# Steering light with magnetic textures

Ioan-Augustin Chioar  Current address: Department of Applied Physics, Yale
University, New Haven 06511, CT, USA.  Department of Physics and Astronomy,
Uppsala University, Box 516, SE-75120 Uppsala, Sweden  Christina Vantaraki
Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120
Uppsala, Sweden  Merlin Pohlit  Department of Physics and Astronomy, Uppsala
University, Box 516, SE-75120 Uppsala, Sweden  Richard M. Rowan-Robinson
Current address: Department of Material Science and Engineering, University of
Sheffield, Sheffield, United Kingdom.  Department of Physics and Astronomy,
Uppsala University, Box 516, SE-75120 Uppsala, Sweden  Evangelos Th.
Papaioannou  Institut für Physik, Martin-Luther-Universität Halle-Wittenberg,
Von-Danckelmann-Platz 2, 06120 Halle, Germany  Björgvin Hjörvarsson
Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120
Uppsala, Sweden  Vassilios Kapaklis<EMAIL_ADDRESS>Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120
Uppsala, Sweden

###### Abstract

We study the steering of visible light using a combination of magneto-optical
effects and the reconfigurability of magnetic domains in Yttrium-Iron Garnet
films. The spontaneously formed stripe domains are used as a field-controlled
optical grating, allowing for active spatiotemporal control of light. We
discuss the basic ideas behind the approach and provide a quantitative
description of the field dependence of the obtained light patterns. Finally,
we calculate and experimentally verify the efficiency of our magneto-optical
grating.

Optical components are used for focusing, filtering, steering and manipulating
the polarization of light. Their properties are typically obtained by the
fabrication of three dimensional objects, having specific refractive indices,
dichroism or birefringerence effectsChen  _et al._ (2020). However, the
mechanical assembly of re-configurable components forming an optical device
can limit its long-term stability and reliability. Flat optics has therefore
been pursued, in an attempt to remedy these shortcomings. The effort has given
rise to a revolution in the field of optics, building upon developments in the
fields of plasmonics, metamaterials and nanofabrication Won (2019); Shaltout
_et al._ (2019a, b); Rho (2020). Metamaterials facilitate the shaping of
optical wavefronts Yu  _et al._ (2011); Yu and Capasso (2014); Chen  _et al._
(2020) through the structuring of near-fields in a designer manner, thereby
offering control of the far-field response Ginis  _et al._ (2020). These can
also be reconfigured, using electric and magnetic fields, temperature,
mechanical as well as chemical reactionsShalaginov  _et al._ (2020).
Magnetically controlled metamaterials are of special relevance in this
context, due to their reconfigurability, flexibility in design and the fast
response of opto-magnetic effects Maccaferri  _et al._ (2020); Zvezdin and
Kotov (2020). Therefore, investigating ways in which tailored magnetic
textures can be harnessed for the design of useful optical responses is of
particular interest Zvezdin and Kotov (2020); Costa-Krämer  _et al._ (2003);
Syouji and Tominaga (2013); Mito  _et al._ (2018); Higashida  _et al._
(2020).In fact, related approaches for magnetic holography recording and the
subsequent steering of light were initiated in the 1970s using garnet
materials Johansen  _et al._ (1971); Lacklison  _et al._ (1973); Scott and
Lacklison (1976); Sauter  _et al._ (1977); Krawczak and Torok (1980); Numata
_et al._ (1980); Sauter  _et al._ (1981); Hansen and Krumme (1984), MnBi
alloysChen  _et al._ (1968); Mezrich (1969); Rüll and Kempter (1976) and EuO
Fan  _et al._ (1969).

In light of the metamaterial approach and opportunities, we here revisit the
use of Yttrium-Iron Garnet (YIG) Johansen  _et al._ (1971); Scott and
Lacklison (1976); Sauter  _et al._ (1981) for obtaining one of the most basic
functions of an optical component: deflection of light. The steering is
obtained by applying an external magnetic field, influencing the spontaneously
formed stripe-like magnetic domains, which in turn affect the intensity of the
angular distribution of the transmitted light. We describe the connection
between the magnetic texture of the YIG film and the deflected light,
establishing a description of the relation between the real space magnetic
domain structure and the reciprocal space, as seen from the scattered light
patternsSchmitte  _et al._ (2003); Grimsditch and Vavassori (2004); Vavassori
_et al._ (2004); Arnalds  _et al._ (2010). We also quantify the efficiency of
the YIG-based grating. The ideas and results discussed here can be utilized
for the design and development of a new generation of flat, reconfigurable and
potentially fast opticsMaccaferri  _et al._ (2020) through the control of
magnetic order and textures at the mesoscaleCosta-Krämer  _et al._ (2003)
using thin film technology and nanolithography Wang  _et al._ (2006); Perrin
_et al._ (2016); Nisoli  _et al._ (2017); Östman  _et al._ (2018); Rougemaille
and Canals (2019); Skjærvø  _et al._ (2020).

