††thanks: Contributed equally to this work††thanks: Contributed equally to
this work††thanks: Contributed equally to this work

# Multi-angle reconstruction of domain morphology with all-optical diamond
magnetometry

Lucio Stefan<EMAIL_ADDRESS>Cavendish Laboratory, University of
Cambridge, J. J. Thomson Avenue, Cambridge, CB3 0HE, UK  The Faraday
Institution, Quad One, Becquerel Avenue, Harwell Campus, Didcot, OX11 0RA, UK
Anthony K. C. Tan  Cavendish Laboratory, University of Cambridge, J. J.
Thomson Avenue, Cambridge, CB3 0HE, UK  Baptiste Vindolet  Université Paris-
Saclay, CNRS, ENS Paris-Saclay, CentraleSupélec, LuMIn, 91190, Gif-sur-Yvette,
France  Michael Högen  Cavendish Laboratory, University of Cambridge, J. J.
Thomson Avenue, Cambridge, CB3 0HE, UK  Dickson Thian  Institute of Materials
Research and Engineering, Agency for Science, Technology and Research
(A*STAR), 138634 Singapore  Hang Khume Tan  Institute of Materials Research
and Engineering, Agency for Science, Technology and Research (A*STAR), 138634
Singapore  Loïc Rondin  Université Paris-Saclay, CNRS, ENS Paris-Saclay,
CentraleSupélec, LuMIn, 91190, Gif-sur-Yvette, France  Helena S. Knowles
Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue,
Cambridge, CB3 0HE, UK  Jean-François Roch  Université Paris-Saclay, CNRS, ENS
Paris-Saclay, CentraleSupélec, LuMIn, 91190, Gif-sur-Yvette, France  Anjan
Soumyanarayanan  Institute of Materials Research and Engineering, Agency for
Science, Technology and Research (A*STAR), 138634 Singapore  Physics
Department, National University of Singapore (NUS), 117551 Singapore  Mete
Atatüre<EMAIL_ADDRESS>Cavendish Laboratory, University of Cambridge, J. J.
Thomson Avenue, Cambridge, CB3 0HE, UK

###### Abstract

Scanning diamond magnetometers based on the optically detected magnetic
resonance of the nitrogen-vacancy centre offer very high sensitivity and non-
invasive imaging capabilities when the stray fields emanating from ultrathin
magnetic materials are sufficiently low ($<10\,\mathrm{mT}$). Beyond this low-
field regime, the optical signal quenches and a quantitative measurement is
challenging. While the field-dependent NV photoluminescence can still provide
qualitative information on magnetic morphology, this operation regime remains
unexplored particularly for surface magnetisation larger than $\sim
3\,\mathrm{mA}$. Here, we introduce a multi-angle reconstruction technique
(MARe) that captures the full nanoscale domain morphology in all magnetic-
field regimes leading to NV photoluminescence quench. To demonstrate this, we
use [Ir/Co/Pt]14 multilayer films with surface magnetisation an order of
magnitude larger than previous reports. Our approach brings non-invasive
nanoscale magnetic field imaging capability to the study of a wider pool of
magnetic materials and phenomena.

##  I Introduction

In the last decade, the negatively-charged nitrogen-vacancy (NV) centre in
diamond has attracted great interest as a versatile quantum sensor for the
investigations of weak-field magnetism which demands high sensitivity,
nanoscale resolution and noninvasiveness. [1, 2, 3, 4, 5]. In the presence of
a magnetic field, the Zeeman splitting of the NV spin can be quantified by
performing optically detected magnetic resonance (ODMR) measurements using
laser and microwave excitation [6]. The single-spin nature of the NV centre
also ensures limited perturbation of the measured system. Further, attaching
an NV-containing diamond platform on a scanning probe [7, 3, 8, 2, 9] enables
scanning NV microscopy (SNVM), which allows for nanoscale noninvasive magnetic
imaging. This technique features a large operating temperature range
(cryogenic to room temperature) and stability in vacuum to ambient conditions
[6, 1, 10]. However, the ODMR measurements are restricted to magnetic fields
below $10\,\mathrm{mT}$ due to the field-induced quenching of the ODMR
contrast, thus preventing the optical readout of the spin splitting [11, 12,
8]. As a consequence, quantitative ODMR-based SNVM has been demonstrated
mainly on magnetic textures in thin films with close to zero surface
magnetisation, such as antiferromagnetic or single layer ferromagnetic
materials [8, 13, 7, 14, 4, 11, 15, 16, 17, 18, 19, 20].

Figure 1:  Effect of Magnetic Field Amplitude and Orientation on NV
Luminescence. (a) Illustration of a diamond probe scanning over a spin texture
(colored cones) with magnetic field lines across the domain boundaries (red
lines). The inset is a schematic of the local magnetic field vector B with
reference to the NV quantisation axis $\hat{u}_{\mathrm{NV}}$ at the tip of
the diamond probe. $\alpha_{B}$ indicates the angle between B and
$\hat{u}_{\mathrm{NV}}$. (b) Labyrinth domain morphology in Ir/Co/Pt
multilayer observed by MFM, exhibiting a zero-field period of
$407\,\mathrm{nm}$ (scale bar: $3\,\mathrm{\mu m}$). (c) Normalized NV
luminescence defined as
$(\mathrm{PL}-\mathrm{PL}_{\rm\min})/(\mathrm{PL}_{\rm\max}-\mathrm{PL}_{\rm\min})$
as a function of $\left|\mathbf{B}\right|$ and $\alpha_{B}$. The corresponding
distributions at various $d_{\mathrm{NV}}$, obtained from simulated magnetic
fields across (b), are overlaid on (c) (Contour lines). The 80th, 60th, and
40th percentiles are indicated with increasingly lighter contour lines. (d)
The histograms of the simulated PL response at the three $d_{\mathrm{NV}}$
values, $60\,\mathrm{nm}$, $100\,\mathrm{nm}$ and $200\,\mathrm{nm}$.
$\overline{\mathrm{PL}}$ indicates the mean PL, $\Delta\mathrm{PL}$ marks the
difference between the 90th and 10th percentile of the PL distribution. (e)
Dependence of $\Delta\mathrm{PL}$ and $\overline{\mathrm{PL}}$ on
$d_{\mathrm{NV}}$. The peak of $\Delta\mathrm{PL}$ marks the optimal distance
for quench-based imaging for the multilayer film of (b). The colored circles
correspond to the three $d_{\mathrm{NV}}$ values considered in panels (c) and
(d).

