Self-Scattering Technique in Physics

• Self-scattering technique is a method where scattering produced within the same physical system is exploited as a defining observable to probe internal dynamics and interactions.
• In astrophysical contexts, it quantifies gravitational encounters among compact dark-matter objects, yielding effective transport cross sections with strong velocity-dependent (v⁻⁴) scaling to explain core formation in dwarf galaxies.
• In radiative transfer and transport simulations, self-scattering underpins polarization diagnostics, null-collision Monte Carlo sampling, and advanced imaging techniques such as scatter ptychography.

“Self-scattering technique” is a context-dependent term used in several technically distinct research programs rather than a single universal method. In the cited literature, it denotes procedures in which scattering generated by the same physical system—or by an intermediate structure illuminated by that system—is exploited to produce an effective interaction, recover hidden dynamical statistics, test a non-scattering hypothesis, compute back-reaction on a scattering trajectory, or synthesize resolution beyond direct viewing. The phrase has been used for gravitational encounters among compact dark-matter objects, scattering of thermal dust emission by other grains, synchrotron self-Compton emission in self-absorbed sources, null-collision Monte Carlo transport, analytic Helmholtz scattering tests, self-force corrections in hyperbolic black-hole scattering, and remote imaging through an observed scatter patch (Loeb, 2022, Kirchschlager et al., 2020, Gao et al., 2012, McDonough et al., 21 Aug 2025, Hovsepyan et al., 18 Jul 2025, Whittall et al., 2023, Long et al., 2024, Gralla et al., 2021, Huang et al., 2022).

1. Terminological scope and common structure

In the cited works, the term always appears with a specific disciplinary meaning. The following usages are explicit in the literature.

Context	Operation called self-scattering	Principal outcome
Compact dark matter	gravitational two-body scattering among compact objects	effective SIDM-like $\sigma/m$
Dust polarimetry	scattering of thermal dust emission by other grains	linear and circular polarization signatures
Self-absorbed synchrotron sources	Compton scattering of synchrotron photons by the same electrons	SSC spectral breaks and two-component spectra
Semiclassical transport	fictitious self-scattering or null-collision channel	exact free-flight statistics for MC BTE solvers
Analytic Helmholtz scattering	complex-wave “self-test” of non-scattering	contradiction arguments for analytic domains
Hyperbolic relativistic scattering	self-force acting on a scattering worldline	correction to trajectory and deflection angle
Scatter ptychography	secondary scattering from an observed intermediate patch	resolution $\delta \approx \lambda R_s/A_s$

This distribution of meanings suggests a family resemblance rather than a single formalism. In each case, a primary field, particle population, or trajectory is not studied only through direct propagation; instead, a secondary scattering process generated within the same physical configuration is made central. The technical role of that secondary process differs sharply by domain. In dark-matter phenomenology it produces an effective transport cross section; in transport Monte Carlo it is a fictitious channel introduced for exact sampling; in analytic PDE scattering it is a contradiction device built from specially chosen interior solutions; in self-force theory it denotes the system’s own field acting back on a scattering orbit; and in imaging it creates an effective aperture located closer to the target (Loeb, 2022, McDonough et al., 21 Aug 2025, Hovsepyan et al., 18 Jul 2025, Whittall et al., 2023, Huang et al., 2022).

A recurrent source of confusion is therefore terminological. “Self-scattering” in dust polarimetry, “self-scattering” in semiclassical Monte Carlo, and “self-scattering” in black-hole self-force calculations do not share a common algorithm. They share only the more abstract pattern that an internally generated or internally sampled scattering process is elevated from a nuisance to the defining observable or computational device.

2. Gravitational self-interaction in dark-matter halos

In one astrophysical usage, the self-scattering technique refers to purely gravitational, two-body encounters among compact dark-matter objects of mass

$M = 10^4 M_4 M_\odot$

moving with velocity dispersion

$v = 10 v_1 \,\mathrm{km\,s^{-1}}.$

For such objects in dwarf-galaxy cores, the transport cross section per unit mass is

$$\frac{\sigma}{m} \approx 10\,\frac{M_4}{v_1^4}\;\mathrm{cm^2\,g^{-1}},$$

which is explicitly presented as the normalization and velocity dependence required in self-interacting dark matter models that alleviate small-scale CDM tensions (Loeb, 2022).

