# Comparative analysis of spin wave imaging using nitrogen vacancy centers and
time resolved magneto-optical measurements

Carolina Lüthi  Physics Department, Technical University of Munich, Garching,
Germany  Lukas Colombo  Physics Department, Technical University of Munich,
Garching, Germany  Franz Vilsmeier  Physics Department, Technical University
of Munich, Garching, Germany  Christian Back  Physics Department, Technical
University of Munich, Garching, Germany  Munich Center for Quantum Science and
Technology (MCQST), Munich, Germany<EMAIL_ADDRESS>
###### Abstract

Spin waves, the fundamental excitations in magnetic materials, are promising
candidates for realizing low-dissipation information processing in
spintronics. The ability to visualize and manipulate coherent spin-wave
transport is crucial for the development of spin wave-based devices. We use a
recently discovered method utilizing nitrogen vacancy (NV) centers, point
defects in the diamond lattice, to measure spin waves in thin film magnetic
insulators by detecting their magnetic stray field. We experimentally
demonstrate enhanced contrast in the detected wavefront amplitudes by imaging
spin waves underneath a reference stripline and phenomenologically model the
results. By extracting the spin wave dispersion and comparing NV center based
spin wave measurements to spin wave imaging conducted through the well-
established time-resolved magneto-optical Kerr effect, we discuss the
advantages and limitations of employing NV centers as spin wave sensors.

††preprint: AIP/123-QED

Spin wavesBloch (1930); Gurevich and Melkov (1996a); Stancil and Prabhakar
(2008), also known as magnons, represent collective excitations of the spins
in a magnetic material. These quasi-particles carry spin angular momentum and
propagate through the material’s magnetic lattice, thereby avoiding heating
associated with charge currents. Their unique properties, including long
coherence times and low dissipation Wang  _et al._ (2023); Kruglyak,
Demokritov, and Grundler (2010); Hortensius  _et al._ (2021); Serga, Chumak,
and Hillebrands (2010); Chumak, Serga, and Hillebrands (2014), as well as
frequencies in the giga- to terrahertz regime and wavelength as small as
several nanometers Cherepanov, Kolokolov, and L’vov (1993); Balashov  _et al._
(2014); Chuang  _et al._ (2014), render spin waves promising candidates for
various applications, such as interference-based ultrafast, nanoscale magnonic
logic circuits, and miniaturized device technologies Hortensius  _et al._
(2021); Chumak  _et al._ (2015); Hahn  _et al._ (2014); Chumak, Serga, and
Hillebrands (2014); Khitun (2012); Khitun, Bao, and Wang (2010); Schneider
_et al._ (2008a); Owens, Collins, and Carter (1985), particularly in the field
of spintronicsHan  _et al._ (2023); Singh  _et al._ (2023); Zhang  _et al._
(2023); Chumak  _et al._ (2015).

Imaging spin waves with high spatial and temporal resolution is essential for
exploiting their potential in technological advancements and understanding
their intricate dynamics. Traditional imaging techniques, such as time-
resolved magneto-optical Kerr effect (TR-MOKE) Vilsmeier, Riedel, and Back ;
Perzlmaier, Woltersdorf, and Back (2008); Farle, Silva, and Woltersdorf
(2013); Au  _et al._ (2011); Bauer  _et al._ (2014); Stigloher  _et al._
(2018), Brillouin light scattering Hillebrands (1999); Demokritov (2001), and
transmission x-ray microscopy Warwick  _et al._ (1998); Sluka  _et al._
(2019), have provided valuable insights into spin wave behavior. However,
these methods often have limitations in terms of spatial resolution and
sensitivity, particularly when studying spin waves in nanoscale systems or
beneath opaque materials, or, in the case of x-ray microscopy, need large and
expensive facilities for their implementation.