To extend the operational range beyond $10\,\mathrm{mT}$, the NV centre can
harness the field-dependent quench of the NV photoluminescence (PL) for
magnetic imaging as demonstrated recently [21, 11, 22, 23]. Quench-based SNVM
monitors the changes in NV PL due to the local magnetic field variation across
a spin texture with the respect to the NV quantization axis. This modality
also offers reduced acquisition time and enables microwave-free non-
perturbative operation [24, 5, 25]. The interpretation of quench-based SNVM
maps can be ambiguous, because of the multiple parameters that influence PL
quenching, such as NV-sample distance, NV axis orientation, sample
magnetization, magnetic domain size or magnetic field noise [26]. Therefore,
this imaging mode has been limited to the mapping of magnetic domain
morphology with surface magnetisation $I_{S}\lesssim 3\,\mathrm{mA}$ [21, 22,
23] (equivalent to $\leavevmode\nobreak\ 2\,\mathrm{nm}$ of $\mathrm{Co}$). In
this report, we reveal distinct quench-based imaging regimes, dependent on the
material parameters, and introduce the Multi-Angle Reconstruction (MARe)
protocol to interpret the domain morphology from quenched SNVM maps. We
demonstrate MARe on [Ir/Co/Pt]14 multilayer film with $12\,\mathrm{mA}$ out-
of-plane surface magnetisation, an order of magnitude larger than the
operational limit of ODMR-based SNVM. Utilising MARe can extend the
applicability of SNVM to a wider range of materials and magnetic regimes.

##  II Quench-Based Imaging in Different Regimes

Figure 1(a) illustrates our experimental setup consisting of a diamond
scanning probe with an NV centre implanted close to the diamond surface at an
NV-sample distance $d_{\mathrm{NV}}$ smaller than $100\,\mathrm{nm}$ [2, 27,
9, 28]. The optical ground state of the NV centre is a spin triplet, with a
quantisation axis $\hat{u}_{\mathrm{NV}}$ along one of the four
crystallographic axes of the diamond lattice [29, 30] and the lowest-energy
state $\ket{m_{s}=0}$ is split from the $\ket{m_{s}=\pm 1}$ states by
$2.87\,\mathrm{GHz}$ [31]. The local magnetic field can be decomposed into
parallel ($\textbf{B}_{\parallel}$) and orthogonal ($\textbf{B}_{\perp}$)
components with respect to $\hat{u}_{\mathrm{NV}}$ (Fig. 1(a) insert). The
$\textbf{B}_{\parallel}$ splits the $\ket{m_{s}=\pm 1}$ states which is
measured by monitoring the ODMR [32]. However, $\textbf{B}_{\perp}$ mixes
these spin states and modifies the branching ratio of the optical transitions
[12]. This results in the quenching of the NV PL and the suppression of the
ODMR contrast (Supplemental Material A), restricting quantitative ODMR-based
imaging to below $\sim 10\,\mathrm{mT}$ [12, 11].

Figure 2:  Quench-based SNVM Imaging Regimes. (a) Different regimes of quench-
based imaging as a function of NV-sample distance ($d_{\mathrm{NV}}$) and
surface magnetisation ($I_{s}$), based on simulated quench images. Little to
no domain morphological information is captured in the No Quench (left greyed
area) and Full Quench (right greyed area) regimes where the PL map is
predominantly bright or dark, respectively. In the Partial Quench regime (area
bounded by dashed lines), field variations are mapped to PL changes resulting
in (b-d) quench images with features indicative of domain boundaries (scale
bar: $1\,\mathrm{\mu m}$). Domain boundaries appear as dark isotropic PL
features (low directionality) for smaller $d_{\mathrm{NV}}$ and $I_{s}$ (b),
and as directional bright features (high directionality) at larger
$d_{\mathrm{NV}}$ and $I_{s}$ (c-d). The orientation of the directionality
depends on $\hat{u}_{\mathrm{NV},\varphi}$ which is the NV axis
$\hat{u}_{\mathrm{NV}}$, projected on the sample surface. The dashed lines
indicate the contour lines for 5% map contrast. (e) Illustration depicting the
NV axis $\hat{u}_{\mathrm{NV}}$, the tilt angle $\vartheta_{\mathrm{NV}}$ from
the normal to the sample surface, and the projection of
$\hat{u}_{\mathrm{NV}}$ in the sample plane, $\hat{u}_{\mathrm{NV},\varphi}$.
The angle $\varphi_{\mathrm{NV}}$ is the angle between
$\hat{u}_{\mathrm{NV},\varphi}$ and the reference axis within the sample
plane. For panels (a-d), $\vartheta_{\mathrm{NV}}{}=54.7^{\circ}$ and
$\varphi_{\mathrm{NV}}{}=0^{\circ}$.

Quench-based SNVM generates a PL intensity map, where regions with strong
$\textbf{B}_{\perp}$ component appear darker. In the limit of modest surface
magnetisation and small NV-sample distance $d_{\mathrm{NV}}$, the domain
boundaries appear dark producing faithful magnetic domain morphology maps.
Therefore, demonstrations are limited to single or bilayer thin film systems
with surface magnetisation $I_{s}\lesssim 3\,\mathrm{mA}$ [21, 22, 23, 11, 8].
Outside this regime, the complex interplay between $d_{\mathrm{NV}}$ and
$I_{s}$, as well as the morphology lenghtscale, on the NV PL obfuscates the
straightforward correspondence of dark regions to domain boundaries.
Therefore, a systematic understanding of quench-based SNVM response is
necessary to retrieve the domain morphology of a magnetic material. To do
this, we first simulate the $d_{\mathrm{NV}}$ dependence of quench-based SNVM
for a known magnetic structure.

Our study involves [Ir($1\,\mathrm{nm}$)/Co(1)/Pt(1)]14 magnetic multilayer, a
room-temperature skyrmion platform with an out-of-plane anisotropy and
$I_{s}=12\,\mathrm{mA}$ (Supplemental Material B) – an order of magnitude
larger than systems studied previously with SNVM. Further, the ambient
stability of the nanoscale spin textures [33, 34] allows us to correlate the
quench-based SNVM images with MFM measurements [35]. Figure 1(b) presents an
MFM image of this film, exhibiting a labyrinth domain morphology with a zero-
field period of $407\,\mathrm{nm}$. Figure 1(c) presents a grey-scale map of
normalised PL intensity simulated as a function of field amplitude
$\left|\textbf{B}\right|$ and field angle $\alpha_{B}$ with respect to the NV
axis $\hat{u}_{\mathrm{NV}}$. To understand how the stray field distribution
of the domain morphology affects the NV PL at various $d_{\mathrm{NV}}$, we
simulate the volumetric field distribution from the MFM map in Figure 1(b)
using the micromagnetics package mumax3 [36] (Supplemental Material C). On
Figure 1(c), we overlay the corresponding $\left|\textbf{B}\right|$ \-
$\alpha_{B}$ distributions of the magnetic field at three different
$d_{\mathrm{NV}}$, at $60\,\mathrm{nm}$ (green contours), $100\,\mathrm{nm}$
(orange) and $200\,\mathrm{nm}$ (blue). (Supplemental Material D). At
$d_{\mathrm{NV}}{}=60\,\mathrm{nm}$ ($200\,\mathrm{nm}$), the NV PL remains
uniformly quenched (unaffected) for the majority of the field distribution,
while $100\,\mathrm{nm}$ $d_{\mathrm{NV}}$ results in strong PL variation.
Figure 1(d) clearly highlights these NV PL variations $\Delta\mathrm{PL}$ via
the corresponding histograms at $d_{\mathrm{NV}}{}=60,\,100$ and
$200\,\mathrm{nm}$. Figure 1(e) presents the $\Delta\mathrm{PL}$ – calculated
as the difference between the 90th and 10th percentile of the NV PL
distribution – as a function of $d_{\mathrm{NV}}$ (solid red curve) alongside
the mean PL (dashed grey curve).