The derivation is the Rutherford analogue for a $1/r$ potential. For two masses $m_1$ and $m_2$ with relative speed $v$ and impact parameter $b$, the hyperbolic scattering angle satisfies

$\delta \approx \lambda R_s/A_s$0

with $\delta \approx \lambda R_s/A_s$1. In the equal-mass case, the $\delta \approx \lambda R_s/A_s$2-degree impact parameter is

$\delta \approx \lambda R_s/A_s$3

and the momentum-transfer cross section is

$\delta \approx \lambda R_s/A_s$4

Using dwarf-core values $\delta \approx \lambda R_s/A_s$5, $\delta \approx \lambda R_s/A_s$6, and $\delta \approx \lambda R_s/A_s$7, the cutoffs are $\delta \approx \lambda R_s/A_s$8–$\delta \approx \lambda R_s/A_s$9 and $M = 10^4 M_4 M_\odot$0, giving $M = 10^4 M_4 M_\odot$1 and hence $M = 10^4 M_4 M_\odot$2–$M = 10^4 M_4 M_\odot$3 (Loeb, 2022).

The central phenomenological feature is the $M = 10^4 M_4 M_\odot$4 scaling. For dwarf galaxies with $M = 10^4 M_4 M_\odot$5, the interaction is strong. For Milky Way velocities $M = 10^4 M_4 M_\odot$6,

$M = 10^4 M_4 M_\odot$7

and for clusters with $M = 10^4 M_4 M_\odot$8,

$M = 10^4 M_4 M_\odot$9

so the effect becomes negligible in high-velocity systems. This is the mechanism proposed to reconcile SIDM-like behavior in dwarfs with the absence of comparable signatures in massive galaxies and clusters (Loeb, 2022).

The same framework gives interaction rates

$v = 10 v_1 \,\mathrm{km\,s^{-1}}.$0

For $v = 10 v_1 \,\mathrm{km\,s^{-1}}.$1, $v = 10 v_1 \,\mathrm{km\,s^{-1}}.$2, and $v = 10 v_1 \,\mathrm{km\,s^{-1}}.$3, the characteristic time is $v = 10 v_1 \,\mathrm{km\,s^{-1}}.$4; for $v = 10 v_1 \,\mathrm{km\,s^{-1}}.$5, $v = 10 v_1 \,\mathrm{km\,s^{-1}}.$6; and for $v = 10 v_1 \,\mathrm{km\,s^{-1}}.$7 at $v = 10 v_1 \,\mathrm{km\,s^{-1}}.$8, $v = 10 v_1 \,\mathrm{km\,s^{-1}}.$9. The paper interprets these values as short enough for collisional relaxation and core formation in dwarfs (Loeb, 2022).

The point-mass approximation requires the physical size $$\frac{\sigma}{m} \approx 10\,\frac{M_4}{v_1^4}\;\mathrm{cm^2\,g^{-1}},$$0 of each object to satisfy $$\frac{\sigma}{m} \approx 10\,\frac{M_4}{v_1^4}\;\mathrm{cm^2\,g^{-1}},$$1, yielding

$$\frac{\sigma}{m} \approx 10\,\frac{M_4}{v_1^4}\;\mathrm{cm^2\,g^{-1}},$$2

for the reference values. The corresponding minimum internal density is

$$\frac{\sigma}{m} \approx 10\,\frac{M_4}{v_1^4}\;\mathrm{cm^2\,g^{-1}},$$3

This is used to argue that the objects must be primordial rather than late-time hierarchical products. The paper discusses primordial black holes and primordial compact dark clumps as possibilities, while emphasizing that compatibility with microlensing, accretion, and dynamical-heating constraints depends on internal structure and formation history (Loeb, 2022).