Figure 1: The magneto-optical grating. (a) Hysteresis loops of the YIG sample
for out-of-plane (upper panel) and in-plane (lower panel) applied magnetic
fields. (b) An in-plane magnetic field couples to the in-plane component of
the magnetization in the YIG film, altering the periodicity $\Lambda$ while
maintaining the ratio $f=s/\Lambda=s^{\prime}/\Lambda^{\prime}$ between the
domains. The full micromagnetic characterization of the YIG film, using
magnetic microscopy is detailed in the Supplementary Videos.

The investigated YIG film ($30\,\rm{mm}\times 3\,\rm{mm}\times 7.3\,\rm{\mu
m}$) was grown by liquid phase epitaxy on a 500 $\mu$m thick Gallium
Gadolinium Garnet (GGG) substrate Agrawal  _et al._ (2014); Schmidt  _et al._
(2020). The angular dependence of the transmitted intensity was determined
using a specially designed magneto-optical diffractometer, based on a
$\theta-2\theta$ goniometer (Huber 424 2-circle goniometer). The sample was
mounted in the center of a quadrupole magnet, providing vectorial magnetic
fields up to 42 mT. The sample was illuminated using a supercontinuum laser
(Fianium SC-400-2), with a wavelength range of 400 - 1100 nm or a
monochromatic laser (Coherent OBIS) with a wavelength of 530 nm and power of
20 mW. Two Glan-Thompson polarizers (Thorlabs GTH10M) were used for setting
the polarization of the incoming beam and for analyzing the rotation in the
detected light. The signal was modulated to allow lock-in detection (SR830)
and recorded using a Si photodiode detector (Thorlabs DET100A). A beam-
splitter was employed for monitoring the intensity of the source, providing
on-the-fly normalization of the intensity of the incoming light. For the field
dependence measurements, the sample was first saturated, ensuring a reset of
the magnetic domain configuration, and thereafter brought to the targeted
field before performing a detector scan. Hysteresis curves were measured for
both in- and out-of-plane applied magnetic fields using a magneto-optical Kerr
effect (MOKE) magnetometer. Finally, a Kerr microscope was used for magnetic
imaging. To image the remanent magnetization state, the samples were first
demagnetized in a time-dependent magnetic field of decaying amplitude. The
microscope data presented here are polar-MOKE (P-MOKE) contrast images in
reflection.

Magnetic stripe domains are formed in the YIG film as shown in Fig. 1,
constituting a one-dimensional grating-like structure for the out-of-plane
magnetization component. The stripe domains can be orientedBručas  _et al._
(2008) along any direction within the sample plane, using external in-plane
magnetic fields (see also Supplementary Videos). The direction of the applied
in-plane magnetic field also affects the magnetic texture by primarily
altering the grating periodicity ($\Lambda$) (Fig. 1b), with the domain ratio
(s) selectively affected, depending on the in-plane field direction and
magnitudeZvezdin and Kotov (2020). For the remainder of this letter, we will
concentrate on the case where in-plane magnetic fields are applied to the YIG
film along the $y$-direction, as defined in Fig. 2.

Figure 2: Schematic description of the magneto-optical scattering setup and
the field dependence of the binary magnetic YIG grating. The YIG film is
illuminated using a supercontinuum laser beam at normal incidence (450 - 900
nm), while placed between two Helmholtz coils, providing a field along the
$y$-direction. The laser beam is linearly polarized before reaching the
sample. An analyzer is positioned in front of the detector, which can be moved
in the $zx$ plane, as depicted on the right side.