Figure 3:  Directional Quench Imaging and morphology reconstruction. (a)
Simulated quenched PL map based on spin texture in Figure 1b, with
$\vartheta_{\mathrm{NV}}{}=54.7^{\circ}$, $\varphi_{\mathrm{NV}}{}=0^{\circ}$
and (b) simulated quenched PL map in the same area but with the NV rotated
$90^{\circ}$ in the sample plane ($\varphi_{\mathrm{NV}}{}=90^{\circ}$). Both
maps are simulated at a NV-sample distance $d_{\mathrm{NV}}{}=77\,\mathrm{nm}$
and surface magnetisation $I_{s}=12\,\mathrm{mA}$ (scale bar: $1\,\mathrm{\mu
m}$). (c) Reconstructed image obtained by summing (a) and (b). (d) Multi-Angle
Reconstruction (MARe) illustrating the domain morphology acquisition based on
multiple N images at various $\varphi_{\mathrm{NV}}$. (e) Coverage of domain
boundaries given as function of $N$ and $\varphi_{max}$. $N$ is the number of
quench images involved in the reconstruction, and are obtained over a range of
$\varphi_{\mathrm{NV}}{}$ ($0^{\circ}$ to $\varphi_{max}$) spaced by
$\Delta\varphi_{\mathrm{NV}}{}=\varphi_{max}/(N-1)$. The reconstruction with
$N=4$ images yields the largest coverage, which saturates above
$\varphi_{max}\simeq 120^{\circ}$.

To assess the operational regime of quench-based SNVM, we need to consider
further the interplay between $I_{s}$ and $d_{\mathrm{NV}}$. As shown in
Figure 2(a), quench-based SNVM can be categorised into different regimes. The
combination of large $d_{\mathrm{NV}}$ and small $I_{s}$ (small
$d_{\mathrm{NV}}$ and large $I_{s}$) results in predominantly bright
(quenched) PL maps. In both the No Quench and the Full Quench regimes, the
lack of PL variation $\Delta\mathrm{PL}$ implies that little to no
morphological information of the underlying spin textures is captured. In
contrast, quench-based SNVM is feasible in the Partial Quench regime (area
bounded by dotted lines in Figure 2(a)) for a limited range of $I_{s}$ and
$d_{\mathrm{NV}}$ combinations. While the Partial Quench regime gives a large
$\Delta\mathrm{PL}$, which is desirable for quench-based SNVM, the resultant
PL maps over an identical spin texture can vary dramatically across this
regime. To highlight this, we simulated quench-based SNVM maps of the same
area in the multilayer film using three different combinations of $I_{s}$ and
$d_{\mathrm{NV}}$ (Fig. 2(b-d)). In general, we observe an evolution from
dark, isotropic features at lower $I_{s}$ and $d_{\mathrm{NV}}$ to bright,
directional features at higher $I_{s}$ and $d_{\mathrm{NV}}$ due to competing
magnetic field contributions above domains and domain boundaries. At lower
$I_{s}$ and $d_{\mathrm{NV}}$ (blue region in Figure 2(a)), the quench image
appears as a uniform bright background with isotropic dark outlines (Fig.
2(b)). This is a result of the strong magnetic field localised at the domain
boundaries which quenches the NV. The NV quench images reported to-date lie in
this region of the parameter space [21, 8, 22, 23] (Supplemental Material D).

Figure 4: Experimental verification of Multi-angle Reconstruction of Domain
Morphology. Experimental quenching map of the same area of Figure 3(a, b) with
$\vartheta_{\mathrm{NV}}{}=60\pm 2^{\circ}$ and (a) with $\varphi=0^{\circ}$
and (b) with $\varphi=90^{\circ}$. The two images are combined to give (c) the
reconstructed domain morphological map with $N=2$. (scale bar: $1\,\mathrm{\mu
m}$). (d) Binarized and magnified image of Figure 1(b) covering the same area
in (a, b and c), with domain boundaries marked in red.

For combinations of larger $I_{s}$ and $d_{\mathrm{NV}}$ values (orange region
in Figure 2(a)), the quench maps generate strikingly different images: panels
(c) and (d) capture highly directional bright and segmented features along the
domain boundaries. In this case, strong off-axis magnetic field above domains
and domain boundaries results in a predominantly dark PL map. However due to
large gradients localised at domain boundaries, there are instances where the
field is aligned closer to $\hat{u}_{\mathrm{NV}}$. This occurs across
portions of domain boundaries orthogonal to the projection of
$\hat{u}_{\mathrm{NV}}$ in the sample plane ($\hat{u}_{\mathrm{NV},\varphi}$),
resulting in directional bright features for panels (c) and (d), highly
dependent on the NV equatorial angle $\varphi_{\mathrm{NV}}$ (Fig. 2(e)).
Notably, this directional behaviour occurs over a significantly larger
parameter space of the Partial Quench regime, well beyond that of panel (a),
and the trend remains valid for different domain periodicity (Supplemental
Material D). It is worth emphasising here that magnetic materials with $I_{s}$
larger than $\sim 3\,\mathrm{mA}$ would inevitably constrain quench-based SNVM
to the directional region of Figure 2(a). Therefore, a protocol that relates
these images with the actual magnetic domain morphology is necessary in order
to extend the operation regime of quench-based SNVM for non-perturbative
investigations of such materials.

##  III Reconstruction of domain morphology - MARe

To reflect the role of $\varphi_{\mathrm{NV}}$ in quench-based SNVM, we
simulate two quench images for $\varphi_{\mathrm{NV}}{}=0^{\circ}$ and
$\varphi_{\mathrm{NV}}{}=90^{\circ}$, displayed in Figure 3(a) and 3(b),
respectively. We set $d_{\mathrm{NV}}$ = $77\,\mathrm{nm}$,
$\vartheta_{\mathrm{NV}}{}=54.7^{\circ}$, and $\sim 12\,\mathrm{mA}$ surface
magnetisation (Supplementary Material E) to reflect our experimental
measurements. The images for both $\varphi_{\mathrm{NV}}$ orientations show
directional segments revealing some features of the domain morphology, but
more importantly these segments are complementary. Therefore, while an image
at a given $\varphi_{\mathrm{NV}}$ remains incomplete, images obtained at
multiple $\varphi_{\mathrm{NV}}$ values can collectively give a significantly
better coverage of the underlying domain morphology, which is the essence of
the proposed imaging protocol. Multi-Angle Reconstruction protocol (MARe)
harnesses the $\varphi_{\mathrm{NV}}$ dependence of PL features to build a
composite map enabling morphological imaging further into the Partial Quench
regime, i.e. in strong-field conditions.