3. Radiative self-scattering in dusty and synchrotron sources

A second major usage appears in radiative transfer. In dusty, optically thick environments, dust self-scattering is the scattering of thermal dust emission by other dust grains within the same region. The local radiation field is anisotropic because of vertical and radial temperature and surface-brightness gradients, and single scattering converts that anisotropy into linear polarization. In the spherical-grain limit, the polarization phase function is

$$\frac{\sigma}{m} \approx 10\,\frac{M_4}{v_1^4}\;\mathrm{cm^2\,g^{-1}},$$4

with the strongest polarization typically near $$\frac{\sigma}{m} \approx 10\,\frac{M_4}{v_1^4}\;\mathrm{cm^2\,g^{-1}},$$5. Kirchschlager and Bertrang analyze how this picture changes for non-spherical grains using the Discrete Dipole Approximation, computing the full $$\frac{\sigma}{m} \approx 10\,\frac{M_4}{v_1^4}\;\mathrm{cm^2\,g^{-1}},$$6 Mueller matrix $$\frac{\sigma}{m} \approx 10\,\frac{M_4}{v_1^4}\;\mathrm{cm^2\,g^{-1}},$$7 for oblate grains of astronomical silicate at $$\frac{\sigma}{m} \approx 10\,\frac{M_4}{v_1^4}\;\mathrm{cm^2\,g^{-1}},$$8 with $$\frac{\sigma}{m} \approx 10\,\frac{M_4}{v_1^4}\;\mathrm{cm^2\,g^{-1}},$$9 dipoles and validity $1/r$0 (Kirchschlager et al., 2020).

The size parameter is

$1/r$1

At $1/r$2, $1/r$3 gives $1/r$4, near the Rayleigh regime, whereas $1/r$5 gives $1/r$6, in the non-Rayleigh regime where spherical and non-spherical predictions diverge strongly. For unpolarized incident light, oblate grains with $1/r$7 show higher $1/r$8 than spheres, especially at forward and backward angles, while for $1/r$9, $m_1$0 can become negative at small and large angles, corresponding to $m_1$1 polarization flips. At $m_1$2 the differences are described as “tremendous”: $m_1$3 increases by approximately a factor $m_1$4 relative to $m_1$5, $m_1$6 by approximately a factor $m_1$7, and the peak polarization is reduced to $m_1$8 with strong dependence on axis ratio and angle (Kirchschlager et al., 2020).

The paper’s central interpretive consequence is that the canonical spherical-grain self-scattering framework tends to infer grain sizes near $m_1$9, or $m_2$0–$m_2$1 at $m_2$2, because larger spheres rapidly lose polarization efficiency once $m_2$3 exceeds unity. Non-spherical, nearly perfectly aligned grains maintain substantial $m_2$4 at $m_2$5 and can reproduce the same observed polarization with larger grains, up to $m_2$6. The paper therefore argues for a necessary re-evaluation of grain sizes derived from (sub-)mm polarization, and it identifies polarization reversals and significant circular polarization as diagnostics of non-spherical scattering and alignment. For linearly polarized incident waves with $m_2$7, single scattering by oblate grains gives $m_2$8 for $m_2$9 and $v$0 for $v$1, whereas circular polarization is negligible for symmetric configurations with unpolarized incident light (Kirchschlager et al., 2020).

A related but distinct radiative usage is synchrotron self-Compton scattering in self-absorbed sources. Here synchrotron photons emitted by a non-thermal electron population are Compton upscattered by the same electrons. The characteristic synchrotron breaks are the self-absorption frequency $v$2, the injection frequency $v$3, and the cooling frequency $v$4. In the Thomson regime, the SSC features occur near

$v$5

In the weak self-absorption regime, $v$6, the electron distribution is not modified by absorption, and the SSC spectrum broadly resembles the synchrotron spectrum but with a low-frequency linear rise,

$v$7

and logarithmic hardening in the high-frequency segments. In the strong absorption regime, $v$8, synchrotron absorption heats low-energy electrons, producing a quasi-thermal pile-up and two-component synchrotron and SSC spectra (Gao et al., 2012).

For the case $v$9, the paper introduces a thermal-like component $b$0 below $b$1 and shows that the SSC spectrum develops two humps, a thermal peak near $b$2 and a non-thermal peak near $b$3. The dominance criterion is

$b$4

with thermal dominance for $b$5 and non-thermal dominance for $b$6. The explicit pile-up condition is formulated by balancing synchrotron-absorption heating against synchrotron plus SSC cooling at $b$7, with the additional requirement $b$8 (Gao et al., 2012).

Taken together, these two radiative literatures use “self-scattering” in a literal sense: the photons being scattered are produced by the same dusty or synchrotron-emitting system whose properties are then inferred from the scattered signal. The controversy is not whether scattering occurs, but which microphysical model maps the scattered observables back to grain size, alignment, absorption strength, or electron distribution.

