Spatial Dwell-Time Resolution Map

• Spatial dwell-time resolution maps are localized representations that couple spatial coordinates with time-dependent exposure parameters to optimize resolution.
• They are applied in focused electron-beam induced deposition, single molecule localization microscopy, and SRP-PHAT, enabling precise control over nonuniform growth and imaging artifacts.
• The approach overcomes common misconceptions by calibrating local discrepancies in geometry, drift, and thermal effects to maintain shape fidelity and resolution.

Searching arXiv for the named concept and closely related usages across domains. Spatial dwell-time resolution map is not a standardized term with a single invariant meaning. In explicit usage, it denotes a calibrated, geometry-aware exposure blueprint for focused electron-beam induced deposition of iron, assigning locally optimized dwell times to a 3D target so that each voxel, branch, angle, or bridge segment receives just enough electron-beam dwell time to grow without collapse, overgrowth, or loss of shape fidelity (Okasha et al., 22 Jul 2025). Closely related constructs appear as a dwell-time-dependent lateral resolution metric in static single molecule localization microscopy, an aliasing-aware spatial sampling criterion for SRP-PHAT maps, position-dependent spatial and temporal resolution plots in 4D resistive silicon detectors, and local spatial resolution maps in atom probe tomography (Shaw et al., 2022, Garcia-Barrios et al., 2024, Menzio et al., 2024, Gault et al., 2015). Taken together, these usages suggest a broader family of spatially resolved representations in which position is coupled to exposure time, temporal separation, induced delay sampling, or local spatiotemporal reconstruction fidelity.

1. Terminological scope and cross-domain usage

The term is used explicitly in Fe FEBID, but related literatures employ adjacent formulations rather than identical wording. In the FEBID setting, the map is a locally optimized dwell-time landscape for 3D growth. In SMLM, the analogous object is a time-interval-dependent estimate of the effective lateral point-spread function. In SRP-PHAT, the relevant construct is a spatial resolution condition derived from local TDOA gradients and GCC bandwidth. In 4D silicon tracking, the closest equivalent is the combination of a spatial resolution map or position-dependent residual map with hit-time resolution versus hit position. In atom probe tomography, advanced spatial distribution maps support local resolution maps across the specimen surface. The quantum dwell-time literature provides the formal notion of dwell time in a spatial region but does not itself introduce a map under this name (Okasha et al., 22 Jul 2025, Shaw et al., 2022, Garcia-Barrios et al., 2024, Menzio et al., 2024, Gault et al., 2015, 0901.1371).

Domain	Mapped quantity	Local dependence
Fe FEBID	Dwell-time distribution for complex 3D shapes	Beam position, slice height, connectivity, thermal resistance
Static SMLM	$\sigma_{xy}(t)$ or effective PSF versus time interval	Time interval / dwell time
SRP-PHAT	Aliasing-free map resolution or admissible GCC bandwidth	$\|\nabla \tau_{kl}(\vec r)\|$, $\Delta r$, source position
RSD 4D tracking	Residuals and hit-time resolution versus position	Hit position across the active matrix
Atom probe tomography	Local in-depth and lateral resolution	ROI position, crystallography, temperature

A recurrent conceptual feature is locality. The mapped quantity is not a single global scalar; it depends on where the system is probed, exposed, reconstructed, or sampled. A second recurrent feature is nonuniformity. Several of the cited works argue explicitly that resolution requirements or effective performance vary across space, whether because of heating, drift accumulation, pairwise delay gradients, crystallographic faceting, or edge effects.

2. Geometry-aware dwell-time mapping in focused electron-beam induced deposition

In "Dwell-Time Model Simulation Assistance for Advancing Iron 3D Nano-Printing of Via Focused Electron Beam Induced Deposition" (Okasha et al., 22 Jul 2025), the spatial dwell-time resolution map is the central control object for Fe FEBID. The motivation is geometry dependence: Fe(CO)$_5$ has slow dissociation kinetics and a narrow process window, so a universal parameter set does not work for Fe. The map is therefore built as a geometry-specific tuning mechanism that converts a CAD-defined 3D structure into a viable print strategy.

