,\text{in}})P^*(\mathbf{k}_{\parallel,\text{in}})P(\mathbf{k}_{\parallel}+\mathbf{k}_{\parallel,\text{in}})}{k_{\bot}(\mathbf{k}_\parallel+\mathbf{k}_{\parallel,\text{in}})} \\ &\cdot\left[R(\mathbf{k}_{\parallel}+\mathbf{k}_{\parallel,\text{in}})e^{-ik_{\bot}(\mathbf{k}_\parallel+\mathbf{k}_{\parallel,\text{in}})z}+e^{ik_{\bot}(\mathbf{k}_\parallel+\mathbf{k}_{\parallel,\text{in}})z}\right]\left[e^{ik_{z,\text{in}}z}+R(\mathbf{k}_{\parallel,\text{in}})e^{-ik_{z,\text{in}}z}\right]. \end{align}\]

1.3 3D transfer function for substrate-epi configuration and axial Kramers–Kronig relation↩︎

In substrate-epi configuration, the object is strictly placed on the upper half-space, rendering the slice-wise transfer function undefined in the lower half-space and thereby preventing a direct 3D transfer function representation in the Fourier domain. To address this, we formally extend the range of the slice-wise transfer function to the full range \((-\infty, \infty)\), and introduce a step function \(S(z)\) to the permittivity distribution \(\Delta \epsilon({\mathbf{r}}) \to S(z)\Delta \epsilon({\mathbf{r}})\), with \[S(z) = \begin{cases} 1, & z < 0,\\ 0.5, & z = 0,\\ 0, & z > 0, \end{cases}\] thereby reformulating Eq. 15 as, \[\label{eq:slice-wiseLinearExtend} \tilde{I}_{\mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{k_\parallel}) \approx \tilde{I}_{0, \mathbf{k}_{\parallel, \mathrm{in}}}+\int_{-\infty}^\infty[H_{\parallel,\text{Re}}(\mathbf{k_\parallel},z)\widetilde{\Delta\epsilon}_{\parallel,\text{Re}}(\mathbf{k_\parallel},z)S(z)+H_{\parallel,\text{Im}}(\mathbf{k_\parallel},z)\widetilde{\Delta\epsilon}_{\parallel,\text{Im}}(\mathbf{k_\parallel},z)S(z)]\cdot\mathrm{d}z.\tag{16}\] This extension preserves the physical meaning of Eq. 15 while enabling the definition of a 3D transfer function via the Parseval–Plancherel identity along \(z\), so that scattering from an object above a reflective substrate can be expressed linearly in 3D Fourier space, \[\label{fbwnhrpj} \tilde{I}_{\mathbf{k}_{\parallel, \mathrm{in}}}({\mathbf{k}}) \approx \tilde{I}_{0, \mathbf{k}_{\parallel,\mathrm{in}}} +\int_{-\infty}^\infty[H_{\text{Re}, \mathbf{k}_{\parallel,\mathrm{in}}}(\mathbf{k})\widetilde{\Delta\epsilon}_{\text{Re}}(\mathbf{k})+H_{\text{Im}, \mathbf{k}_{\parallel,\mathrm{in}}}(\mathbf{k})\widetilde{\Delta\epsilon}_{\text{Im}}(\mathbf{k})]\cdot\mathrm{d}k_z,\tag{17}\] where \(\widetilde{\Delta\epsilon}_{\text{Re}/\text{Im}} \equiv \mathscr{F}(\Delta\epsilon_{\text{Re}/\text{Im}})\) with \(\mathscr{F}\) denoting 3D Fourier transform, and \[\begin{align} \tag{18} H_{\mathrm{Re}, \mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{k}) &= \frac{ik_0^2}{2}\left[ H_{0, \mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{k}) - H^*_{0, \mathbf{k}_{\parallel, \mathrm{in}}}(-\mathbf{k}) \right],\\ \tag{19} H_{\mathrm{Im}, \mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{k}) &= -\frac{k_0^2}{2}\left[ H_{0, \mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{k}) + H^*_{0, \mathbf{k}_{\parallel, \mathrm{in}}}(-\mathbf{k}) \right], \end{align}\] with \[\begin{align} H_{0, \mathbf{k}_{\parallel, \mathrm{in}}}(\mathbf{k}) &= \frac{R^*({\mathbf{k}_{\parallel, \mathrm{in}}})P^*({\mathbf{k}_{\parallel, \mathrm{in}}})P(\mathbf{k}_{\parallel} + \mathbf{k}_{\parallel, \mathrm{in}} ) }{k_{\bot}({\mathbf{k}_{\parallel}}+{\mathbf{k}_{\parallel, \mathrm{in}}})}\\ &\cdot\left\{ R(\mathbf{k}_{\parallel}+\mathbf{k}_{\parallel,\text{in}})\delta[k_z - k_{\bot}(\mathbf{k}_\parallel+\mathbf{k}_{\parallel,\text{in}})+k_{z,\text{in}} ] + R(\mathbf{k}_{\parallel,\text{in}})\delta[k_z + k_{\bot}(\mathbf{k}_\parallel+\mathbf{k}_{\parallel,\text{in}})-k_{z,\text{in}} ] \right.\\ &\left. + \delta[k_z + k_{\bot}(\mathbf{k}_\parallel+\mathbf{k}_{\parallel,\text{in}})+k_{z,\text{in}} ] + R(\mathbf{k}_{\parallel,\text{in}})R(\mathbf{k}_{\parallel}+\mathbf{k}_{\parallel,\text{in}})\delta[k_z - k_{\bot}(\mathbf{k}_\parallel+\mathbf{k}_{\parallel,\text{in}})-k_{z,\text{in}} ] \right\}. \end{align}\] Eq. 3 linearly links the 3D spectrum of permittivity contrast with the 2D measurement spectrum. The presence of the step function acts as a constraint in the real space, which in turn enforces an axial Kramers–Kronig relation on the real and imaginary part of \(\widetilde{\Delta\epsilon}\) in the Fourier domain, \[\begin{align} \mathrm{Im}[\widetilde{\Delta \epsilon }({\mathbf{k}})] &= \frac{1}{\pi} \mathrm{p.v.}\int_{-\infty}^\infty \frac{\mathrm{Re}[\widetilde{\Delta \epsilon }({\mathbf{k}}_{\parallel}, k'_z)]}{k_z - k'_z} \mathrm{d}k'_z = \mathscr{H}_z \mathrm{Re}[\widetilde{\Delta \epsilon }({\mathbf{k}})],\\ \mathrm{Re}[\widetilde{\Delta \epsilon }({\mathbf{k}})] &= -\frac{1}{\pi} \mathrm{p.v.}\int_{-\infty}^\infty \frac{\mathrm{Im}[\widetilde{\Delta \epsilon }({\mathbf{k}}_{\parallel}, k'_z)]}{k_z - k'_z} \mathrm{d}k'_z = -\mathscr{H}_z \mathrm{Im}[\widetilde{\Delta \epsilon }({\mathbf{k}})], \end{align}\] where p.v. denotes the Cauchy principal value, and \(\mathscr{H}_z\) is the Hilbert transform along \(k_z\). It can be formulated as a compact form by applying Hilbert transform to the real and imaginary part of \(\widetilde{\Delta \boldsymbol{\epsilon} }({\mathbf{k}})\), respectively, \[i\mathscr{H}_z \widetilde{\Delta \epsilon }({\mathbf{k}}) =i\{ \mathscr{H}_z \mathrm{Re}[\widetilde{\Delta \boldsymbol{\epsilon} }({\mathbf{k}})] + i \mathscr{H}_z \mathrm{Im}[\widetilde{\Delta \epsilon }({\mathbf{k}})] \} = \widetilde{\Delta \epsilon }({\mathbf{k}}),\] which equivalently yields to an identity, \[\label{KKidentity} \widetilde{\Delta \epsilon }({\mathbf{k}}) = \frac{1}{2}(1+i\mathscr{H}_z )\widetilde{\Delta \epsilon }({\mathbf{k}}).\tag{20}\] In the forward process, the object is confined to the upper half-space, so applying the operator, \((1+i\mathscr{H}_z )/2\), doesn’t change the scattering result. In contrast, during inversion, this operator incorporates the axial Kramers–Kronig relation, enforcing the recovered 3D spectrum to satisfy this relation in the Fourier domain and thus remain confined to the upper half-space in real space.