The overlapping features in the PL maps obtained at different
$\varphi_{\mathrm{NV}}$, e.g. $0^{\circ}$ and $90^{\circ}$ as in panels (a)
and (b) of Figure 3 allow us to perform an initial image registration to
compensate for the domain outline shift caused by
$\vartheta_{\mathrm{NV}}{}\neq 0^{\circ}$ (Supplemental Material F).
Subsequently, the maps are normalised and summed to yield a MARe image, as
displayed in Figure 3(c), revealing a larger fraction of the domain boundaries
with just two values of $\varphi_{\mathrm{NV}}$. To quantify the domain
boundary coverage, we integrate the product of the domain outlines from the
MFM image (Fig. 1(b)) with the binarized MARe image. In order to maximise the
fraction of domain boundaries covered by the protocol, we consider $N\geq 2$
images taken at different $\varphi_{\mathrm{NV}}$ values ranging from
$0^{\circ}$ to $\varphi_{\mathrm{NV}}^{\rm max}$ spaced equally by
$\Delta\varphi_{\mathrm{NV}}{}=\varphi_{\mathrm{NV}}^{\rm max}/(N-1)$. Figure
3(d) illustrates the MARe scheme for $N=4$ and $\varphi_{\rm
max}=120^{\circ}$, which corresponds to 4 quench-based SNVM images with each
obtained at $40^{\circ}$ relative angle. The corresponding MARe image clearly
captures an increased fraction of the domain morphology.

Figure 3(e) presents the calculated fraction of domain boundary coverage for
MARe with $N=2,3$ and $4$ (black, blue and red curve). For $N=2$ ($3$) the
maximum coverage reaches $81\%$ ($94\%$) at $\varphi_{\rm max}=80^{\circ}$
($100^{\circ}$). Extending MARe to $N=4$ further improves the coverage
reaching a maximum of $\sim 98\%$. This shows that even for $N\leq 4$, the
MARe protocol is capable of recovering the domain morphology with near-unity
coverage.

Figure 4 presents our experimental demonstration of domain morphology mapping
using MARe on the [Ir/Co/Pt]14 multilayer. To obtain quench-based SNVM images
we use a (100) diamond probe containing a single NV centre with
$\vartheta_{\mathrm{NV}}{}=60\pm 2\,\mathrm{{}^{\circ}}$ and
$d_{\mathrm{NV}}{}=77\pm 3\,\mathrm{nm}$ (Supplementary Material E). The
combination of the $d_{\mathrm{NV}}$ ($77\,\mathrm{nm}$) value and
$I_{s}\leavevmode\nobreak\ (12\,\mathrm{mA})$ of the [Ir/Co/Pt]14 multilayer
yields directional quench images according to Figure 2(a). Figures 4(a) and
4(b) show experimental quench images acquired at
$\varphi_{\mathrm{NV}}{}=0^{\circ}$ and $\varphi_{\mathrm{NV}}{}=90^{\circ}$,
respectively, on the same area used for simulating Figure 3(a) and 3(b)
(Supplemental Material B). The domain boundary coverage of each of these
images is $60\begin{subarray}{c}+13\\\ -11\end{subarray}\%$, in line with the
simulations and there is good agreement between the simulated and the measured
images for both orientations (Supplemental Material G). Figure 4(c) is the
corresponding $N=2$ MARe image showing matching bright features with the
highlighted domain boundaries of the binarised MFM image displayed in Figure
4(d). The experimentally achieved domain boundary coverage is improved to
$71\begin{subarray}{c}+12\\\ -15\end{subarray}\%$ – an enhancement beyond the
single frame coverage of $\sim 60\%$. The deviation from the simulated
coverage is due to the nonlinearity of the experimental map, as well as image
thresholding and registration operations (see Supplementary Material H).
Another reason for this deviation might be due to perturbations of the domain
morphology induced by MFM scanning. As the experimental protocol includes MFM
scans performed before and after each quench-based SNVM map, we do observe
local perturbations due to MFM that could potentially lead to deviations from
the unperturbed images captured by quench-based SNVM (see Supplemental
Material I). Nonetheless, the experimental demonstration of MARe extends the
operational range of non-invasive quench-based SNVM into the Partial Quench
regime.

##  IV Outlook

Our work methodically evaluates quench-based SNVM in terms of characteristic
NV and magnetic material properties. We establish a predictive scheme
involving MFM, micromagnetics and NV photodynamics simulations, which yields
images in excellent agreement with experimentally acquired data. We find two
regimes of quench imaging where morphological information is captured. The
first regime corresponds to mostly bright PL maps with dark outlines tracing
the domain boundaries, which corresponds to materials of low magnetisation
($I_{s}<3\,\mathrm{mA}$). The second regime, which has not been reported to-
date, results in PL maps with directional segmented features with strong
$\hat{u}_{\mathrm{NV},\varphi}$ dependence. We established a multi-angle
reconstruction scheme, herein named as MARe, to enable domain morphology
mapping with near-unity coverage for the second regime. The experimentally
validated MARe protocol extends quench-based SNVM imaging of out-of-plane spin
textures to magnetic systems with $I_{s}>3\,\mathrm{mA}$. Furthermore, the
scheme to identify the imaging regimes can be generalized to complex magnetic
textures, thus enabling the forecast of the attainable SNVM modes. We
anticipate that these insights, alongside tools developed for prediction,
interpretation and reconstruction, will stimulate the adoption of quench-based
SNVM as a non-perturbative nanoscale magnetometry to a wider pool of
materials, thereby furthering the development of quantitative quench-based
SNVM imaging.

##  V Acknowledgements

This work was performed at the Cambridge Nanoscale Sensing and Imaging Suite
(CANSIS), part of the Cambridge Henry Royce Institute, EPSRC grant
EP/P024947/1. We further acknowledge funding from EPSRC QUES2T (EP/N015118/1)
and from the Betty and Gordon Moore Foundation. This work was also supported
by the Faraday Institution (FIRG01) and by the SpOT-LITE programme (Grant Nos.
A1818g0042, A18A6b0057), funded by Singapore’s RIE2020 Initiatives. A. K. C.
Tan acknowledges funding from A*STAR, Singapore. B. Vindolet acknowledges
support by a PhD research Grant of Délégation Générale de l’Armement. J.-F.
Roch thanks Churchill College and the French Embassy in the UK for supporting
his stay at the Cavendish Laboratory.