The growth model is introduced as a Gaussian-shaped deposition law,

$$\text{Deposition model} = G_r \, e^{-\frac{r^2}{2\sigma^2}} \, e^{-K R_T},$$

where $G_r$ is the vertical growth rate at the beam center at base deposition temperature, $K$ is a thermal-resistance scaling factor, $\sigma$ is the standard deviation describing lateral spread, and $R_T$ is a geometry-dependent factor capturing thermal resistance and beam-induced heating. The map is constructed in two calibration stages: initial calibration on vertical nanowires, followed by refinement using angled, bridged, and multi-branch 3D structures. The workflow uses quantitatively calibrated Monte Carlo / FEBID growth simulations, with measured nanowire heights, widths, and growth directions supplying the calibration targets.

The optimized experimental conditions are reported as $20\ \text{kV}$, $\|\nabla \tau_{kl}(\vec r)\|$0, Fe(CO)$\|\nabla \tau_{kl}(\vec r)\|$1 preheated to $\|\nabla \tau_{kl}(\vec r)\|$2, and chamber pressure $\|\nabla \tau_{kl}(\vec r)\|$3 mbar. Under these conditions, the growth rate reaches about $\|\nabla \tau_{kl}(\vec r)\|$4–$\|\nabla \tau_{kl}(\vec r)\|$5, and the spatial morphological resolution reaches roughly $\|\nabla \tau_{kl}(\vec r)\|$6. Average nanowire widths in the calibration stage are about $\|\nabla \tau_{kl}(\vec r)\|$7–$\|\nabla \tau_{kl}(\vec r)\|$8. The calibration logic is explicit: $\|\nabla \tau_{kl}(\vec r)\|$9 is extracted empirically from height-versus-time curves, $\Delta r$0 from measured time-dependent width, and $\Delta r$1 from heat dissipation / growth-length relations using a trial-and-error numerical solution. The reported rearranged equation for $\Delta r$2,

$\Delta r$3

is described in the text as the form used for solving the previously unsolved calibration relation; the notation is acknowledged there to be somewhat irregular.

The resulting dwell-time map is strongly nonuniform. Dwell time varies with beam position $\Delta r$4, slice height or layer index, local bridge length or branch connectivity, and local thermal resistance. For 3D fabrication, the target is sliced perpendicular to the growth direction, and each slice receives its own beam coordinates and dwell times. This nonuniformity is physically necessary because FEBID is non-local: deposition depends not only on the primary beam spot but also on secondary electrons, backscattered electrons, forward-scattered electrons, and the local history of previously grown material.

The 3D parameter-space graphs in figures 2(d), 4(c), 5(c), and 6(d) visualize how total dwell time depends on $\Delta r$5, $\Delta r$6, and $\Delta r$7. The authors interpret three paths in this parameter space as increasing $\Delta r$8 to allow lateral pixel growth, decreasing $\Delta r$9 to enable angled structures, and decreasing $_5$0 when heat dissipation is not required. This functions as an inverse-design interface: the target structure determines the required navigation through parameter space rather than the other way around. The practical significance is continuous growth across complex Fe nanoarchitectures, including tetrapods, spirals, rings, flower-like geometries, and $_5$1 bridges, with improved shape fidelity and reduced collapse risk.

3. Time-resolved lateral resolution in static single molecule localization microscopy

In "A lateral resolution metric for static single molecule localization microscopy images from time-resolved pair correlation functions" (Shaw et al., 2022), the relevant construct is a dwell-time-dependent lateral resolution metric for static samples. The method reformulates pair-correlation analysis so that the effective PSF is estimated as a function of time interval between localization events. The motivation is that the experimentally relevant resolution is not just single-localization precision; it is the effective precision with which pairwise distances between labeled objects can be measured once drift, imperfect drift correction, blinking, and possible slow motion are included.