1.4 Discrete form of 3D transfer function↩︎

2 Additional reconstruction information↩︎

2.1 Additional information of the simulation validation of the reconstruction framework↩︎

In the simulation validation, a synthetic object consisting of three beads covered by a thin spherical shell is used. The top bead and upper half-shell are pure absorption, while the bottom beads and lower half-shell are pure phase, as shown in Fig. 6 (a). Because the recovered spectrum is non-Hermitian, symmetric spectral regions must be combined to decouple 3D phase and absorption. The FS band is already in a symmetric region, so its real and imaginary parts correspond directly to phase and absorption from FS. As shown in Fig. 6 (b), phase and absorption features are separated in real and imaginary part of \(\Delta \epsilon_{\mathrm{FS}}\), respectively. Due to the missing-cone problem, beads and shell are elongated along \(z\)-direction, with the top and bottom shell interfaces unresolved. The BS bands, in contrast, are symmetrically located at positive and negative high-\(k_z\) frequencies. To decouple phase and absorption of a non-Hermitian object from BS, both bands must be used. Using both BS bands, the real and imaginary parts encode high-\(k_z\) phase and absorption, respectively, though the resulting \(z\)-profiles exhibit oscillations near \(4k_0\) (left two panels of Fig. 6 (c)). From the oscillatory profiles, the previously missing top-absorption and bottom-phase shell interfaces are recovered and separated in the BS reconstruction, providing information complementary to FS. When only a single BS band is isolated for the inverse Fourier transform, phase and absorption mix in the high-\(k_z\) components, but the absolute value produces a smooth axial envelope [right panel of Fig. 6 (c)].

2.2 Additional information of C. elegans reconstruction↩︎

The recovery of 3D FS and BS information is performed using the proposed band-limited reconstruction algorithm. The in-plane spatial frequency sampling, \(\Delta \mathbf{k}_{\parallel}\), is determined by the field of view, and the axial sampling, \(\Delta k_z\), corresponds to a depth range of \(-30~\mu\)m to \(30~\mu\)m. The threshold \(\alpha\) in truncated singular value decomposition is set as \(\mathrm{max}(7\times 10^{-2}, 5\times 10^{-2}\sigma_\mathrm{max})\). The reconstruction were performed on a Nvidia L40S 48 GB graphics card. Initialization of the inversion operator, \(\mathbf{A}^{+}\), took 300 s, while each reconstruction required 5 s. Notably, the inversion operator \(\mathbf{A}^{+}\) recovers the object’s 3D Fourier spectrum and, once initialized for a given measurement configuration, can be reused for reconstructing different blocks or subsequent measurements.

2.3 Reconstruction of breast cancer cells↩︎

References↩︎