##  Appendix A Simulation of the NV photodynamics

To capture the photodynamics of the NV centre, we use of a seven-state model
which includes the ground-state and excited-state fine structure of the NV
centre (Fig. A). The strain splitting is $E_{\rm gs}=E_{\rm es}\approx 0$,
where the subscripts gs and es indicate the optical ground state and the
optical excited state, respectively. At zero-field, the levels $\ket{i}$ with
$i=0,1,2$ are split by $D_{\rm gs}=2.87\,\mathrm{GHz}$ in the optical ground
state while the levels of the excited state $\ket{i}$, $i=3,4,5$, are split by
the excited state zero-field splitting $D_{\rm es}=1.42\,\mathrm{GHz}$. The
transition rates from level $\ket{i}$ to the level $\ket{j}$ are denoted as
$\gamma_{ij}$. The decay rates are defined as in the work by Tetienne et al.
[12]: we assume $\gamma_{30}=\gamma_{41}=\gamma_{52}=\gamma_{r}$,
$\gamma_{46}=\gamma_{56}$, and $\gamma_{61}=\gamma_{62}$. The spin non-
conserving transitions from the excited state are assumed to be forbidden.
Optical excitation pumps the ground state populations to the excited state but
stimulated emission is neglected, the laser being off-resonant and the
vibrational relaxation decay time being short. The values used for the
numerical simulations are taken from the works by Robledo et al. and Tetienne
et al. [37, 12] (Tab. A). Within the assumption of Markovian noise:

$\dfrac{\mathrm{d}\rho(t)}{\mathrm{d}t}=-\frac{i}{\hbar}\left[\mathscr{H},\rho\right]-\frac{1}{2}\sum_{k=0}^{m}\left(L_{k}^{\dagger}L_{k}\rho+\rho
L_{k}^{\dagger}L_{k}\right)+\sum_{k=0}^{m}L_{k}\rho L_{k}^{\dagger}\;\,$ (1)

where $\mathscr{H}$ is the magnetic-field dependent Hamiltonian describing the
seven-state system, $\rho$ is the density operator and $L_{k}$ are the Kraus
operators which describe the $m$ photon emission or absorption processes. We
work in the approximation of microwave excitation rate weaker than the laser
pumping, hence $T_{2}^{*}$ dephasing is neglected. The laser pump is described
as an incoherent absorption process. The Kraus operators then can either take
the form:

$L_{k}^{abs}=\sqrt{\gamma_{ji}}\,\outerproduct{i}{j},\;i=\left(3,4,5\right),j=\left(0,1,2\right)\;\,$
(2)

or

$L_{k}^{em}=\sqrt{\gamma_{ij}}\,\outerproduct{j}{i},\;i=\left(3,4,5,6\right),\,j=\left(0,1,2\right)\;.$
(3)

Extra Kraus operators can be added if incoherent microwave driving is included
in the model:

$L_{k}^{mw}=\sqrt{\gamma_{ij}^{mw}}\outerproduct{i}{j},\;i,j=\left(0,1,2\right),i\neq
j\;.$ (4)

The steady-state PL rate is proportional to the sum of the steady-state
populations in the excited state:

$\Gamma_{\mathrm{PL}}\propto\sum_{i=3}^{5}\rho_{ii}\,.$ (5)

| Decay rate (MHz)
---|---
$\gamma_{r}$ | 65
$\gamma_{36}$ | 11
$\gamma_{46}=\gamma_{56}$ | 80
$\gamma_{60}$ | 3
$\gamma_{61}=\gamma_{62}$ | 3
Table S1: Photodynamics parameters. The previously reported [37, 12] decay
rates used in the 7-state model for the NV magnetic field-dependent
photodynamics. Figure S5: Schematics of the NV seven-level system. Seven-level
system used to capture the NV photodynamics, for an arbitrary magnetic field.
In general, off-axis magnetic fields couple the zero-field eigenstates and
allow for spin-flip transitions which modify the zero-field photodynamics.
Green lines represent laser excitation, red lines optical decay and purple
lines non-radiative decay. Figure S6: Magnetic field-dependent NV
photodynamics. (a) Changes in steady state population under continuous green
excitation of the triplet ground state and singlet state of an NV as a
function of a magnetic field, $B_{\perp}$, orthogonal to the NV axis
$\hat{u}_{\mathrm{NV}}$. (b) The corresponding quench response of the NV PL
(blue curve) with increasing $B_{\perp}$ due to larger shelving state
population (shown in (a)). ODMR contrast (red curve) is also reduced due to
the decrease in population difference between $\ket{0}$ and $\ket{\pm 1}$
(shown in (a)).

Magnetic field components orthogonal to $\hat{u}_{\mathrm{NV}}$ (off-axis)
couple the different spin states, modifying the branching ratio of the
transitions [12] and altering the steady-state populations of the levels (Fig.
S6(a)). On one hand, this leads to a reduction of the ESR contrast (Fig. S6c),
because of the reduced population difference between the $\ket{0}$ and
$\ket{\pm 1}$ levels (Fig. S6(b)). On the other hand, this leads to the
quenching of the PL [8, 12, 38], due to a larger population getting trapped in
the singlet state (Fig. S6(b)). This effect leads to a trade-off between
magnetic field amplitude ($\propto 1/d_{\mathrm{NV}}{}$) and spatial
resolution ($\propto d_{\mathrm{NV}}{}$) when imaging small spin textures.

##  Appendix B Material Properties and Preparations

The multilayer stack of [Ir($1\,\mathrm{nm}$)/ Co(1)/ Pt(1)]14 were deposited
on thermally oxidised Si wafers by DC magnetron sputtering. Additional
fabrication information is found in previous studies [34]. Relevant properties
of the Ir/Co/Pt stack are shown in Table B. The surface magnetisation $I_{s}$
is given by $M_{s}\cdot t_{\rm eff}$, where $t_{\rm eff}$ is the effective
magnetic thickness which is the number of repetition multiplied by the
thickness of the magnetic layer. In this case, $t_{\rm eff}$=14 nm, and hence
$I_{s}$=12.3 mA (Table. B). The $I_{s}$ of various systems studied with
quenched SNVM is given in Table B for comparison. The zero-field magnetic
domains are stabilised by demagnetising the sample. This results in labyrinth
morphology with a period, $P$=407 nm (Fig. S9(a)).

The sample is marked with a wirebonder (Fig. S7) which allows us to image the
same area of interest (yellow box in Fig. S7) using two techniques (SNVM and
MFM) on separate platforms. MFM is always carried out before and after
quenched SNVM, to ensure that the morphology of the probed area remains
identifiable and the features are largely unchanged.

$\mathbf{M_{s}}$ $(\mathrm{MA/m})$ |  $\mathbf{K_{eff}}$ $(\mathrm{MJ/m^{3}})$ |  $\mathbf{D}$ $(\mathrm{mJ/m^{2}})$
---|---|---
0.881 |  0.474 |  1.25
Table S2: Material Properties. The saturation magnetisation $M_{s}$, effective anisotropy $K_{eff}$ and DMI strength $D$ of [Ir/Co/Pt]14 film. Material System |  $\mathbf{I_{s}}$ $(\mathrm{mA})$
---|---
$14\times$ Ir/Co/Pt |  $12.3$
Pt/CFA/MgO/Ta [22] CFA: Co2FeAl |  1.8
Pt/FM/Au/FM/Pt [21] FM: Ni/Co/Ni |  2.6
Pt/Co/NiFe/IrMn [23] |  1.7
Table S3:  Material Systems. The surface magnetisation $I_{s}$ of various
systems studied with quenched SNVM compared to [Ir/Co/Pt]14.  Figure S7:
Marked Sample. Microscopic image of a marked area of the sample surface with a
MFM probe in view. The marking is achieved using a wirebonding tip, and the
area of interest probed by quenched SNVM, micromagnetics and MFM in the main
text is highlighted in yellow.