The layout of the magneto-optical scattering experimental setup is illustrated
in Fig. 2. The Faraday effect acting upon the light (polarized along the
$y$-direction) transmitted through the YIG film results in a rotation of the
polarization. Having domains of opposite out-of-plane magnetization components
yields rotations of opposite signs. The interference between the light with
opposite rotation of the polarization gives rise to a diffraction pattern,
closely resembling that of a conventional optical grating. However, the
interference arises from the phase difference of the partial waves and not a
modulation in transmitted intensity along the grating direction. Assuming a
linear-response regime, we can further calculate the intensity of the
transmitted diffracted beam though the YIG film. Defining $F$ as the Faraday
rotation and $d$ as the film thickness, the rotation will be $\varphi=Fd$.
Domains of opposite magnetization, rotate the polarization in opposite
directions ($\varphi_{1}=+Fd$ for $M^{+}$ and $\varphi_{2}=-Fd$ for $M^{-}$),
resulting in a periodic modulation of the electric field components.
Consequently, a maximum achievable efficiency in terms of change in beam power
can be estimated, knowing the attenuation coefficient $a$ and by using (see
Supplemental Material for full derivation)Haskal (1970):

$\eta_{max}=\frac{4}{\pi^{2}}e^{-2}\left(\frac{2F}{a}\right)^{2}sin^{2}\frac{\pi
s}{\Lambda}$ (1)

For the YIG film used here: $F=2200$ deg/cm (experimentally determined, see
Supplemental Material), $a$ = 1417 cm-1 (measured absorption coefficient, see
Supplemental Material), resulting in $\eta_{max}=1.6\times 10^{-4}$, for a
wavelength of $\lambda$ = 530 nm. This value is comparable to, yet higher than
certain reported results in the literature, for example MnBi magnetic
gratingsMezrich (1969); Chen  _et al._ (1968); Tanaka  _et al._ (1972); Rüll
and Kempter (1976), yet smaller than the values reported for Bi-substituted
garnet materialsLacklison  _et al._ (1973); Scott and Lacklison (1976); Sauter
_et al._ (1977). It is worth noting that these improvements can imply
additional chemical synthesis complexity and an intricate interplay between
the magnetic properties and film thickness which impacts the angular
deflection window and magneto-optical efficiencyLacklison  _et al._ (1973);
Scott and Lacklison (1976). The actual experimental value of the efficiency
for our YIG film was determined to be $\eta_{exp}=1.47(6)\times 10^{-4}$, in
reasonable agreement with the calculated value.

Figure 3: Wavelength dependence of the magneto-optical scattering and the
remanent magnetic state, where $\Lambda\approx 3.2$ $\mu$m. (a) The
diffraction peaks move when the laser wavelength is changed, as in
conventional optical grating. (b) Fitting the angular position of the first
order peak against the laser wavelength yields the reciprocal lattice unit
value $q_{0}=q/(2\pi)=1/\Lambda$ (see Supplementary Material for details)
shown in the inset, which closely follows the observed periodicity as observed
by magneto-optical microscopy (Fig. 1).

Fig. 3a displays the wavelength and angular dependence of the transmitted
light and in Fig. 3b we show the angular dependence of the intensity at two
wavelengths (660 and 530 nm) along with the position and the intensity of the
first diffraction peak. We note the close to perfect scaling of the angular
position of the peak and the wavelength of the incoming light as well as the
strong wavelength dependence of the intensity of the diffracted light,
reminiscent of the YIG intrinsic magneto-optical activity (see Supplemental
Material). The in-plane field dependence of the diffracted light is
illustrated in Fig. 4. As the applied field is increased, the repeat distance
of the magnetic stripes decreases, leading to an increase of the diffraction
angle for any given order, while a decrease of the intensity is also recorded.
The latter can be traced to Fig. 1b, originating from a reduction in the
P-MOKE contrast as the field increases.

The stripe domains disappear, as does the diffraction, when the sample is
saturated. Starting from remanence and with the field applied parallel to the
sample surface, a linear dependence of the angular position of the first order
diffracted beam with the applied field strength is observed, almost the whole
way up to magnetic saturation, as shown in the inset of Fig. 4. At the same
time the intensity of the diffracted beam is generally decreasing with the
increase in applied field, ultimately reaching zero at magnetic saturation.
The decrease in the diffracted intensity originates from a reduction in the
out-of-plane magnetization, thus decreasing the difference in the rotation of
the polarization angle in the stripe domains. Note that the azimuthal rotation
of the in-plane applied field starting from saturation results in the rotation
of the scattering plane, since the stripe domains form parallel to the new
field direction (see Supplementary Video).