The formalism begins with a space-time point process $_5$2, with pair autocorrelation $_5$3. Under translation invariance in space and time, and often rotational invariance in space, it is written as $_5$4. For static molecules, the observed autocorrelation is decomposed as

$_5$5

where $_5$6 is the mean density of molecules, $_5$7 is the emitter temporal autocorrelation, $_5$8 is the distinct-pair structural term, and $_5$9 is the effective spatial broadening kernel. The key estimator subtracts a long-time reference,

$$\text{Deposition model} = G_r \, e^{-\frac{r^2}{2\sigma^2}} \, e^{-K R_T},$$0

and uses the time-dependent part of the pair autocorrelation to recover the shape of the effective PSF at each time interval. A Gaussian model is then assumed,

$$\text{Deposition model} = G_r \, e^{-\frac{r^2}{2\sigma^2}} \, e^{-K R_T},$$1

so that $$\text{Deposition model} = G_r \, e^{-\frac{r^2}{2\sigma^2}} \, e^{-K R_T},$$2 becomes the practical resolution estimate.

The interpretation of $$\text{Deposition model} = G_r \, e^{-\frac{r^2}{2\sigma^2}} \, e^{-K R_T},$$3 is explicitly temporal. Short time intervals reflect intrinsic localization precision. Increasing $$\text{Deposition model} = G_r \, e^{-\frac{r^2}{2\sigma^2}} \, e^{-K R_T},$$4 with time indicates residual drift, diffusion, or other temporal degradation. A plateau in $$\text{Deposition model} = G_r \, e^{-\frac{r^2}{2\sigma^2}} \, e^{-K R_T},$$5 is identified as a good indicator that drift correction is working. The paper also emphasizes that experimentally obtained images typically have effective PSFs broader than expected from localization precision alone because additional uncertainty is introduced by drift and drift correction algorithms.

The method is positioned as complementary to Fourier Ring Correlation rather than a replacement for it. FRC jointly reflects localization precision and spatial sampling density or structure coverage, whereas the pair-correlation PSF metric reports more directly on distance-measurement precision. This distinction is illustrated by the reported datasets: in drift-corrected simulation the weighted average estimated resolution is $$\text{Deposition model} = G_r \, e^{-\frac{r^2}{2\sigma^2}} \, e^{-K R_T},$$6 nm while FRC is $$\text{Deposition model} = G_r \, e^{-\frac{r^2}{2\sigma^2}} \, e^{-K R_T},$$7 nm; for Alexa647 DNA origami nanorulers the average estimated resolution is $$\text{Deposition model} = G_r \, e^{-\frac{r^2}{2\sigma^2}} \, e^{-K R_T},$$8 nm and FRC is $$\text{Deposition model} = G_r \, e^{-\frac{r^2}{2\sigma^2}} \, e^{-K R_T},$$9 nm; for NUP210 in fixed primary mouse neurons the average resolution estimate is $G_r$0 nm and FRC is $G_r$1 nm; for F-actin in CH27 B cells the average resolution estimate is $G_r$2 nm and FRC is $G_r$3 nm.

The method is subject to explicit assumptions and failure modes. It assumes an effectively static sample, uncorrelated blinking statistics between molecules, and an approximately Gaussian effective PSF in the lateral plane. If the effective PSF broadens strongly at long times, subtracting $G_r$4 can bias the inferred width toward artificially narrow values. The practical warning criterion

$G_r$5

is given as an indicator that long-time subtraction may be distorting the estimate. In this literature, therefore, a spatial dwell-time resolution map is most naturally understood as $G_r$6 plotted against time interval.

4. Spatial-spectral sampling constraints in SRP-PHAT localization

In "Analytical model for the relation between signal bandwidth and spatial resolution in Steered-Response Power Phase Transform (SRP-PHAT) maps" (Garcia-Barrios et al., 2024), the problem is posed as a sampling question. The steered-response power map is

$G_r$7

with

$G_r$8

Each grid point samples PHAT-weighted GCC functions at the TDOAs induced by the candidate source position. The central issue is whether the spatial grid is too coarse for the GCC bandwidth, causing aliasing in the SRP map.

