# Temperature dependence of the electron and hole Landé $g$-factors in CsPbI3
nanocrystals in a glass matrix

Sergey R. Meliakov1, Evgeny A. Zhukov2,1, Vasilii V. Belykh2, Mikhail O.
Nestoklon2, Elena V. Kolobkova3,4, Maria S. Kuznetsova5, Manfred Bayer2,
Dmitri R. Yakovlev2,1 1P.N. Lebedev Physical Institute of the Russian Academy
of Sciences, 119991 Moscow, Russia  2Experimentelle Physik 2, Technische
Universität Dortmund, 44227 Dortmund, Germany  3ITMO University, 199034 St.
Petersburg, Russia  4St. Petersburg State Institute of Technology, 190013 St.
Petersburg, Russia  5Spin Optics Laboratory, St. Petersburg State University,
198504 St. Petersburg, Russia

###### Abstract

The coherent spin dynamics of electrons and holes in CsPbI3 perovskite
nanocrystals in a glass matrix are studied by the time-resolved Faraday
ellipticity technique in magnetic fields up to 430 mT across a temperature
range from 6 K up to 120 K. The Landé $g$-factors and spin dephasing times are
evaluated from the observed Larmor precession of electron and hole spins. The
nanocrystal size in the three studied samples varies from about $8$ to $16$
nm, resulting in exciton transition varying from $1.69$ to $1.78$ eV at the
temperature of 6 K, allowing us to study the corresponding energy dependence
of the $g$-factors. The electron $g$-factor decreases with increasing
confinement energy in the NCs as result of NC size reduction, and also with
increasing temperature. The hole $g$-factor shows the opposite trend. A model
analysis shows that the variation of $g$-factors with NC size arises from the
transition energy dependence of the $g$-factors, which becomes strongly
renormalized by temperature.

Keywords: Perovskite nanocrystals, CsPbI3, coherent spin dynamics, electron
and hole $g$-factors, time-resolved Faraday ellipticity.

##  I Introduction

The successful synthesis of colloidal nanocrystals (NCs) from the lead halide
perovskites has greatly increased the possibilities for tailoring their
material properties [1, 2], enhancing their attractiveness for applications in
photovoltaics, optoelectronics, electronics, and beyond [3, 4, 5, 6]. Lead
halide perovskite NCs are commonly synthesized by colloidal chemistry in
solutions and can be composed of hybrid organic-inorganic or fully-inorganic
materials. In comparison to the hybrid organic-inorganic perovskite NCs, the
fully-inorganic NCs made, e.g., of CsPbI3, CsPbBr3, or CsPbCl3, show
considerably higher stability in ambient conditions. Further enhanced
stability is gained when the NCs are synthesized in a glass matrix from a melt
[7, 8, 9, 10, 11, 12], which facilitates also their usage in optoelectronic
applications.

Lead halide perovskite NCs demonstrate also interesting spin-dependent
properties [13], which have been addressed by several optical and magneto-
optical techniques. Among them are optical orientation and optical alignment
[14], polarized emission in magnetic field [15], time-resolved Faraday/Kerr
rotation [16, 17, 18, 19, 20], time-resolved differential transmission [21],
and optically-detected nuclear magnetic resonance [20]. The reported spin
dynamics cover wide temporal ranges from a few picoseconds at room temperature
[21] up to tens of nanoseconds for the spin coherence [20] and spin dephasing
[17] times and up to sub-milliseconds for the longitudinal spin relaxation
times [12] at cryogenic temperatures. Among the above-mentioned techniques,
time-resolved Faraday/Kerr rotation is particularly informative, as it
provides comprehensive data on the electron and hole Landé $g$-factor, spin
coherence, spin dephasing and longitudinal spin relaxation. It can reveal
signals from long-living electrons and holes provided by photocharging of NCs
[17, 19]. In CsPbBr3 NCs, carrier spin coherence up to room temperature [16,
19] and its optical manipulation have been demonstrated [18]. The spin mode-
locking effect reported recently for CsPb(Cl,Br)3 NCs in a glass matrix
demonstrates that advanced protocols of coherent spin synchronization can be
implemented in perovskite NCs [20].