Figure 4: Field dependence of the magneto-optical scattering for in-plane
applied fields along the $y$-direction (Fig. 2). All patterns were recorded
using a wavelength of 530 nm. The inset shows the field dependence of the
position and the intensity of the first diffraction peak. The intensities have
been normalized to the first diffraction order in absence of an applied
magnetic field (0 mT).

Finally, we describe the temporal response of the YIG magneto-optical
diffraction. For this purpose, we used the experimental protocol illustrated
in Fig. 5a. The time-dependent response depends on the field protocol as well
as the position of the detector. Having chosen a detector angle within the
angular position window of the first order peaks, we applied a sine-wave
magnetic field on top of a field offset, effectively driving the sample
between its saturated and remanent states. This results in a time dependency,
as exemplified in the left column of Fig. 5c. Here we notice the large
difference in response, solely arising from the choice of the detector angle.
In fact, the relative positioning of the detector within the first order peak
window produces signals with mixed spectral load and possibilities for beam
modulation options involving tunable weighing of the harmonic content.
Changing the amplitude and the sign of the applied field, i.e. performing
partial or extended loops, is expected to add another degree of freedom,
allowing further tailoring of the spatiotemporal steering of light. An
interesting outlook also resides in the use of driving frequencies beyond the
current quasi-static framework.

Figure 5: Frequency modulation using magneto-optical gratings. (a) Combining
the detector angular placement with a field oscillation at a given amplitude
and frequency, it is possible to alter the spectral intensity content of the
signal as shown in (b) by changing the beam sweeping sequence over the
detector (c). In all cases shown here, an oscillating field of 5 mT amplitude,
5 mT offset and 0.1 Hz frequency was used, with the laser wavelength set to
530 nm. The recorded intensities are for the first diffraction order and
detector angles of $\theta_{1}=8.8^{\circ}$, $\theta_{2}=10.2^{\circ}$ and
$\theta_{3}=11.0^{\circ}$. The intensities have been normalized to the highest
intensity of the first harmonic (0.1 Hz). This highlights the tunable harmonic
content, enabling modulation on frequencies above the field driving frequency.

The concepts discussed here can be used when designing magnetically controlled
flat optical devices. A foreseeable major potential for improvement lies in
the field of magnetic metamaterialsSkjærvø  _et al._ (2020); Nisoli  _et al._
(2017); Maccaferri  _et al._ (2020), where the necessary magnetic domain
structures can be designed and engineered utilizing lithography. For
reasonably large diffraction angles to be achieved, the width of the domains
must be comparable or larger than the wavelength of the light, for which
nanopatterned magnetic metamaterials offer an ideal settingCosta-Krämer  _et
al._ (2003). This can be done in combination with conventional magnetic
materials rather than targeting specific magnetic materials with the required,
intrinsic domain structures, such as the YIG presented here. Additionally, a
variety of magnetic materials suitable for the fabrication of such
metamaterials exhibit all-optical switching properties, where ultra-fast laser
pulses may be used to set the magnetic state in nanoarrays Rowan-Robinson  _et
al._ (2021); Mishra  _et al._ (2021) or films of these materialsMangin  _et
al._ (2014); Ciuciulkaite  _et al._ (2020); Ksenzov  _et al._ (2021). In this
way, light can be acted upon by metamaterial architectures, but also be used
to set this action by writing-in the necessary mesoscopic magnetic structure.
Advanced design and control of such metamaterials will allow for more
intricate schemes of light controlWang  _et al._ (2020); Ksenzov  _et al._
(2021) in terms of the scattering, but also over properties of light such as
angular and orbital momentaBeth (1935, 1936); Allen  _et al._ (1992); Woods
_et al._ (2021), holding strong promises for information technology related
applications.

## Supplementary Material

Supplemental material includes a detailed derivation of the magneto-optical
grating equations and efficiency, along with experimental data on the photon
energy-dependent Faraday rotation and absorption coefficient. We further
present details on the calculation of the scattering patterns for real-space
magnetic microscopy data. Two videos are included, presenting the field
dependence of the magnetic domain structure alongside the resulting reciprocal
space patterns, and the experimentally observed light beam deflections while
applying magnetic fields.