The bridge between spatial resolution and delay sampling follows from a first-order Taylor approximation:

$G_r$9

so that the local delay increment satisfies

$K$0

Requiring aliasing-free GCC sampling yields the sufficient condition

$K$1

and therefore

$K$2

The paper explicitly interprets this as a spatial dwell-time resolution criterion: at each candidate location, the grid must be fine enough that moving to a neighboring point does not advance the induced delay by more than the delay-sampling limit set by GCC bandwidth.

A key point is that the result does not rely on the far-field assumption and does not depend on any specific array topology. The derivation is pairwise and geometric, so arbitrary arrays are handled microphone-pair by microphone-pair. Near the array, especially when the source may lie between microphones, the TDOA gradient can be large; the simplified conservative bound becomes

$K$3

Far from the array, the gradient becomes smaller and depends more weakly on direction, so the spatial sampling requirement relaxes.

The practical consequence is explicitly nonuniform map design. A spatial resolution map can be constructed by evaluating $K$4 at each point and for each microphone pair, then adapting the effective GCC bandwidth pointwise. For a fixed grid, the GCC bandwidth should be reduced where the TDOA changes too quickly across space. The normalized bandwidth-limited GCC is introduced to preserve main-peak height, but the paper cautions that this normalization tends to amplify sidelobes. That can be beneficial near the array in some cases but can also hurt performance in reverberation. The resulting view is that SRP-PHAT should be designed as a matched spatial-spectral sampling problem rather than as independent choices of map resolution and signal bandwidth.

5. Position-dependent 4D resolution in resistive silicon detectors

In "First test beam measurement of the 4D resolution of an RSD 450 microns pitch pixel matrix connected to a FAST2 ASIC" (Menzio et al., 2024), the paper does not present a separate map under the wording "spatial dwell-time resolution map." The closest equivalent is the combination of a spatial resolution map or position-dependent residual map with a hit-time resolution versus hit position plot. Together these show nearly uniform 4D performance across the full active surface of an RSD2 $K$5 pitch, 7-pixel matrix read out by FAST2A.

The device under test is an RSD/AC-LGAD sensor from the second FBK RSD production, with a $K$6 electrode matrix of $K$7 pitch, although only 14 electrodes were wire-bonded, corresponding to 7 active pixels. The electrodes are cross-shaped with narrow arms, allowing charge sharing in both $K$8 and $K$9 while preserving an effectively $\sigma$0 fill factor. The active area covered by the matrix is about $\sigma$1. The test was performed at the DESY T24 beam line with a $\sigma$2 electron beam and an EUDET2 telescope as reference.

The spatial resolution map was derived by reconstructing tracks with the EUDET telescope, selecting good events with a track pointing to one of the 7 pixels, reconstructing the DUT hit position with DPC and ST, comparing the reconstructed DUT position to the telescope track position, and building 2D residual distributions and position-dependent resolution profiles across the pixel surface. The preferred method for the best spatial performance is ST using signal amplitude, $\sigma$3, rather than signal area. The hit-position resolution is decomposed as

$\sigma$4

with

$\sigma$5

The measured resolution versus signal size is fitted with

$\sigma$6

The observed spatial behavior is central. The reconstructed position correlates very well with the telescope position and does so seamlessly across pixel boundaries. Residuals do not show dead zones at pixel borders. Points with larger residuals cluster near pixel edges, where the resolution worsens somewhat and where the non-Gaussian tails originate, but this is interpreted as edge degradation rather than inefficiency. At $\sigma$7 bias, using $\sigma$8, the reported DUT+track residual widths are $\sigma$9 and $R_T$0. With telescope resolution $R_T$1, the intrinsic RSD spatial resolution reaches about $R_T$2, roughly $R_T$3 of the $R_T$4 pitch; the conclusion also quotes about $R_T$5 of the pitch. Across bias voltages, resolution is below $R_T$6 even at the lowest gain, and for $R_T$7 it reaches a plateau of about $R_T$8 once the gain is above roughly $R_T$9. The fitted constant term dominating the asymptotic resolution is $20\ \text{kV}$0.