A key parameter in spin physics is the Landé $g$-factor, which determines the
Zeeman splitting of charge carriers. We showed recently experimentally and
theoretically that in bulk lead halide perovskites the electron, hole, and
exciton $g$-factors follow universal dependences on the band gap energy [22,
23]. In NCs, additional mixing of the band states by confinement gives a
considerable contribution to the electron $g$-factor causing deviations from
the universal dependence for bulk, but the mixing has only a weak effect on
the hole $g$-factor [24], as predicted theoretically and confirmed
experimentally in low-temperature measurements at $T=5$ K for CsPbI3 NCs in
glass. It is interesting to extend these experiments to higher temperatures,
which could provide additional information on the band structure influence. At
first glance, one would expect that the $g$-factor behaves according to the
temperature shift of the band gap energy. However, it was shown for GaAs and
CdTe semiconductors that the temperature dependence of the electron $g$-factor
may have also other strong contributions [25, 26, 27, 28], which origin is not
yet fully clarified even for conventional semiconductors.

In this paper, we study the coherent spin dynamics of electrons and holes in
perovskite CsPbI3 NCs in glass by time-resolved Faraday ellipticity. The spin
dynamics are measured in the temperature range of $6-120$ K, from which the
electron and hole $g$-factors as well as the spin relaxations times are
evaluated. By analyzing NCs of different sizes in three samples at different
temperatures, we measure the spin properties across a range of optical
transition energies exceeding 100 meV. Our model analysis explains the
observed qualitative trends for the carrier $g$-factor dispersion and its
variation with temperature. Experiment and theory are also in good
quantitative agreement for the hole $g$-factor, but show significant
deviations for the electron $g$-factor, which origins need to be understood
further.

##  II Experimental results

We study experimentally a set of CsPbI3 NCs embedded in a fluorophosphate
glass matrix, namely, three samples with different NC sizes covering the range
of $8-16$ nm. We label them in this paper as the samples #1, #2 and #3. The
photoluminescence and absorption spectra of these samples at the temperature
of $T=6$ K are presented in the ESI Figure S1. The rather wide size dispersion
in each sample together with the change between the samples allows us to cover
the spectral range of exciton transitions of $1.69-1.78$ eV at 6 K (Figure
1d), see also Ref. 24, where samples from the same synthesis procedure were
investigated. By tuning the laser photon energy, we selectively address NCs
with a specific exciton transition energy corresponding to a specific NC size.
For that, spectrally narrow laser pulses with 1 meV width and 1.5 ps duration
are used for excitation.

The time-resolved Faraday ellipticity (TRFE) technique is used to measure the
coherent dynamics of electron and hole spins in a magnetic field to determine
their parameters. This all-optical pump-probe technique exploits polarized
laser pulses [29, 30], where spin-oriented carriers are photogenerated by the
circularly-polarized pump pulses and the dynamics of their spin polarization
are detected through the change of the ellipticity of the linearly polarized
probe pulses [31, 32]. The presented experiments are performed at cryogenic
temperatures in the range of $6-120$ K. Magnetic fields up to 430 mT are
applied in the Voigt geometry, perpendicular to the light wave vector.

###  II.1 Electron and hole spin dynamics at $T=6$ K

Figure 1: Coherent spin dynamics in CsPbI3 NCs at $T=6$ K. (a) TRFE dynamics
(black solid line) of sample #1, measured in $B=350$ mT at the laser photon
energy $E_{\text{L}}=1.703$ eV. Red dashed line shows a fit to the signal,
using eq. (2). The three lower traces show the decomposed contributions to the
signal. (b) Magnetic field dependences of the electron (red circles) and hole
(blue triangles) Larmor precession frequencies. Fits to the experimental data
with eq. (1) (solid lines) gives $|g_{\rm e}|=2.33$ and $|g_{\rm h}|=0.14$.
(c) Magnetic field dependences of the spin dephasing times $T_{2}^{*}$ for
electrons (red circles) and holes (blue triangles). Solid lines show fits with
eq. (3). (d) Spectral dependences of the electron FE amplitude $S_{\rm 0,e}$
for the studied samples #1 (green circles), #2 (red open circles), and #3
(black circles) in $B=430$ mT. Lines are guides to the eye. (e) Spectral
dependences of the electron (circles) and hole (triangles) $g$-factors in the
three studied samples. Solid lines give the calculations for CsPbI3 NCs from
Ref. 24.

An example of the TRFE dynamics measured on sample #1 at $T=6$ K for the laser
photon energy $E_{\rm L}=1.703$ eV is shown at the top of Figure 1a. The
dynamics are measured with the magnetic field $B=350$ mT applied perpendicular
to the light wave vector (Voigt geometry). The TRFE signal contains two
oscillating and one nonoscillating component. The oscillations are observed
only in finite external magnetic field and we assign them to coherent spin
precession of the charge carriers with the Larmor precession frequency
$\omega_{\text{L}}$, which is determined by the $g$-factor and scales with the
magnetic field $B$ according to:

$\omega_{\text{L}}=|g|{{\mu}_{\text{B}}}B/{\hbar}.$ (1)