###### Acknowledgements.

The authors acknowledge support from the Knut and Alice Wallenberg Foundation
(project 2015.0060), STINT (project, KO2016-6889) and the Swedish Research
Council (project 2019-03581). The authors would like to express their
gratitude to Prof. G. Andersson for providing guidance with the Kerr
microscopy measurements. V.K. would like to thank Prof. P. M. Oppeneer and
Prof. Alexandre Dmitriev for fruitful discussions.

## Data availability

The data that support the findings are available from the corresponding
authors upon request.

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Supplemental Material: Spatiotemporal steering of light using magnetic
textures

## Equations of the magneto-optical grating

Supplementary FIG. 1: Schematic representation of a binary magnetic grating
with a periodicity of $\Lambda$ and fraction $f=s/\Lambda$.

We construct the magnetic domain grid by using a repetition of the unit box
function $U(x)$, defined as

$U(x)=\begin{cases}1,&x\in[-\frac{1}{2},\frac{1}{2}]\\\
0,&x\notin[-\frac{1}{2},\frac{1}{2}]\end{cases}\vspace{2mm}$ (S1)

which we can further generalize to include an arbitrary offset ($x_{0}$) and
an arbitrary box width ($\Lambda$):

$U\left(\frac{x-x_{0}}{\Lambda}-\frac{1}{2}\right)=\begin{cases}1,&x\in[x_{0},x_{0}+\Lambda]\\\
0,&x\notin[x_{0},x_{0}+\Lambda]\end{cases},\Lambda>0\vspace{2mm}$ (S2)

We can now write the grid as an offset sum of the base, which we in turn
define as an asymmetric square wave, with widths of $f\Lambda$ and
$(1-f)\Lambda$, respectively:

$G(x,\Lambda,f,N)=\sum_{m=0}^{N-1}U\left(\frac{x-m\Lambda}{f\Lambda}-\frac{1}{2}\right)-\sum_{m=0}^{N-1}U\left(\frac{x-(m+f)\Lambda}{(1-f)\Lambda}-\frac{1}{2}\right)\vspace{2mm}$
(S3)

The fraction $f=\frac{s}{\Lambda}\in(0,1)$, in accordance with Supplementary
Fig. 1. Fourier transforming Equation S3 with respect to x yields:

$\begin{split}\mathcal{G}(q,\Lambda,f,N)&=\sum_{m=0}^{N-1}\left(\frac{2}{q}\cdot
e^{iq\Lambda f\left(\frac{m}{f}+\frac{1}{2}\right)}\cdot
sin\left(\frac{q\Lambda f}{2}\right)\right)\\\
&=\sum_{m=0}^{N-1}\left(\frac{2}{q}\cdot
e^{iq\Lambda(1-f)\left(\frac{m+f}{1-f}+\frac{1}{2}\right)}\cdot
sin\left(\frac{q\Lambda(1-f)}{2}\right)\right)\end{split}\vspace{3mm}$ (S4)

To further simplify, we introduce $q_{0}=\frac{q}{2\pi}$ and therefore arrive
at:

$\mathcal{G}(q_{0},\Lambda,f,N)=\frac{1}{\pi q_{0}}\cdot e^{i\pi q_{0}\Lambda
f}\cdot\left(\frac{e^{i2\pi q_{0}\Lambda N}-1}{e^{i2\pi
q_{0}\Lambda}-1}\right)\cdot\left[sin(\pi q_{0}\Lambda f)-e^{i\pi
q_{0}\Lambda}\cdot sin(\pi q_{0}\Lambda(1-f))\right]\vspace{2mm}$ (S5)

The light intensity $I$ will be proportional to:

$\mathcal{I}(q_{0},\Lambda,f,N)=\mathcal{G}\cdot\mathcal{G}^{*}=\frac{1}{\pi^{2}q_{0}^{2}}\cdot\Bigg{(}\frac{sin^{2}(\pi
q_{0}\Lambda N)}{sin^{2}(\pi q_{0}\Lambda)}\Bigg{)}\cdot\\\ \Big{[}sin^{2}(\pi
q_{0}\Lambda f)+sin^{2}\big{[}\pi q_{0}\Lambda(1-f)\big{]}-2\cdot sin(\pi
q_{0}\Lambda f)\cdot sin\big{[}\pi q_{0}\Lambda(1-f)\big{]}\cdot cos(\pi
q_{0}\Lambda)\Big{]}\vspace{2mm}$ (S6)

From Equation S6, the zeroth order ($q_{0}=0$) diffraction peak intensity is
given by:

$\mathcal{I}_{0}(q_{0}=0,\Lambda,f,N)=\Lambda^{2}N^{2}\cdot(1-2f)^{2}=\mathcal{I}_{in}\cdot(1-2f)^{2}\vspace{2mm}$
(S7)

where $\mathcal{I}_{in}=\Lambda^{2}N^{2}$ can be considered as corresponding
to an initial/incident intensity. As expected, for the case of a symmetric
base square wave ($f=\frac{1}{2}$), the zeroth order diffracted intensity
vanishes, $\mathcal{I}_{0}(0,\Lambda,f=\frac{1}{2},N)=0$, along with all even
orders.