Timing is reconstructed from the four electrode times corrected for position-dependent delay,

$20\ \text{kV}$1

with intrinsic timing resolution written as

$20\ \text{kV}$2

In the simplified jitter-dominated limit,

$20\ \text{kV}$3

The best measured temporal resolution is $20\ \text{kV}$4 ps. The paper states that this is about $20\ \text{kV}$5 ps worse than the intrinsic RSD time resolution and is dominated by the FAST2 ASIC resolution. The temporal resolution versus hit position is uniform over the pixel surface at the highest gain point, indicating that the delay correction template works properly. In this context, the position-dependent 4D map documents a detector with about $20\ \text{kV}$6 spatial resolution, about $20\ \text{kV}$7 ps temporal resolution, $20\ \text{kV}$8 fill factor, and homogeneous response across the matrix.

6. Local spatial resolution maps in atom probe tomography

In "Spatial resolution in atom probe tomography" (Gault et al., 2015), the central problem is definitional. The presence of atomic planes in a reconstruction had often been treated as sufficient proof of good spatial resolution, but the paper argues that this is insufficient and introduces advanced spatial distribution maps as a local, specimen-specific quantification framework. The resolution map in this literature is explicitly local rather than global.

Advanced SDMs interrogate the local neighborhood of atoms in the reconstructed volume. The paper distinguishes z-SDMs, which are 1D distributions along the analysis direction and are used to measure spacing and sharpness of atomic planes in depth, from xy-SDMs, which are 2D histograms perpendicular to the analysis direction and are used to assess lateral arrangement and blur within a plane. Because the ROI can be moved around a pole or crystallographic direction, the same procedure yields a map of local resolution across the specimen surface.

The in-depth resolution is defined from the width of the central peak in a z-SDM, fitted by a Gaussian:

$20\ \text{kV}$9

For the lateral resolution, a radial distribution analysis is performed on the xy-SDM and the first-neighbor peak is fitted with a Gaussian; the width of that peak is interpreted as the lateral resolution. The interpretation assumes a Gaussian pulse-spread function for the microscope or reconstruction system. The paper also emphasizes that APT resolution is anisotropic: depth resolution is better than lateral resolution. For pure metals, depth resolution can be below $\|\nabla \tau_{kl}(\vec r)\|$00 nm, while lateral resolution is typically below $\|\nabla \tau_{kl}(\vec r)\|$01 nm.

The map construction is procedural. One chooses a pole or crystallographic feature, computes an SDM for a small ROI centered on that pole, fits the relevant peak to obtain a local resolution value, moves the ROI away from the pole, and repeats. The resulting distribution shows that a given crystallographic plane family is well resolved only within the corresponding facet around the pole and deteriorates rapidly as the ROI moves away from the center. The paper therefore treats local resolution as a function of specimen surface orientation and reconstruction state rather than as a universal microscope constant.

Crystallography and temperature are major control variables. For in-depth resolution, the peak width varies linearly with interplanar spacing and is modeled as

$\|\nabla \tau_{kl}(\vec r)\|$02

For lateral resolution, the reported fit is

$\|\nabla \tau_{kl}(\vec r)\|$03

capturing the dependence on crystallographic packing. The paper also gives a temperature-dependent lateral broadening model of the form

$\|\nabla \tau_{kl}(\vec r)\|$04

and concludes that lateral resolution degrades at higher temperature. The broader conclusion is that resolution is specimen-specific: it depends on tip radius, crystallographic orientation, local facet geometry, temperature, pulse fraction and field conditions, and possibly surface shape evolution during analysis. The common misconception addressed directly here is that seeing atomic planes suffices to claim a known or universal resolution; the paper rejects that interpretation.