4. Null-collision self-scattering in semiclassical transport

In charge-transport simulation, the self-scattering technique—explicitly also called the null-collision method—is a way to generate statistically exact free-flight times and scattering events for carriers whose microscopic scattering rates depend on energy and momentum while keeping Monte Carlo sampling simple. The starting point is a semi-classical Boltzmann transport equation with mechanism-resolved rates

$b$9

so that the true free-flight distribution is the nonhomogeneous Poisson-process density

$\delta \approx \lambda R_s/A_s$00

The self-scattering construction embeds this time-dependent process inside a homogeneous Poisson sampler of constant rate $\delta \approx \lambda R_s/A_s$01 satisfying $\delta \approx \lambda R_s/A_s$02 (McDonough et al., 21 Aug 2025).

A fictitious self-scattering channel is added with rate

$\delta \approx \lambda R_s/A_s$03

so the augmented rate is constant. One samples a candidate event time from

$\delta \approx \lambda R_s/A_s$04

accepts a physical scattering with probability $\delta \approx \lambda R_s/A_s$05, and otherwise performs a null event that leaves the state unchanged. If a physical event occurs, mechanism $\delta \approx \lambda R_s/A_s$06 is selected with conditional probability

$\delta \approx \lambda R_s/A_s$07

The paper shows that this thinning procedure reproduces the exact NHPP statistics implied by the full energy- and momentum-dependent rates and correctly recovers both the total relaxation time and the relative fractions of each mechanism (McDonough et al., 21 Aug 2025).

The method is exact because the homogeneous Poisson process of rate $\delta \approx \lambda R_s/A_s$08 becomes, after time-dependent thinning, an NHPP of intensity $\delta \approx \lambda R_s/A_s$09. The paper provides an explicit series proof in terms of Erlang or Erlang-like mixtures and derives

$\delta \approx \lambda R_s/A_s$10

with

$\delta \approx \lambda R_s/A_s$11

This addresses a common misunderstanding about the method: although the simulation algorithm begins by sampling from a constant rate, the recovered free-flight statistics are not those of a constant-relaxation-time model; they are those of the full, time-dependent total rate (McDonough et al., 21 Aug 2025).

The implementation details are central. The paper discusses global bounds, local or adaptive bounds, and piecewise-constant bounds in $\delta \approx \lambda R_s/A_s$12-space. It emphasizes that $\delta \approx \lambda R_s/A_s$13 must always be enforced, so any detected violation $\delta \approx \lambda R_s/A_s$14 requires increasing $\delta \approx \lambda R_s/A_s$15, resampling, or substepping. Failure modes include highly peaked rates in time or $\delta \approx \lambda R_s/A_s$16-space, discontinuities or large gradients, nonphysical negative rates from interpolation or tabulation, and overly large $\delta \approx \lambda R_s/A_s$17, which causes excessive null events (McDonough et al., 21 Aug 2025).

Validation is performed in $\delta \approx \lambda R_s/A_s$18-doped Si for both energy-independent and energy-dependent rates. Within narrow energy bins where rates are approximately constant, free-flight histograms fit

$\delta \approx \lambda R_s/A_s$19

One cited example is $\delta \approx \lambda R_s/A_s$20 phonon absorption events in the $\delta \approx \lambda R_s/A_s$21–$\delta \approx \lambda R_s/A_s$22 bin, fitted with $\delta \approx \lambda R_s/A_s$23. For constant rates, the recovered total analytic relaxation time is $\delta \approx \lambda R_s/A_s$24, with fitted values within $\delta \approx \lambda R_s/A_s$25 for energies below $\delta \approx \lambda R_s/A_s$26. For energy-dependent rates, the fitted $\delta \approx \lambda R_s/A_s$27 follows the analytic total $\delta \approx \lambda R_s/A_s$28 closely, and the mechanism fractions track $\delta \approx \lambda R_s/A_s$29 over energy in a $\delta \approx \lambda R_s/A_s$30-flight test at $\delta \approx \lambda R_s/A_s$31, $\delta \approx \lambda R_s/A_s$32, and $\delta \approx \lambda R_s/A_s$33 (McDonough et al., 21 Aug 2025).