Here, ${\mu}_{\text{B}}$ is the Bohr magneton and $\hbar$ is the reduced
Planck constant. The decay of the oscillations is described by the spin
dephasing time $T_{2}^{*}$, which in an inhomogeneous ensemble of NCs is
commonly determined by the spread of Larmor precession frequencies. The dashed
red line in Figure 1a shows a fit to the spin dynamics using the following
function:

$A_{\rm
FE}(t)\propto\sum\limits_{i=\text{e,h}}S_{\text{0,i}}\cos(\omega_{\text{L,i}}t)\exp(-t/{T_{\text{2,i}}^{*}})+\\\
+S_{\text{1}}\exp(-t/{\tau_{\rm no}}).$ (2)

Here, $S_{\text{0,e}}$ and $S_{\text{0,h}}$ are the initial light-induced spin
polarizations of electrons and holes, respectively. $S_{1}$ is the initial
spin polarization corresponding to the nonoscillating component.
$T_{\text{2,e}}^{*}$ and $T_{\text{2,h}}^{*}$ are the electron and hole spin
dephasing times, $\tau_{\rm no}$ is the decay time of the nonoscillating
component. The three contributions after decomposition are shown in Figure 1a.
One can see that the nonoscillating component with $\tau_{\rm no}=1.3$ ns has
a very small amplitude so that we do not consider it any further in our
analysis. We assign the fast and slow oscillating components to electron
($\omega_{\text{L,e}}=73.3$ rad/ns, $T_{\text{2,e}}^{*}=100$ ps) and hole
($\omega_{\text{L,h}}=4.0$ rad/ns, $T_{\text{2,h}}^{*}=200$ ps) spin
precession, respectively. We base this assignment on the known dependences of
the carrier $g$-factors on the band gap energy in lead halide perovskite bulk
crystals [22] and on the confinement energy in nanocrystals [24]. The spin
dynamics measured in different magnetic fields on samples #1 and #2 are shown
in the ESI Figures S3 and S4.

The magnetic field dependences of the Larmor precession frequencies are shown
in Figure 1b. As expected, they show linear dependences on the magnetic field
strength without any offset for extrapolation to zero field which may arise
from exchange interaction effects between electron and hole. According to eq.
(1), the slopes of these dependences correspond to the absolute values of the
electron and hole $g$-factors $|g_{\rm e}|=2.33$ and $|g_{\rm h}|=0.14$. It
was shown in Ref. 24, that in CsPbI3 NCs the electron $g$-factor is positive,
while the hole $g$-factor crosses zero in about the studied spectral range and
can be either negative or positive. The TRFE technique does not provide direct
information on the $g$-factor sign. It can, in principle, be identified
through dynamic nuclear polarization detected via TRFE, as demonstrated for
FA0.9Cs0.1PbI2.8Br0.2 crystals, but is very sensitive to temperature and was
observed only at $T=1.6$ K [33]. We show below that the analysis of the
spectral dependence of the hole $g$-factor allows us to conclude that at the
energy of 1.703 eV, $g_{\rm h}$ is negative, i.e. $g_{\rm h}=-0.14$.

It is worth to highlight that the magnetic field dependences of the Larmor
precession frequencies shown in Figure 1b have no offset at zero magnetic
field. Therefore, we can safely assign the measured spin dynamics to
independent resident electrons and holes confined in the NCs and not to
carriers bound in excitons. More arguments along that line can be found in
Refs. 17, 24. The resident carriers in the NCs can appear from long-living
photocharging, where either the electron or the hole from a photogenerated
electron-hole pair escapes from the NC. As a result, some NCs in the ensemble
are charged with electrons, some NCs with holes, while the rest stays neutral.

The magnetic field dependences of the spin dephasing times of electrons
($T_{\text{2,e}}^{*}$) and holes ($T_{\text{2,h}}^{*}$) are shown in Figure
1c. With magnetic field growing from 90 mT to 430 mT, both times decrease,
namely, for the electrons from 260 ps to 90 ps and for the holes from 350 ps
to 200 ps. This behavior is typical for inhomogeneous spin ensembles with a
finite $g$-factor spread $\Delta g$ and can be described by the following
expression [29]:

$\frac{1}{T_{2}^{*}(B)}{\approx}\frac{1}{{T_{2}^{*}}(0)}+\frac{{{\Delta}g}{{\mu}_{\text{B}}}B}{\hbar}.$
(3)

Here, ${T_{2}^{*}}(0)$ is the spin dephasing time at zero magnetic field.
Fitting the experimental data with eq. (3), yields $\Delta g_{\text{e}}=0.25$
for the electrons and $\Delta g_{\text{h}}=0.07$ for the holes. Thus, the
relative spreads are $\Delta g_{\text{e}}/g_{\text{e}}=11\%$ and $\Delta
g_{\text{h}}/g_{\text{h}}=50\%$.