7. Quantum dwell time as a formal background for spatial mapping

In "Dwell-time distributions in quantum mechanics" (0901.1371), dwell time is defined for a spatial region

$\|\nabla \tau_{kl}(\vec r)\|$05

through the self-adjoint operator

$\|\nabla \tau_{kl}(\vec r)\|$06

where $\|\nabla \tau_{kl}(\vec r)\|$07 is the projector onto the region. The paper emphasizes that dwell time is an interval observable, not an arrival-time observable. Its distribution for a pure state $\|\nabla \tau_{kl}(\vec r)\|$08 is

$\|\nabla \tau_{kl}(\vec r)\|$09

The mean dwell time is literally the time-integrated probability of being in the region:

$\|\nabla \tau_{kl}(\vec r)\|$10

The formal structure becomes distinctly nonclassical at the level of higher moments. The second moment contains an interference term with no classical counterpart,

$\|\nabla \tau_{kl}(\vec r)\|$11

and the paper argues that the second moment is characteristically quantum. For a free particle in $\|\nabla \tau_{kl}(\vec r)\|$12, the dwell-time operator has two eigenvalues for each $\|\nabla \tau_{kl}(\vec r)\|$13,

$\|\nabla \tau_{kl}(\vec r)\|$14

so a highly monochromatic packet yields a bimodal dwell-time distribution. This differs sharply from the classical free dwell time $\|\nabla \tau_{kl}(\vec r)\|$15.

The paper also derives a direct relation between dwell-time moments and flux-flux correlations at the region boundaries. For the stationary Pollak–Miller flux-flux correlation function, the first moment yields the dwell time and the second moment yields the second dwell-time moment. This is important because it ties a spatially defined temporal quantity to measurable boundary currents rather than to an internal clock variable. Operationally, the mean dwell time can also be accessed indirectly by weak absorption:

$\|\nabla \tau_{kl}(\vec r)\|$16

The paper notes, however, that this procedure reliably measures only the average dwell time in the weak-absorption regime, while direct extraction of the full distribution and higher moments remains difficult.

A plausible implication is that a genuinely quantum spatial dwell-time resolution map would need to encode more than a mean residence time. The paper itself suggests that such a map should represent mean dwell time, boundary-driven moment structure, the interference correction to $\|\nabla \tau_{kl}(\vec r)\|$17, region-size and phase dependence such as $\|\nabla \tau_{kl}(\vec r)\|$18 oscillations and low-$\|\nabla \tau_{kl}(\vec r)\|$19 anomalies, and the operational distinction between mean-time measurements and higher-moment measurements. In that sense, the quantum formalism supplies the most explicit meaning of "dwell time," even though the map concept is more fully operationalized in the imaging, acoustics, detector, and nanofabrication literatures.

8. Common themes, distinctions, and recurring misconceptions

Across these literatures, the mapped object is always local and rarely reducible to a single number. In Fe FEBID, locality is geometric and thermal: exposure must adapt to branch length, connectivity, and heating. In static SMLM, locality is temporal: the effective lateral PSF broadens with time interval and can plateau when drift correction becomes effective. In SRP-PHAT, locality is geometric and sampling-theoretic: required grid spacing depends on the local TDOA gradient. In RSD 4D tracking, locality is positional within the sensor plane: residuals worsen near edges without producing dead zones. In APT, locality is crystallographic and specimen-specific: resolution varies with ROI position, surface orientation, and temperature.

Several misconceptions recur. One is that a visible structure is equivalent to a known resolution. APT explicitly rejects the inference from visible atomic planes to a universal resolution. Another is that nominal point precision or nominal bandwidth alone determines usable resolution. SMLM shows that effective PSF can exceed localization precision once drift and correction artifacts are included, while SRP-PHAT shows that bandwidth and map resolution are coupled rather than independent. A third is that homogeneous activity implies homogeneous performance only by inspection. The RSD measurements distinguish between seamless activity across pixel boundaries and modest edge degradation that broadens residual tails without creating inefficiency. In Fe FEBID, the corresponding misconception is that a universal process recipe can be transferred across geometries; the reported result is the opposite.

Taken together, these works support a precise but nonuniversal understanding of a spatial dwell-time resolution map. It is not a single field-independent formalism. Rather, it is a family of local representations that couple spatial coordinates to a time-like control or response variable—electron-beam dwell time, time interval between localizations, induced propagation delay sampling, hit-time correction, or residence time in a region—and then use that coupling to quantify or optimize resolution. This suggests that the concept is best treated as methodological rather than lexical: a map of where spatiotemporal fidelity is preserved, limited, or must be actively controlled.