5. Self-tests and self-force in wave and orbital scattering

In analytic scattering theory for the two-dimensional Helmholtz equation, the self-scattering technique is a contradiction method for proving that a given incident wave must scatter from a sufficiently regular inhomogeneity. The setting is a bounded domain $\delta \approx \lambda R_s/A_s$34 with constant index $\delta \approx \lambda R_s/A_s$35 and an incident field $\delta \approx \lambda R_s/A_s$36 satisfying

$\delta \approx \lambda R_s/A_s$37

Assuming, contrary to the intended conclusion, that the incident wave is non-scattering, the paper derives the boundary identity

$\delta \approx \lambda R_s/A_s$38

for any $\delta \approx \lambda R_s/A_s$39 solving $\delta \approx \lambda R_s/A_s$40 in $\delta \approx \lambda R_s/A_s$41. It then chooses “self-test” functions $\delta \approx \lambda R_s/A_s$42 with $\delta \approx \lambda R_s/A_s$43 on the complex characteristic set

$\delta \approx \lambda R_s/A_s$44

reducing the non-scattering hypothesis to the vanishing of an oscillatory-exponential boundary integral $\delta \approx \lambda R_s/A_s$45 for all large $\delta \approx \lambda R_s/A_s$46 (Hovsepyan et al., 18 Jul 2025).

The decisive step is steepest descent in the complexified boundary parameter $\delta \approx \lambda R_s/A_s$47 for analytic or piecewise analytic boundaries. With $\delta \approx \lambda R_s/A_s$48 and an admissible contour through a simple saddle point $\delta \approx \lambda R_s/A_s$49, the first main theorem yields

$\delta \approx \lambda R_s/A_s$50

with

$\delta \approx \lambda R_s/A_s$51

If $\delta \approx \lambda R_s/A_s$52, the assumption $\delta \approx \lambda R_s/A_s$53 is contradicted, so scattering is forced. When $\delta \approx \lambda R_s/A_s$54, a second-order expansion gives

$\delta \approx \lambda R_s/A_s$55

and $\delta \approx \lambda R_s/A_s$56 again implies scattering. The method is applied to true ellipses, nonconvex analytic domains, cardioids, deltoids, and corners; the disk is exceptional and admits nonscattering Herglotz waves at transmission eigenvalues (Hovsepyan et al., 18 Jul 2025).

In relativistic scattering theory, “self-scattering” refers instead to back-reaction from a field sourced by the scattered particle itself. For a scalar charge $\delta \approx \lambda R_s/A_s$57 on a hyperbolic geodesic in Schwarzschild spacetime, the total angle is written

$\delta \approx \lambda R_s/A_s$58

The frequency-domain formulation for this problem must handle a continuous $\delta \approx \lambda R_s/A_s$59 spectrum, slowly convergent radial integrals, and the failure of external extended homogeneous solutions because the source is not compactly supported outside the particle radius on unbound motion. The paper therefore uses internal EHS only, one-sided mode-sum regularization, Green-function reconstruction, integration by parts, and analytic tail corrections. The practical line-integral formulas for the correction are

$\delta \approx \lambda R_s/A_s$60

$\delta \approx \lambda R_s/A_s$61

The implementation is validated internally and against time-domain calculations, with percent-level agreement for a benchmark at $\delta \approx \lambda R_s/A_s$62 once large-$\delta \approx \lambda R_s/A_s$63 truncation is controlled (Whittall et al., 2023).

A further development isolates the near-separatrix singular structure. For test-particle scattering off Schwarzschild, there is a critical impact parameter $\delta \approx \lambda R_s/A_s$64 such that scattering occurs for $\delta \approx \lambda R_s/A_s$65 and plunge for $\delta \approx \lambda R_s/A_s$66. The geodesic deflection diverges as

$\delta \approx \lambda R_s/A_s$67

while the first-order scalar self-force correction diverges more strongly,

$\delta \approx \lambda R_s/A_s$68

The coefficient $\delta \approx \lambda R_s/A_s$69 is extracted numerically and fitted over $\delta \approx \lambda R_s/A_s$70 by

$\delta \approx \lambda R_s/A_s$71

This input is then used to resum the post-Minkowskian series through a function $\delta \approx \lambda R_s/A_s$72 that preserves the known 4PM content at large $\delta \approx \lambda R_s/A_s$73 but enforces the correct logarithmic and pole singularities near the separatrix. The resulting resummed angle agrees with numerical self-force data to roughly the percent level even in the strong-field regime (Long et al., 2024).