In a similar fashion we can estimate the intensity of the first order beam
($q_{0}=\frac{1}{\Lambda}$), which is proportional to:

$\mathcal{I}_{1}(q_{0}=\frac{1}{\Lambda},\Lambda,f,N)=\left[\frac{2\Lambda
N}{\pi}\cdot sin(\pi
f)\right]^{2}=\mathcal{I}_{in}\cdot\bigg{(}\frac{2}{\pi}\bigg{)}^{2}sin^{2}(\pi
f)\vspace{2mm}$ (S8)

Taking now into account the polarization profiles for the $x$\- and
$y$-directions and assuming a normal incidence $y$-polarized beam onto a
sample of thickness $d$, Faraday rotation coefficient $F$ and absorption
coefficient $\alpha$, we get:

$I_{0}=I_{0}^{x}+I_{0}^{y},$

with

$\begin{split}I_{0}^{y}&=I_{in}\cdot e^{-\alpha d}\cdot cos^{2}(Fd),\\\
I_{0}^{x}&=I_{in}\cdot(1-2f)^{2}\cdot e^{-\alpha d}\cdot sin^{2}(Fd)\\\
I_{1}&=I_{1}^{x}=I_{in}\cdot\Big{(}\frac{2}{\pi}\Big{)}^{2}\cdot sin^{2}(\pi
f)\cdot e^{-\alpha d}\cdot sin^{2}(Fd)\end{split}\vspace{3mm}$ (S9)

as there is a grating structure along the $x$-direction and no grating along
the $y$-direction. Following this, the magneto-optical grating efficiency can
be determined:

$\eta=\frac{I_{1}}{I_{in}}=\frac{4}{\pi^{2}}\cdot sin^{2}(\pi f)\cdot
e^{-\alpha d}\cdot sin^{2}(Fd)$ (S10)

This result is identical to the expression provided by HaskalHaskal (1970).
While there are various different approachesFan  _et al._ (1969); Haskal
(1970); Mezrich (1970), the unit box function summation method is very
versatile in defining the grid, which could be made to include finite-sized
domain walls or mesoscopic magnetic textures, as well.

Finally, for small Faraday rotation angles the optimized thickness is
$d=2/\alpha$ and the maximum efficiency is thus given by:

$\eta_{max}=\frac{4}{\pi^{2}}e^{-2}\left(\frac{2F}{\alpha}\right)^{2}sin^{2}(\pi
f)$ (S11)

## Faraday spectra

In order to perform magneto-optical scattering measurements at the visible
wavelength where the YIG magneto-optical activity is largest, we performed
wavelength dependent measurements of the Faraday rotationSchmidt  _et al._
(2020). The measured spectrum is shown in Supplementary Fig. 2 highlighting
the maximum response of the films in the green wavelength region.

Supplementary FIG. 2: Faraday rotation of the YIG film. Spectra dependence of
the Faraday rotation in the visible range. A distinct maximum for wavelengths
corresponding to green wavelength can be seen. Based on this a laser
wavelength of 530 nm was chosen for the scattering studies presented in the
main text.

A mercury lamp was used as a broadband white light source, which in
conjunction with a Newport Cornerstone monochromator, allowed the wavelength
to be swept in 10 nm increments. The output light was focused onto the YIG
sample with polarization parallel to the short width of the YIG strip i.e.
parallel to the stripe domains. The YIG sample was located within an
electromagnet providing fields in the range -1.2 to 1.2 T. In the Faraday
configuration the transmitted light is measured, and the Faraday rotation was
extracted using an analyser - photo-elastic modulator (PEM) combination on the
transmitted beam path, as outlined in the Hinds instruments PEM application
note Oakberg (2010). A Hamamatsu H11901-20 photomultiplier tube was used as
the photodetector. For each wavelength the out-of-plane magnetic field was
swept between a saturating field of $\pm$ 300 mT and a hysteresis loop was
recorded, from which the Faraday rotation was extracted as half the optical
rotation between positive and negative magnetic saturated states.