##  Appendix C Micromagnetic simulations

The magnetic field above the spin texture was obtained via Mumax3 simulations.
For the study of quenched imaging in various regimes (Fig. 2 in main text),
The multilayer film is modelled using the effective medium method [39] so as
to reduce computation resources. The simulation grid consists of $256\times
256\times 128$ cells spanning $10\,\mathrm{\mu m}\times 10\,\mathrm{\mu
m}\times 384\,\mathrm{nm}$ (cell size: $\approx 39\,\mathrm{nm}\times
39\,\mathrm{nm}\times 3\,\mathrm{nm}$). The first 14 layers are modelled with
an effective saturation magnetisation $M_{\rm eff}=M_{s}/3$ and the volume
above as non-magnetic spacers. The simulation is further refined for
comparison with experiments (Fig. 3 and 4 in main text) with each cell layer
corresponding to $1\,\mathrm{nm}$ of Ir, Co, or Pt. Maintaining the same grid
size, this reduces the total simulated height to $128\,\mathrm{nm}$.
Similarly, the Pt, Ir layer and the volume above the multilayer film are
modelled as non-magnetic spacers. Differing from the effective medium model,
the Co layer has the experimentally obtained magnetisation $M_{s}$. In both
cases, the simulated non-magnetic volume above the multilayer film allows us
to retrieve the magnetostatic field environment above the spin texture (Fig.
S8) via Mumax3. The magnetisation distribution used in the simulation is based
on segmenting a MFM image into up and down domains by image thresholding (Fig.
S9).

Figure S8: Simulated Magnetic Field. (a-c) Magnetic field components $B_{x}$,
$B_{y}$, $B_{z}$, at $d_{\mathrm{NV}}=77\,\mathrm{nm}$ above the sample
surface, simulated based on the magnetisation distribution in Figure S9(b).
(Scale bar: $1\,\mathrm{\mu m}$) Figure S9: Image Thresholding. (a) MFM image
of sample surface (highlighted in Figure S7) showing labyrinth domains at zero
field. (b) Corresponding binary image after thresholding process, yielding
up/down magnetisation used for simulations in Figure S8. (Scale bar:
$1\,\mathrm{\mu m}$)

##  Appendix D Analysis of Quenched Imaging Regimes

The diagram in Figure 2 of the main text is constructed based on the
directionality of the observed PL features and the contrast of quenched images
with different combinations of surface magnetisation $I_{s}$ and NV-sample
distance $d_{\mathrm{NV}}$. The directionality of the PL features is
determined from the auto-correlation of the quenched image (Fig. S10). The
directionality is defined as $1-r_{min}/r_{max}$, where $r_{\rm min}$ and
$r_{\rm max}$ are respectively the minor and the major axis of an elliptical
Gaussian fit to the cross-correlation peak. A directionality equal to zero
indicates isotropic features (Fig. S11(a)), and a value increasing to unity
implies increasing anisotropy. The PL contrast is given as $1-P_{10}/P_{90}$,
where $P_{x}$ is the $x^{th}$ percentile of the PL distribution of each
quenched image (Fig. S11(b)).

Apart from the films magnetisation and $d_{\mathrm{NV}}$, we expect the
magnetic field distribution be heavily influenced by the domain periodicity.
We show here that the quench imaging regimes put forward in the main paper
remain valid at different P, with appropriate scaling of $d_{\mathrm{NV}}$ and
M. We define the scaled $d_{\mathrm{NV}}$ as
$d_{\mathrm{NV}}{}^{\prime}=d_{\mathrm{NV}}{}\times(P/P_{0})^{S_{d}}$, and
scaled M as $I_{s}^{\prime}=I_{s}/I_{s,0}\times(P/P_{0})^{S_{i}}$ where
$P_{0}=407\,\mathrm{nm}$ and $I_{s,0}=12.3\,\mathrm{mA}$ correspond to the
value for our sample [Ir($1\,\mathrm{nm}$)/ Co(1)/ Pt(1)]14. Scaling factor
$S_{d}$ and $S_{i}$ are empirically determined to be $-1$ and $-0.8$
$(\sim\sqrt{2/3})$. The analytical derivation is however beyond the scope of
the paper. The scaled directionality maps at varying $P$ are given in Figure
S12. We also include films studied by Gross et al. [21] and Rana et al. [23]
in this framework (Fig. S13). The framework is in good agreement with the work
of Gross et al. which observed isotropic PL features. In the study of Rana et
al., we are unable to resolve the directionality of the features observed.
However, we expect the quenched imaging regime to deviate from our framework
as our simulation model does not include exchange bias present in their film.

Figure S10: Quenched image autocorrelation and directionality. (a) Quenched
images at $d_{\mathrm{NV}}{}=12\,\mathrm{nm}$ and $I_{s}=1.6\,\mathrm{mA}$ and
(b) at $d_{\mathrm{NV}}{}=78\,\mathrm{nm}$ and $I_{s}=10.5\,\mathrm{mA}$.
(Scale bar: $\mathrm{1\mu m}$) (c, d) Autocorrelation maps of panels a, b,
respectively. Quenched maps with low directionality display a a cross-
correlation peak with circular simmetry. When the directionality increases,
the peak becomes elliptic. Figure S11: Details on Quenched Imaging Regimes.
(a) The directionality of PL features and (b) the PL contrast of a quenched
image given as a function of $d_{\mathrm{NV}}$and $I_{s}$.  Figure S12: Scaled
Quenched Imaging Regimes at Varying Domain Periodicity. The directionality of
PL features as a function of scaled $I_{s}$ ($I_{s}^{\prime}$) and scaled
$d_{\mathrm{NV}}$ ($d_{\mathrm{NV}}{}^{\prime}$) at domain periodicity, (a)
$P=200\,\mathrm{nm}$, (b) $P=400\,\mathrm{nm}$, (c) $P=600\,\mathrm{nm}$ and
(d) $P=800\,\mathrm{nm}$. The similar directionality picture indicates that
the quench imaging regimes remain valid across different $P$ with appropriate
scaling to $I_{s}$ and $d_{\mathrm{NV}}$.  Figure S13: Overview of Quenched
Imaging on Thin Films. Previous studies involving quenched imaging of thin
films are plotted on the scaled directionality map. The position on the map is
based on the $d_{\mathrm{NV}}$, $I_{s}$, and $P$ in each study.