In recent years, the emergence of nitrogen-vacancy (NV) centers in diamond as
sensitive magnetic field sensors has opened new avenues for spin wave imaging
Bertelli  _et al._ (2020); Purser  _et al._ (2020); Koerner  _et al._ (2022);
Tetienne  _et al._ (2013). NV centers, point-like defects in the diamond
lattice consisting of a substitutional nitrogen atom adjacent to a vacancy,
exhibit remarkable properties, including high sensitivity to static and
fluctuating magnetic fields Maletinsky  _et al._ (2012); Appel  _et al._
(2015); Rondin  _et al._ (2014), nanoscale spatial resolution Balasubramanian
_et al._ (2008); van der Sar  _et al._ (2015), even below opaque
materialsBertelli  _et al._ (2021), and optical addressability Doherty  _et
al._ (2013). By using an ensemble of spins instead of a single NV center, the
sensitivity can be further enhanced by a scaling of $\sqrt{N}$, where $N$ is
the number of NV centersPurser  _et al._ (2020).

In this work, we discuss the use of an ensemble of shallowly implanted NV
centers as a means to measure spin waves in a magnetic thin film following the
seminal work reported in van der Sar  _et al._ (2015), in this case the low
damping ferrimagnetic insulator yttrium iron garnet (YIG) Cherepanov,
Kolokolov, and L’vov (1993); Serga, Chumak, and Hillebrands (2010). We
demonstrate how the contrast in the imaged spin wave amplitude can be
enhanced, allowing us to image spin waves for a long travelling distance.
Furthermore, we phenomenologically model these measurements to explain the
observed features in the NV spin wave images. By comparing the spin wave
measurements obtained through NV center detection to spin wave imaging
acquired with conventional TR-MOKE measurements, we discuss the advantages and
limitations of utilizing NV centers for spin wave imaging, as well as the
potential of integrating NV center measurements with TR-MOKE imaging into a
unified setup.

Figure 1: a) Sample configuration: A diamond chip containing a layer of NVs is
positioned on a 200 nm thick YIG film grown on a gadolinium gallium garnet
substrate. A stripline S1 excites spin waves in the sample. A second
perpendicular stripline S2, separated from S1 by a $\mathrm{SiO_{2}}$ layer,
is used to generate an Oerstedfield $B_{\mathrm{oe}}$ that oscillates at the
same frequency as the spin wave strayfield. An external magnetic field $B$ is
applied along S1 at an angle of $\phi=35^{\circ}$ relative to the sample
plane. b) Photoluminescence spectrum of the NV centers in dependence of the
external magnetic field $B$ and the drive frequency. Transitions in the ground
($\omega_{\pm},\omega_{\perp}$) and excited states ($\omega_{\mathrm{ex}}$)
correspond to a peak in the signal. Due to the handedness of the circularly
polarized spin wave stray field, the $\omega_{-}$ transitions are enhanced for
frequencies above the ferromagnetic resonance (FMR) limit. Figure 2: a)
Spatially resolved PL signal measured above the YIG film, when a spin wave
with frequency $f_{\mathrm{sw}}=2.24$ GHz, corresponding to the external field
$B=22.5$ mT, is excited by S1. The X-axis is located 10 $\mathrm{\mu}$m away
from the center of S1, and the center of S2 is located at $X=40~{}\mathrm{\mu
m}$. $B$ is applied along S1 at an angle of $\phi=35^{\circ}$ relative to the
sample plane. The microwave current is split between S1 and S2 to additionally
generate an Oersted field $B_{\mathrm{oe}}$ oscillating at $f_{\mathrm{sw}}$,
which results, together with the strayfield of the spin wave, in a standing
wave of the total magnetic field above the YIG film. b) Simulated PL signal
corresponding to the same spatial position as in a). The phase shift of the
simulated wavefronts at the edges of S2 occur due to the change in sign of
$B_{\mathrm{oe}}$ from the left side to the right side of S2. c) Normalized
TR-MOKE signal, when a spin wave with frequency $f_{\mathrm{sw}}=2.24$ GHz is
excited by S1. The external field is applied along S1, which is approximately
10 $\mathrm{\mu}$m away from the X-axis.