In a complementary weak-field PM treatment near a Newtonian star, the self-force is computed analytically for scalar, electromagnetic, and gravitational cases on a straight-line trajectory. In gravity, a matter-mediated force generated by the star’s response appears at the same order as the gravitational self-force and is essential for gauge-invariant observables. The combined result yields the full 2PM scattering angle

$\delta \approx \lambda R_s/A_s$74

reproducing Westpfahl’s momentum transfer and Damour’s radiative angular-momentum flux (Gralla et al., 2021).

6. Secondary-aperture imaging and scatter ptychography

In coherent imaging, the relevant technique is scatter ptychography. A target is illuminated coherently, the reflected or transmitted field propagates to an intermediate scatterer, and the camera images the intensity on a patch of that scatterer rather than the target itself. Phase retrieval on multiple such intensity measurements reconstructs the complex field on the scatter plane and then the target. The central resolution law is

$\delta \approx \lambda R_s/A_s$75

where $\delta \approx \lambda R_s/A_s$76 is the target–scatterer distance and $\delta \approx \lambda R_s/A_s$77 is the diameter of the observed scatter patch. Direct viewing scales as

$\delta \approx \lambda R_s/A_s$78

or, when the camera is pixel-limited,

$\delta \approx \lambda R_s/A_s$79

The corresponding improvement factors are

$\delta \approx \lambda R_s/A_s$80

The physical interpretation is that the scatterer acts as a secondary effective aperture located closer to the target (Huang et al., 2022).

The forward model propagates an object field $\delta \approx \lambda R_s/A_s$81 to the scatter plane by angular spectrum or Fresnel propagation. In exact angular-spectrum form,

$\delta \approx \lambda R_s/A_s$82

The measured data are intensities

$\delta \approx \lambda R_s/A_s$83

with $\delta \approx \lambda R_s/A_s$84 indexing distinct object–scatterer configurations. In the reported experiment, two axial displacements of the target, $\delta \approx \lambda R_s/A_s$85 and $\delta \approx \lambda R_s/A_s$86, were sufficient; one plane was insufficient (Huang et al., 2022).

Reconstruction is performed by a multi-plane error-reduction algorithm of Gerchberg–Saxton type, together with a multistage angular spectrum method for forward and backward propagation. For each plane,

$\delta \approx \lambda R_s/A_s$87

followed by back-propagation and support enforcement. In the experiment, the reconstruction used the central $\delta \approx \lambda R_s/A_s$88 region, 200–500 iterations, and, for amplitude masks, a real and nonnegative object constraint (Huang et al., 2022).

The reported geometry used $\delta \approx \lambda R_s/A_s$89, $\delta \approx \lambda R_s/A_s$90, $\delta \approx \lambda R_s/A_s$91, $\delta \approx \lambda R_s/A_s$92, and an observed scatter patch $\delta \approx \lambda R_s/A_s$93–$\delta \approx \lambda R_s/A_s$94. The camera had $\delta \approx \lambda R_s/A_s$95, f/1.6, aperture diameter $\delta \approx \lambda R_s/A_s$96, detector size $\delta \approx \lambda R_s/A_s$97, and pixel pitch $\delta \approx \lambda R_s/A_s$98. The estimated scatter-ptychography resolution was

$\delta \approx \lambda R_s/A_s$99

the reconstruction resolved approximately $M = 10^4 M_4 M_\odot$00 features, and the direct-view pixel-limited resolution was approximately $M = 10^4 M_4 M_\odot$01–$M = 10^4 M_4 M_\odot$02 per pixel, corresponding to an experimental improvement of about $M = 10^4 M_4 M_\odot$03 (Huang et al., 2022).

This imaging usage makes the term “self-scattering” geometrically rather than dynamically literal. The target is not observed through a direct aperture alone; it is observed through the field it has itself launched onto a secondary scattering surface. The method is advantageous when coherent illumination is available, the observed patch $M = 10^4 M_4 M_\odot$04 is large, $M = 10^4 M_4 M_\odot$05 is small relative to $M = 10^4 M_4 M_\odot$06, photon flux is sufficient, and at least two diverse measurements can be acquired. Under those conditions, self-scattering is converted from a visibility loss mechanism into the device that sets the effective numerical aperture (Huang et al., 2022).