##  Appendix E NV Sensor Characterisation

We use a 3-axis Helmholtz coil to apply an external magnetic field
$\mathbf{B}$ at varying $\varphi$ and $\vartheta$, with a fixed field strength
$\lvert\mathbf{B}\rvert=1\,\mathrm{mT}$. We obtain the ODMR spectra by
recording the integrated PL intensity of the NV centre as we sweep the
microwave (MW) frequency. In the presence of magnetic field, the ODMR spectrum
displays a splitting of the $\ket{m_{s}=+1}$ and $\ket{m_{s}=-1}$ states due
to the Zeeman effect (Fig. S14(a)). This splitting is proportional to the
projection of the magnetic field on the NV axis $\hat{u}_{\mathrm{NV}}$. The
ODMR spectrum is first obtained as a function of $\varphi$ while fixing
$\vartheta=90\,\mathrm{{}^{\circ}}$ (Fig. S14(b)). The Zeeman splitting is
maximum when $\varphi=\varphi_{\mathrm{NV}}$ which in our case is
$\varphi_{\mathrm{NV}}=93\pm 2\,\mathrm{{}^{\circ}}$. Next, we vary
$\vartheta$ while fixing $\varphi=\varphi_{\mathrm{NV}}$ (Fig. S14(c)).
Similarly, the maximum splitting occurs when
$\vartheta=\vartheta_{\mathrm{NV}}$ which we obtain to be
$\vartheta_{\mathrm{NV}}=60\pm 2\,\mathrm{{}^{\circ}}$.

We determine the NV-sample distance $d_{\mathrm{NV}}$ by measuring with our
diamond tip the stray field emitted across the edge of a [Ta/CoFeB/MgO] strip.
The out-of-plane magnetic hysteresis is characterised by a MagVision Magneto-
Optical Kerr Effect (MOKE) microscope (Vertisis Technology) in the polar
sensitivity mode and shows that the magnetisation remains saturated at
remanence (Fig. S15). The Zeeman shift of the ODMR spectrum across the edge at
remanence is given in Figure S16(a) (blue dashed curve) and is fitted (red
curve) following the procedure devised by Hingant et al. [40] to retrieve the
$d_{\mathrm{NV}}$. We repeat the measurement numerous times along the edge at
$50\,\mathrm{nm}$ spacing, and the extracted values are averaged (Fig.
S16(b)). The diamond tip used in this work has a
$d_{\mathrm{NV}}=77.5\,\mathrm{nm}\pm 3\,\mathrm{nm}$.

Figure S14: Axis measurements of the NV probe. (a) ODMR spectrum obtained
under an external magnetic field. We can observe the splitting of the
$\ket{m_{s}=+1}$ and $\ket{m_{s}=-1}$ due to the Zeeman effect. We measure a
splitting of $54\,\mathrm{MHz}$ which corresponds to a field felt by the NV of
about $1\,\mathrm{mT}$. (b) Measurement of $\varphi_{\mathrm{NV}}$. We fix
$\vartheta$ and $\varphi$ is varying. When the ODMR splitting is maximum,
$\varphi=\varphi_{\mathrm{NV}}$. (c) Measurement of $\vartheta_{\mathrm{NV}}$.
We fix $\varphi$ and $\vartheta$ is varying.
$\vartheta=\vartheta_{\mathrm{NV}}$ when the ODMR splitting reaches its
maximum value.  Figure S15: Calibration Strip Characterisation. (a) Intensity
of polar MOKE signal of a [Ta/CoFeB/MgO] strip as a function of an out-of-
plane magnetic field. (b) Topography image of [Ta/CoFeB/MgO] calibration strip
(scale bar: $10\,\mathrm{\mu m}$) Figure S16: Calibration of the NV probe. (a)
We represent on this plot the topography of the edge of a CoFeB magnetic
stripe (in brown) and the measured Zeeman shift of the NV ODMR spectrum (in
blue) due to the magnetic field emitted at the edge of the stripe. We deduce
the value of $d_{\mathrm{NV}}$ from the fit (in red) of the Zeeman shift
experimentally measured. (b) Histogram distribution of all the NV-sample
distances we measured. The average value is
$d_{\mathrm{NV}}=77.5\,\mathrm{nm}$ and the standard deviation is
$\sigma_{d_{\mathrm{NV}}}\simeq 3\,\mathrm{nm}$.

##  Appendix F Quenched Imaging with [111] NV Centre

Figure S17: Quenching with [111] NV centres. Quenching maps obtained with NVs
with $\vartheta_{\mathrm{NV}}{}=0^{\circ}$ on the same area as Figure 2 in the
main text (scale bar: $1\,\mathrm{\mu m}$). The different maps correspond to
(a) $I_{s}=3.1\,\mathrm{mA}$, $d_{\mathrm{NV}}{}=30\,\mathrm{nm}$, (b)
$I_{s}=9.2\,\mathrm{mA}$, $d_{\mathrm{NV}}{}=84\,\mathrm{nm}$, and (c)
$I_{s}=15.4\,\mathrm{mA}$, $d_{\mathrm{NV}}{}=162\,\mathrm{nm}$, which
correspond to the parameters of Figure 2(b-d) of the main text. (d-f) 2D
cross-correlation maps of the images in (a-c) with the domain boundaries. The
negative correlation at zero displacement indicates a low PL at the boundary.
If the displacement increases, the correlation is positive, corresponding to
the bright PL observed within the domains.

The discussion in the main text focuses on NV centres found in commercially
available (100) diamond tips. Quenched maps obtained with NVs with
$\vartheta_{\mathrm{NV}}{}=54.7^{\circ}$ on samples with out-of-plane magnetic
anisotropy give rise to different imaging regimes, as explained in the main
text. Notably, there is a range of $d_{\mathrm{NV}}$ and $I_{s}$ where the
quenched maps directionally highlight the domain boundaries. The
directionality is due to the non-zero angle between the NV axis and the
magnetic anisotropy. Hence, this effect is not present when using NV centres
pointing along the [111] axis (i.e. $\vartheta_{\mathrm{NV}}{}=0^{\circ}$),
hosted in (111)-oriented diamond tips, which have been recently reported [41].
The simulations shown in Figure S17(a-c), which have been taken at the same
$d_{\mathrm{NV}}$ and $I_{s}$ of Figure 2(b-d) of the main text, respectively.
At low magnetisation (Fig. S17(a)), the NV PL is quenched along the domain
boundaries (cross-correlation in Fig. S17(d)), resulting in a bright image
with dark outlines. At higher magnetisation (Fig. S17(b)) the quenching still
traces the domain boundaries (cross-correlation in Fig. S17(e)), but also
expands further within the domain area. The thin bright lines correspond to
the innermost areas of the domains, where the magnetic field is mainly
orthogonal to the sample surface and thus aligned with the NV axis. In Figure
S17(c), the combination of large magnetisation and high $d_{\mathrm{NV}}$
gives an image similar to Figure S17(b), but with lower resolution.