We excite spin waves in a 200 nm thick YIG thin film grown by liquid phase
epitaxy on a gadolinium gallium garnet substrate by a microwave current sent
through a stripline (S1) fabricated onto the YIG surface. The diamond chip is
placed on the YIG sample with the NV layer facing towards the magnetic
surface, as shown in Fig. 1. a). Additional information on the diamond chip
and the sample can be found in the Supplementary Material A. As the spin waves
propagate right (left)-wards, they generate a circularly polarized field with
handedness that drives the $\omega_{-}(\omega_{+})$ ESR transition of the NVs,
resulting in a spin-dependent photoluminescence (PL) signalBertelli  _et al._
(2020); van der Sar  _et al._ (2015); Andrich  _et al._ (2017). In Fig. 1. b)
the NV photoluminescence signal, obtained via lock-in detection and measured
approximately 10 $\mathrm{\mu}$m away from the center of S1, is shown in
dependence of the external magnetic field $B$ and the microwave current
frequency. The NV setup is described in detail in the Supplementary Material
B. Here, $B$ is applied along S1 at an angle of $\phi=35^{\circ}$ relative to
the sample plane, aligning it with one of the four possible orientations of
the NV axes. The peaks in the PL signal belong to the ESR transitions in the
ground ($\omega_{\pm},\omega_{\perp}$) and excited states
($\omega_{\mathrm{ex}}$), where only the PL signal of NV centers aligned with
$B$ result in a linear dependence on $B$. As the NVs are not only driven by
the microwave field of S1 but additionally also by $B_{\mathrm{sw}}$, the
$\omega_{-}$ transitions are enhanced for frequencies above the ferromagnetic
resonance (FMR) limit.

To gain the phase sensitivity to image individual wave fronts of the spin
waves, a second perpendicular reference stripline (S2), isolated from S1 by an
insulating $\mathrm{SiO_{2}}$ layer, is used (Fig. 1. a)). A microwave current
that generates an Oersted field $B_{\mathrm{oe}}$ oscillating at the same
frequency as $B_{\mathrm{sw}}$ is sent through S2. The superposition of
$B_{\mathrm{oe}}$ with the spin wave stray field $B_{\mathrm{sw}}$ results in
a standing wave pattern in the total magnetic field that drives the NV ESR
with a spatial periodicity equal to the spin wave wavelength Bertelli  _et
al._ (2020). In Fig. 2. a), the PL signal for a spatial scan over the sample
is shown. The X-axis, aligned with S1, is located approximately 10
$\mathrm{\mu}$m away from the center of S1. Furthermore, we scanned over S2,
which is aligned with the Y-axis, with its center located at
$X=40~{}\mathrm{\mu m}$. We find an enhanced contrast in the imaged spin wave
amplitude directly above S2. Additionally, we observe a $180^{\circ}$ phase
shift of the wave fronts at the edges of S2 and a light bending of the wave
fronts towards the center of S2.

To explain these characteristics, we phenomenologically model our PL signal by
calculating the time-averaged magnitude of the perpendicular time dependent
magnetic field $B_{\mathrm{tot_{\perp}}}$ relative to the NV axis. Thereby, we
take into account that ESR transitions can only be driven by rf fields
perpendicular to their quantization axis. Furthermore, we average over time,
as we measure a time independent PL signal at a fixed position on the sample,
which scales with the magnitude of the standing wave in the total
perpendicular magnetic field. We use a coordinate system where the X-axis is
aligned with S1, the Y-axis with S2, and the Z-axis with the normal vector of
the sample plane. In $B_{\mathrm{tot_{\perp}}}$, we consider the spin wave
strayfield above the film $B_{\mathrm{sw}}$ as well as the Oersted field
$B_{\mathrm{oe}}$ of S2:

$B_{\mathrm{tot_{\perp}}}(t)=B_{\mathrm{sw}}(t)+B_{\mathrm{oe}}(t).$ (1)

The time dependent magnetic fields $B_{\mathrm{sw}}(t)$ and
$B_{\mathrm{oe}}(t)$, as well as further details about the simulation, are
given in the Supplementary Material C. In Fig. 2. b), the simulated PL signal
corresponding to the same spatial position on the sample as the measurement
depicted in Fig. 2. a) is shown. The observed phase shift at the edges of S2
is attributed to the change in sign of $B_{\mathrm{oe}}$ from the left side to
the right side of S2. Moreover, we find an enhanced contrast in the PL signal
above S2 due to the comparable magnitudes of $B_{\mathrm{sw}}(t)$ and
$B_{\mathrm{eo}}(t)$ in the vicinity of S2. However, reproducing the bending
of the wavefronts towards the center of S2 requires a more advanced model,
which is beyond the scope of this study.