Figure 1d shows the spectral dependence of the electron FE amplitude
$S_{\text{0,e}}$ for the three studied samples. These profiles can be
considered as those of exciton absorption in the inhomogeneous NC ensembles.
They have different widths with the narrowest distribution in sample #1 and
the largest distribution in sample #3. The traces of three samples overlap
spectrally with each other so that they continuously cover the spectral range
of $1.69-1.78$ eV at $T=6$ K, in correspondence to the variation of NC sizes
from about $8$ to $16$ nm [24], where the smallest NCs have the highest energy
due to strongest carrier quantum confinement.

The spectral dependences of the electron and hole $g$-factors are shown in
Figure 1e. Both dependences show monotonic changes with increasing energy. The
electron $g$-factor decreases from $+2.34$ to $+1.72$ and the hole $g$-factor
increases starting from the negative value of $-0.19$ and reaching the
positive value of $+0.36$. The experimental results are in good agreement with
the predictions of the tight-binding calculations from Ref. 24 shown by the
solid lines, which account for the band mixing provided by electron and hole
quantum confinement in CsPbI3 NCs. Note, that in Ref. 24 we measured the same
CsPbI3 NCs, but with a laser system having a high pulse repetition rate of 76
MHz, which has about three orders of magnitude smaller peak power. This
limited the spectral range where the spin parameters could be reliably
determined, e.g., for the hole $g$-factor it was only $1.68-1.72$ eV compared
to $1.68-1.78$ eV in the present study, which limits systematic studies in the
earlier work.

As mentioned above, from the Larmor precession frequency measured in TRFE
experiments one can evaluate only the magnitude of the $g$-factor, but not its
sign. The question about the sign is of particular importance for the hole
$g$-factor, which is expected to cross zero in the considered spectral range.
In the ESI Figure S5 we plot the dependence of $|g_{\rm h}|$ on the laser
photon energy $E_{\text{L}}$, revealing a nonmonotonous behavior: $|g_{\rm
h}|$ decreases in the spectral range from about 1.69 eV to 1.71 eV, but
increases in the range from 1.71 eV to 1.78 eV. As the monotonic dependence
was theoretically predicted [24], we suggest that $g_{\rm h}<0$ for
$E_{\text{L}}<1.71$ eV, crosses zero at $E_{\text{L}}\approx 1.71$ eV, and
$g_{\rm h}>0$ for $E_{\text{L}}>1.71$ eV. We use this reasoning for plotting
the data in Figure 1e.

Figure 2: Temperature dependence of the spin dynamics in sample #1 for $B=430$
mT. (a) TRFE traces at various temperatures from 15 to 90 K. (b) Zoom into
dynamics from panel (a) at 15 K, 25 K, and 55 K to highlight the hole Larmor
precession. (c) Temperature dependence of electron (red circles) and hole
(blue triangles) FE amplitude. (d) Temperature dependence of electron and hole
spin dephasing time $T_{2}^{*}$. Temperature dependence of electron (e) and
hole (f) $g$-factors and corresponding laser photon energies (g).

###  II.2 Temperature dependence of spin dynamics

We next turn to the spin dynamics measured at various temperatures in order to
extract the temperature dependences of the spin related parameters in the
CsPbI3 NCs. As the temperature changes, the band gap of the material also
changes and one needs to tune the laser accordingly to ensure that NCs of the
same size are addressed at the different temperatures. This is difficult for
strongly inhomogeneous samples, like for sample #3. Therefore, we start with
the basic temperature trends for the most homogeneous sample #1. In what
follows, we measure the complete spectral dependences at various temperatures
for all samples.

Figures 2a and 2b show the spin dynamics in sample #1 measured at various
temperatures. The laser photon energy is adjusted to the maximum of the TRFE
amplitude for each temperature, as shown in Figure 2g. At $T\leq 15$ K the FE
dynamics consist of electron, hole and nonoscillating components. The
amplitude $S_{1}$ of the nonoscillating component decreases with temperature,
and at $T\geq 55$ K it is not detected, while the pronounced electron spin
oscillations are seen up to $T=90$ K. The hole spin oscillations almost
disappear around $T=25$ K, when the laser photon energy is equal to 1.71 eV.
Figure 1e shows that the hole $g$-factor crosses zero around this energy. With
further temperature increase, the hole spin oscillations become prominent
again and are observed up to 65 K. Note that on another spot of this sample #1
we can measure spin oscillations up to 120 K, see the ESI Figure S6.