Supplementary FIG. 3: Real and reciprocal space. Graphical illustration of the
connection between Kerr microscopy and computed reciprocal space patterns,
presented in the main text. The example shown here relates a Kerr microscope
image for the remanent magnetic state of the YIG film, with the resulting
reciprocal space map resulting from a 2D Fourier transformation. The right
panel illustrates the expected scattering pattern and related detector angles
for a wavelength of $\lambda$ = 530 nm.

## Linking real- to reciprocal-space

Utilizing the recorded magneto-optical Kerr effect images from our microscope,
we computed the respective reciprocal space maps for all the applied magnetic
field values. We further compared these to the actual recorded magneto-optical
scattering patterns as shown in Fig. 4 of the main article text. Supplementary
Fig. 3 graphically depicts the process of computing the scattering patterns
from the microscopy data.

Supplementary FIG. 4: Field dependence maps of the magneto-optical scattering,
determined by magnetic microscopy and magneto-optical scattering. Left: Maps
computed from Kerr microscopy images for applied fields applied along the $y$
direction in the range of -8 to 8 mT. Right: Actual field dependence map
resulting from magneto-optical scattering data.

Starting from an in-plane saturated state, a series of Kerr microscopy images
were recorded during a full hysteresis sweep, highlighting the evolution of
the magnetic domain texture in an externally applied field. We performed a
Discrete Fourier Transform for each of these images, obtaining reciprocal
space maps for the corresponding magnetic domain textures, with distinctive
peaks arising from the periodic features of the domain configuration and shown
in Supplementary Fig. 4. Selecting an appropriate region of interest along the
grid vector direction, a projected, one-dimensional reciprocal signal is
generated, which can then be directly compared to the data obtained from the
light scattering measurements. Furthermore, by tracking the position of the
first order peaks as a function of applied field, see Supplementary Fig. 5(a),
the reciprocal lattice unit can be directly extracted. Note that there is an
offset between the two field sweep directions, which is attributed to the
remanence of the Kerr microscope’s electromagnet poles. Assuming a symmetric
linear response of the peak position with the applied field around remanence,
a fact also confirmed by the light scattering measurements (see inset of Fig.
4 of the main text), an absolute value fit function is used to extract the
field offsets. The fitting window is centered around the directly recorded
remanence and its extent was defined by taking the field span for which the
best overall match was achieved for the three fitting parameters using the two
datasets independently.

The real lattice parameters are obtained by inverting the reciprocal lattice
units and their dependence on the applied fields is represented Supplementary
Fig. 5(b). The step-like shape of both plots of Supplementary Fig. 5 is
reminiscent of the discrete nature of the peak positions, measured in number
of pixels.

Supplementary FIG. 5: (a) The reciprocal lattice unit (r.l.u) dependence on
the applied field for the two field sweeps, going from negative to positive
saturation (red, up-triangle data) and from positive back to negative
saturation (blue, down-triangle data). The inset provides the fitting function
used, while the table contains the fit parameters extracted for each sweep,
respectively. The vertical purple lines mark the extent of the fitting window.
(b) The evolution of the real lattice parameter as a function of applied
field. Supplementary FIG. 6: Optical absorption coefficient of the YIG film
studied in this work.

## YIG optical absorption

Supplementary Fig. 6 shows the optical absorption coefficient for the YIG
sample. Values of the coefficient from this dataset were used for the
theoretical estimates of maximum grating efficiency discussed in the main
text.

## References

## References

  * Haskal (1970) H. Haskal, IEEE Transactions on Magnetics 6, 542 (1970).
  * Fan  _et al._ (1969) G. Fan, K. Pennington, and J. H. Greiner, Journal of Applied Physics 40, 974 (1969).
  * Mezrich (1970) R. Mezrich, IEEE Transactions on Magnetics 6, 537 (1970).
  * Schmidt  _et al._ (2020) G. Schmidt, C. Hauser, P. Trempler, M. Paleschke, and E. T. Papaioannou, Physica Status Solidi (b) 257, 1900644 (2020).
  * Oakberg (2010) D. T. C. Oakberg, “Hinds Instruments: Application note, Magneto-optic Kerr Effect” (2010).