In the following, we compare our NV measurements to TR-MOKE measurements in
Damon-Eshbach (DE) configurationDamon and Eshbach (1961), where the external
field is applied in the YIG thin film plane along S1. TR-MOKE measurements
involve synchronizing the rf excitation and optical probing pulses. Similar to
stroboscopic techniques, a time-independent phase relationship between
excitation and probing is typically established to capture a stationary image.
To achieve this synchronization, a pulsed Ti:Sa-laser with a repetition rate
$f_{\mathrm{rep}}=80$ MHz is employed, while the excitation frequency
$f_{\mathrm{ex}}$ of the spin waves fulfils $f_{\mathrm{ex}}=n\cdot
f_{\mathrm{rep}}$, with $n$ being an integer. On the other hand, to detect
spin waves with frequencies unequal to multiples of $f_{\mathrm{rep}}$, we use
super-Nyquist sampling MOKE Dreyer  _et al._ (2021). In that case, we can tune
the rf excitation to any intermediate frequency $f_{\mathrm{ex}}=n\cdot
f_{\mathrm{rep}}+\epsilon$, where $\epsilon$ is a rational number.
Demodulating the Kerr signal at frequency $\epsilon$ enables direct extraction
of the real and imaginary components of the magnetic rf-susceptibility,
thereby providing phase-resolved measurements of the spin precession Dreyer
_et al._ (2021) (See Supplementary Material B for further information
concerning the MOKE setup). The lock-in detected Kerr signal
$S_{\mathrm{Kerr}}$ is then normalized to the topographic signal
$S_{\mathrm{Topo}}$ to gain an enhanced contrast in the measured spin
wavefronts. Fig. 2. c) shows a TR-MOKE measurement of propagating spin waves.
As direct contact with the magnetic surface is needed, the measurement is
taken far away from S2. In Supplementary Material E, we additionally show spin
wave measurements with different wavelengths.

To extract the spin wave dispersion from both measurement techniques, we
adjust the external static magnetic field $B$ such that the spin wave
frequency coincides with the $\omega_{-}$ NV ESR frequencies (The mathematical
equations to calculate the dispersion are given in Supplementary Material D).
This allows us to probe spin waves with varying wavelength, detectable by both
measurement techniques. In the NV measurements, the field is applied along S1
and one of the NV quantization axes. Given that the external field is always
less than 30 mT, significantly smaller than the saturation magnetization of
YIG $\mu_{0}M_{\mathrm{S}}\thickapprox 185$ mT, the static magnetization of
the YIG film tilts only slightly out of plane, at an angle
$B\mathrm{sin}(\phi)/(\mu_{0}M_{\mathrm{S}})\leq 5.3^{\circ}.$ Hence, direct
comparison between the NV measurements and the in-plane MOKE measurements is
feasible. Additionally, in the Supplementary Material F, we demonstrate that
within this range of magnetic fields, MOKE measurements in DE configuration
and with the field applied along the NV axis yield consistent wavelengths
within the error margin. To facilitate a direct comparison of the measured
wavelengths, we transpose the in plane fields of the TR-MOKE measurements into
the direction of the NV center axis. In Fig. 3, the theoretical dispersion
curve (red line), alongside the fitted wavelength of the NV measurement (blue
dots) and the MOKE measurements (green dots) are shown, which are in good
agreement with each other (See supplementary material G for information about
the fit model.). This consistency demonstrates the reliability and equivalence
of the two measurement techniques.

Figure 3: Dependence of the wavelength of the excited spin waves on the
external magnetic field. The theoretical dispersion curve (red line) is in
good agreement with the fitted wavelength from the NV measurements (blue dots)
and the TR-MOKE measurements (green dots).