##  Appendix G Domain Coverage Estimation

In order to estimate the percentage of domain boundaries covered by the
simulated quenched maps, we first binarize the selected MFM images with Otsu
thresholding [42] (a portion is shown Fig. S18(a)) and detect the boundaries
with the Canny algorithm. The quenched maps are simulated from the stray
fields obtained with Mumax3, as explained above. We first simulate the
quenched maps with NVs at $\vartheta_{\mathrm{NV}}{}=54.7^{\circ}$ and
different $\varphi_{\mathrm{NV}}{}$ (Fig. S18(b) for
$\varphi_{\mathrm{NV}}{}=0^{\circ}$). The single images are then registered to
the domain boundaries with the ECC image alignment algorithm [43], in order to
compensate for the small shift from the domain boundaries induced by the non-
zero angle between the magnetic anisotropy and $\vartheta_{\mathrm{NV}}$. The
images are then combined as explained in the main text (Fig. S18(c) for $N=4$
and $\varphi_{max}=180^{\circ}$). Additionally, we simulate a quenched map
with an NV at $\vartheta_{\mathrm{NV}}{}=0^{\circ}$, which exhibits no shift
and no $\varphi_{\mathrm{NV}}$-dependence (Fig. S18(d)). The images are then
binarized using local gaussian thresholding (Fig. S18(e-g)). The binarized
images are multiplied to the domain boundaries and integrated to yield the
coverage. For experimental quenched maps, the above estimation protocol
includes additional thinning and dilation of the binarized images before
multiplication. The thinning and dilation process ensures that local
deviations between the binarized experimental quenched maps obtained via SNVM
and the domain outline retrieved from MFM images are accounted in the coverage
estimation. These local deviations are largely due to experimental map
nonlinearity, suboptimal image threshold and registration conditions, and MFM
perturbation. To estimate the experimental coverage error, we use the 3
smallest structuring element – a 3 pixel wide diamond, 3 pixel wide square and
5 pixel wide diamond – for binary dilation. The coverage value is given by the
estimation protocol using a 3 pixel wide square dilation structuring element
while the coverage bounds are given by the diamond structuring elements.

Figure S18: Estimation of the domain coverage. (a) Portion of the binarised
MFM scan (background) and domain edges (red pixels) obtained via Canny edge
detection (scale bar: $1\,\mathrm{\mu m}$). Quenched maps of the same area,
where (b) is the map taken with an NV with
$\vartheta_{\mathrm{NV}}{}=54.7^{\circ}$ and
$\varphi_{\mathrm{NV}}{}=0^{\circ}$, (c) is the reconstructed image with $N=4$
at $\varphi_{max}=180^{\circ}$ (see main text), and (d) is acquired with an NV
with $\vartheta_{\mathrm{NV}}{}=0^{\circ}$. (e-g) are the images obtained by
binarising (b-d), respectively.

##  Appendix H  Directionality, image reconstruction and boundary coverage

Figure S19: Directionality and image reconstruction (a,b) Simulated quenched
maps with $\varphi_{\mathrm{NV}}{}=0^{\circ}$ and
$\varphi_{\mathrm{NV}}{}=90^{\circ}$ and (c) MARe image with $N=10$ and
$\varphi_{max}=180^{\circ}$. (Scale bar: $1\,\mathrm{\mu m}$). (d-f)
2-dimensional cross-correlations between the quenched images in (a-c) and the
domain boundaries. For single images (d,e), the cross-correlation shows a
positive correlation shifted from the origin in the direction opposite to
$\hat{u}_{\mathrm{NV},\varphi}$, the projection of $\hat{u}_{\mathrm{NV}}$ in
the sample plane. This indicates that the bright outlines are highly
directional and do not occur exactly on top of the domain boundaries. On the
contrary, the cross-correlation of the MARe map (f) is isotropic, indicating
that most of the boundaries are uniformly covered. (g) Simulations of domain
boundary coverage to include $N$ beyond $N=4$.

We can further analyze the properties of quenched maps by studying the
2-dimensional cross-correlation between the maps and the domain boundaries. We
do this by simulating the quenched maps starting from the MFM image (Fig.
S9(a-b)), as described above and in the main text. We then calculate the
cross-correlation between the maps and the domain boundaries obtained from the
MFM map with the Canny edge detection algorithm. The cross-correlation plots
(Fig. S19(d-e)) show a non-uniform positive correlation peak which is shifted
from the origin. The shift is opposite to the direction of the direction of
$\hat{u}_{\mathrm{NV},\varphi}$. This indicates that the directional features
on average do not occur on top of the domain boundaries. The shift originates
from the non-zero tilt of $\hat{u}_{\mathrm{NV}}$ from the normal to the
sample plane. This has important consequences for the MARe scheme, since the
shift needs to be compensated with image registration algorithms before
combining the images (Sec. G). In Figure S19(c) we show a MARe image with
$N=10$ and $\varphi_{max}=180^{\circ}$. The corresponding 2D cross-correlation
(Fig. S19(f)) displays a circularly symmetric peak, indicating that the
reconstructed map is non-directional.

We also study the option of using more than four images ($N>4$) to reconstruct
the domain morphology. We show the result of the simulations in Figure S19(g).
As presented in the main text, $N=4$ achieves a peak coverage of about $0.97$
at $\varphi_{max}\approx 120^{\circ}$. The maximum coverage for $N=5$ is
$\approx 0.98$ at $\varphi_{max}\approx 180^{\circ}$. For $N>5$, the coverage
reaches a peak value of $\approx 1$.

##  Appendix I Imaging with Minimal Perturbation

As explained in the main text and in previous studies [22, 21, 23], quenched
SNVM enables perturbation-free imaging of spin textures. We present here a
comparison of repeated quenched SNVM and MFM scans over the same area, which
highlight the non-perturbative advantage of quenched SNVM. We first obtain two
consecutive quenched images over an area on the sample (Fig. S20(a,b)), and
thereafter another two consecutive MFM images over the same area (Fig. S20(c,
d)). By comparing the quenched and MFM images, we observed areas (circled in
Fig. S20(a-e)) showing non-perturbative consecutive quenched imaging (Fig.
S20(a,b)), but were subsequently perturbed by consecutive MFM scans (Fig.
S20(c,d)). In addition, the quenched image simulated (Fig. S20(e)) from the
MFM image in Figure S20(d) shows markedly different PL features compared to
experiments (Fig. S20(a,b)) at the vicinity of the highlighted areas (circled
in Figure S20(a), (b) and (e)), reinforcing the non-perturbative advantage of
quenched SNVM over conventional MFM. In our case, the MFM probe used for the
comparison is a low moment variant from Asylum Research, Oxford Instruments
(ASYMFMLM-R2). These observations are however not exhaustive in nature and
requires a statistical approach to determine the degree of perturbation
induced by MFM over quenched SNVM. A rigorous characterisation is highly non-
trivial and involves a vast parameter space including various magnetic
material parameters, different laser intensities utilised during quenched SNVM
and numerous low-moment probe options for MFM.

Figure S20: Evidence of non-perturbative imaging. Sequential study of the
domain morphology by (a,b) consecutive quenched SNVM imaging, followed by
(c,d) consecutive MFM. Areas of perturbation due to consecutive MFM imaging
are circled (c, d), while no visible changes are observed in the corresponding
areas in the consecutive quenched images (a, b). (e) Simulated quenched image
based on second MFM scan 2 (d) shows dissimilar PL features in the circled
vicinity as compared to experiments (a, b). (Scale bar: 500 nm)

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