In conclusion, the utilization of NV centers presents several advantages in
the measurement of spin waves. Their high resolution capabilities extend
beyond the diffraction limit, enabling precise characterization of spin wave
phenomena. Moreover, their ability to measure through opaque materials offers
unique opportunities for non-invasive investigation Bertelli  _et al._ (2021);
Borst  _et al._ (2023). Additionally, NV centers exhibit remarkable
sensitivity to magnetic fields, enhancing their utility in detecting subtle
magnetic variations. However, it is worth noting some limitations of NV
centers, including the lack of time resolution and the requirement of a
standing wave for effectively imaging wavefronts. Additionally, NV centers are
limited to probing only spin waves at the NV ESR frequencies. On the other
hand, Kerr microscopy offers high time resolution and the flexibility to probe
a wide range of frequencies. However, direct contact with the magnetic surface
is necessary, and the spacial resolution is ultimately limited by diffraction.
To overcome these limitations and harness the complementary strengths of both
techniques, we propose a combined approach. By integrating NV center
measurements with Kerr microscopy, one can benefit from high-resolution
imaging, non-invasive probing through opaque materials, high sensitivity to
magnetic fields, high time resolution, and flexibility in probing frequencies.
This integrated approach promises to advance the capability to probe spin wave
dynamics and enable new insights in the field.

###### Acknowledgements.

This work was supported by the Bayerisches Staatsministerium für Wissenschaft
und Kunst through project IQSense via the Munich Quantum Valley (MQV) and by
the DFG via the Munich Center for Quantum Science and Technology (MCQST,
EXC2111).

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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##  Appendix A Diamond substrate and YIG sample

###  A.1 Diamond substrate

We use a type IIa single-crystal diamond of size 0.5 mm $\times$ 0.5 mm
$\times$ 100 $\mu$m grown by chemical vapour deposition, with Nitrogen content
$<$ 1 ppm, surface roughness $<$ 2 nm Ra, and (100)-orientation (Applied
Diamond). It was implanted with $\mathrm{{}^{15}N^{+}}$ ions at an energy of 4
keV, a dose of $1.2\times 10^{13}$ ions$/\mathrm{cm^{2}}$, and a tilt angle of
7∘ (performed by CuttingEdge Ions), resulting in an implantation depth of 9
nm. Following this, it underwent annealing in a vacuum environment of $\sim
10^{-6}$ mbar. The annealing process involved a 45-minute ramp to 900∘C,
followed by 3 hours at 900∘C $\pm 10^{\circ}$C and a 2 h ramp to room
temperature. To remove the graphitic layer that formed during the annealing,
it was acid cleaned during 4 h by a boiling 3-acid mix consisting of 3 ml
sulfuric acid ($\mathrm{H_{2}SO_{4}}$), 3 ml nitric acid ($\mathrm{HNO_{3}}$),
and 3 ml perchloric acid ($\mathrm{HClO_{4}}$).

###  A.2 YIG sample

We use a 200 nm thick YIG thin film grown on a gadolinium gallium garnet
substrate via liquid phase epitaxy. The thin film posses a saturation
magnetization $\mu_{0}M_{\mathrm{S}}=0.185$ mT and an exchange stiffness
$A_{\mathrm{ex}}=3.7\cdot 10^{-12}\frac{\mathrm{J}}{\mathrm{m}}$. To excite
spin waves a stripline S1 with a with of 5 $\mu$m was fabricated on top of the
YIG film by optical lithography and electron beam evaporation of Ti(5
nm)/Au(100 nm). A second stripline S2 of Ti(5 nm)/Au(200 nm) with width 30
$\mu$m as well as a 150 $\mu$m thick $\mathrm{SiO_{2}}$ layer were fabricated
on top of S1 by optical lithography and electron beam evaporation.

##  Appendix B Measurement setup

###  B.1 NV center setup

The setup used for the NV center measurements was an in-house built confocal
microscope. The NV centers were optically excited by a 515 nm laser (iBeam-
smart-515s, Toptica), which was focused to a diffraction-limited spot by an
objective with a numerical aperture of 0.9 and a magnification of 63. Before
the objective, a polarizer in combination with a $\lambda/2$-plate were used
to rotate the polarization of the laser light, maximizing the contrast in the
$\omega_{\pm}$. The NV PL was gathered by the same objective, separated from
the excitation light by a dichroic short pass mirror (cut-off wavelength 600
nm, Edmund Optics) and a long-pass filter (cut-on wavelength 590 nm).
Microwaves for driving the NVs and spin waves were generated using a microwave
generator (N5183A MXG, Agilent). To simultaneously send a microwave current
through the pair of striplines S1 and S2 depicted in Fig. 1 a), the microwave
excitation was divided using a power combiner (ZFRSC-123-S+, Mini-Circuits).
To measure the PL signal with a lock-in amplifier the microwave signal was
modulated by a coaxial switch (139-ZASWA-2-50DRA, Mouser Electronics). The PL
photons were collected by an avalanche photodiode and, after passing a voltage
amplifier, measured by the lock-in amplifier. Each data point was measured
during 1 s at the 3th filter order. All measurements were conducted at room
temperature.

###  B.2 TR-MOKE setup

The TR-MOKE measurements utilized a custom-built confocal microscope. A pulsed
800 nm Ti:Sa-Laser was employed. The beam then passed through a pellicle beam
splitter with 92% transmission. Given that the sample’s magnetization would
influence the light’s polarization, a polarizing prism was employed for
polarization component analysis. A Wollaston prism split the returning beam
into two linearly polarized components, each directed towards a differential
detector with two photodiodes. To synchronize the laser with the excitation,
the laser’s cavity length was adjusted to match the repetition rate with a
global frequency reference. This adjustment was facilitated by a control loop
incorporating a photodiode detecting generated pulses, mixed with the
reference signal to produce a down-converted signal for use by a phase lock
loop and proportional-integral-derivative controller. The reference signal,
derived from a 10 MHz rubidium atomic clock underwent frequency multiplication
to achieve the desired 80 MHz value. This reference signal was also utilized
for an arbitrary waveform generator producing MW signals sent to the sample’s
stripline S1 (Fig. 1 a)). The output of the lock-in is referred to as the Kerr
signal $S_{\mathrm{Kerr}}$, while the summed signal of the two photodiodes is
referred to as the topographic signal $S_{\mathrm{Topo}}$.

##  Appendix C Simulation method

To simulate the PL signal resulting from the standing wave pattern of the
total magnetic field above the YIG thin film, the time-averaged magnitude of
total magnetic stray field perpendicular to the NV axis
$<|B_{\mathrm{tot}_{\perp}}|>_{t}$ was calculated, where
$B_{\mathrm{tot_{\perp}}}(t)$ is given by

$B_{\mathrm{tot_{\perp}}}(t)=(B_{\mathrm{sw}}(t)+B_{\mathrm{oe}}(t))_{\perp}.$
(2)

The time dependent spin wave stray field $B_{\mathrm{sw}}(t)$ is given by
Bertelli  _et al._ (2020)

$B_{\mathrm{sw}}(t)=-B_{\mathrm{sw}}\mathrm{Re}[e^{i(k_{y}y-\omega
t)}(\hat{\mathbf{y}}+i\mathrm{sgn}(k_{y})\hat{\mathbf{z}})],$ (3)

$B_{\mathrm{sw}}=\mu_{0}m_{0}(1+\mathrm{sgn}(k_{y})\eta)|\mathbf{k}|de^{-k_{y}z_{0}}$,
where $\mathbf{k}$ is the wave vector, $\omega=2\pi f$ is the frequency,
$\eta$ is the degree of ellipticity, and $t$ is the time. For the parameters,
we used $m_{0}=0.8M_{\mathrm{S}}$ for the magnetization, $z_{0}=1~{}\mu$m for
the distance between the NV layer and the YIG surface, and $\eta=1$ for the
ellipticity. Furthermore, the time-dependent Oersted field of S2 is given by
Appel  _et al._ (2015)

$\displaystyle B_{\mathrm{oe}}(t)$
$\displaystyle=-C\bigg{(}\mathrm{atan}(\frac{w-2x}{2z_{0}})+\mathrm{atan}(\frac{w+2x}{2z_{0}}))\hat{\mathbf{x}}$
(4) $\displaystyle+\frac{1}{2}\big{(}-\mathrm{log}((w-2x)^{2}+4z_{0}^{2})$
$\displaystyle+\mathrm{log}((w+2x)^{2}+4z_{0}^{2})\big{)}\hat{\mathbf{z}}\bigg{)}\mathrm{cos}(\omega
t),$

where $w=30~{}\mu$m is the width of S2 and $C=-(\mu_{0}Jh)/(2\pi w)$. Here,
$J=0.08~{}\mathrm{mA}/(\mu\mathrm{m})^{2}$ is the microwave current density
sent through S2, and $h=250$ nm is the distance between the thin film surface
and S2.

##  Appendix D Calulation of the dispersion

The spin wave dispersion for free magnetization boundary conditions is given
by Bertelli  _et al._ (2020)

$\displaystyle\omega$
$\displaystyle=\gamma\mu_{0}M_{\mathrm{S}}\sqrt{\Omega_{\mathrm{H}}+\alpha_{\mathrm{ex}}k^{2}+1-f(|k_{y}|)}$
(5)
$\displaystyle\cdot\sqrt{\Omega_{\mathrm{H}}+\alpha_{\mathrm{ex}}k^{2}+(k_{y}^{2}/k^{2})f(|k_{y}|)},$

with $\Omega_{\mathrm{H}}=B\mathrm{cos}(\phi)/\mu_{0}M_{\mathrm{S}}$, the
exchange stiffness $\alpha_{\mathrm{ex}}$, the saturation magnetization
$M_{\mathrm{S}}$, the angle $\phi=35^{\circ}$, and

$f(|k_{y}|)=1-\frac{1}{|k_{y}|d}+\frac{1}{|k_{y}|d}\mathrm{exp}(-|k_{y}|d),$
(6)

where $d=200$ nm is the YIG thickness.

##  Appendix E Wavefront measurements in dependence of the external field

Figures 5 and 6 provide additional images of the observed spin wave
wavefronts, captured by the NV setup and the TR-MOKE setup respectively, under
various external fields resulting in different excitation frequencies and
wavelengths.

##  Appendix F TR-MOKE dispersion for fields applied in DE configuration and
along the NV axis

In Fig. 4, the spin wave dispersion extracted for the case where the external
magnetic field is applied in DE configuration (green dots) and along S1 at an
angle of $\phi=35^{\circ}$ relative to the sample plane, aligning it with one
of the four possible NV center axes direction (blue dots), is plotted against
the theory curve (red line). The deviation in the fitted wavelengths arises
from errors in the magnetic field calibration. However, within the bounds of
this error, the measurements closely match the expected theoretical curve,
demonstrating the negligible effect of the perpendicular component of the
external field with respect to the thin film plane.

Figure 4: Dependence of the wavelength of the excited spin waves if the
external magnetic field is applied in DE configuration (green dots) and along
S1 at an angle of $\phi=35^{\circ}$ relative to the sample plane. The
theoretical dispersion curve (red line) is in good agreement with the fitted
wavelengths.

##  Appendix G Fitting method

The wavefronts $A(y)$ of the NV- and TR-MOKE measurements were both fitted by

$A(y)=Ae^{y-y_{0}}\mathrm{sin}\bigg{(}\frac{2\pi}{\lambda}y+\phi_{0}\bigg{)}+a_{\mathrm{off}}+ya_{0},$
(7)

where $A$ is the wavefront amplitude, $y_{0}$ the distance of the X-axis from
S1, $\lambda$ the wavelength of the spin wave, $\phi_{0}$ a phase constant,
$a_{\mathrm{off}}$ a constant for the signal offset, and $a_{0}$ the slope of
the signal.

Figure 5: Spatial PL signal measured above the YIG film, when spinwaves
excited by the frequency $f$, depending on the external applied field
strength, travel through the YIG thin film.  Figure 6: Spatial TR-MOKE signal
measured above the YIG film, when spinwaves excited by the frequency $f$,
depending on the external applied field strength, travel through the YIG thin
film.