The temperature dependences of the FE amplitudes and spin dephasing times,
evaluated by fitting the spin dynamics with eq. (2), are presented in Figures
2c and 2d, respectively. The FE amplitudes of electrons and holes are close to
each other in the whole temperature range and decrease with increasing
temperature. The electron spin dephasing time shortens from 90 ps at $T=6$ K
to 50 ps at 90 K. The hole spin dephasing time decreases from 200 ps at $T=6$
K to 80 ps at 55 K. We suggest that the shortening of the spin dephasing times
is due to acceleration of the carrier spin relaxation via their interaction
with phonons at elevated temperatures.

###  II.3 Temperature dependence of $g$-factors

The temperature dependences of the electron and hole $g$-factors are shown in
Figures 2e and 2f. They are measured at the maximum of the FE amplitude, which
in sample #1 shifts from 1.703 eV at 6 K to 1.759 eV at 90 K, see Figure 2g.
The electron $g$-factor decreases with temperature from +2.34 to +1.82. The
hole $g$-factor is equal to $-0.14$ at $T=6$ K, crosses zero at about 30 K
($E_{\text{L}}\approx 1.710$ eV), and reaches $+0.34$ at 65 K
($E_{\text{L}}=1.725$ eV). This behavior is in line with the trend in the
spectral dependences of the $g$-factors shown in Figure 1e. Namely, with
increasing energy, independent of whether achieved by laser energy tuning or
temperature shift, the electron $g$-factor decreases and the hole $g$-factor
increases.

For closer insight into the mechanisms underlying the temperature dependence
of the carrier $g$-factors we measure the spectral dependences of the
$g$-factors at various temperatures. These results for sample #3 with the
largest spectral broadening are shown in Figure 3a for temperatures of 6, 50,
80, and 120 K. The corresponding spectral profiles of the FE amplitude are
given in Figure 3b. One sees that the total covered energy range increases up
to $1.68-1.83$ eV due to tuning with temperature. All gained experimental
results on the carrier $g$-factors for the three studied samples and several
temperatures from 6 to 120 K are collected in Figure 4. They are in agreement
with the conclusions drawn for sample #1. Namely, the electron $g$-factor
decreases with increasing energy, while the hole $g$-factor increases.

Figure 3: Spectral and temperature dependences of the charge carrier
$g$-factors in CsPbI3 NCs of sample #3. (a) Spectral dependences of the
electron (circles) and hole (triangles) $g$-factors measured at the
temperatures of 6 K (black), 50 K (purple), 80 K (red), and 120 K (green).
Solid lines are calculations for $T=6$ K from Ref. 24. (b) Electron TRFE
amplitudes at the corresponding laser photon energies and temperatures.
Figure 4: Spectral dependences of the electron (a) and hole (b) $g$-factors in
samples #1 (open triangles), #2 (closed triangles) and #3 (squares) at the
temperatures of 6 K (black symbols), 30 K (blue symbols), 50 K (purple
symbols), 80 K (red symbols), and 120 K (green symbols). Solid lines are
calculations for $T=6$ K from Ref. 24.

##  III Modeling and discussion

Let us analyze the factors that contribute to the temperature dependence of
the electron and hole $g$-factors in CsPbI3 NCs. As starting point we take the
universal dependence of carrier $g$-factors on the band gap found
experimentally and confirmed theoretically for lead halide perovskite bulk
crystals at $T=5$ K [22]. We recently extended these studies to address the
role of quantum confinement of charge carriers in NCs on the $g$-factors. The
calculations predict that in NCs the hole $g$-factor follows the universal
dependence, but the electron $g$-factor significantly deviates from the bulk
trend [24]. We demonstrated this expectation experimentally for CsPbI3 NCs at
$T=5$ K in Ref. 24 and confirmed it in the present study for a much wider
spectral range, see Figure 1e. At cryogenic temperatures, good agreement
between experiment and theory is found, which is remarkable, as the band
parameters for lead halide perovskite crystals and NCs are not known with high
precision.

To model the temperature modifications of the $g$-factors, information on the
temperature dependence of the band parameters, the band gap energy, the band
dispersion determining the carrier effective masses, the band mixing at finite
wave vectors for confined carriers, etc. need to be known. Also carrier-phonon
polaron formation may contribute. At present, the corresponding available
information is limited. Also, the understanding of the band structure details
of the lead halide perovskite crystal and NCs are far from being complete.
Therefore, it is not possible to perform precise calculations for the
temperature dependence of the $g$-factors. Instead, we perform estimations
based on the available band structure information, which details can be found
in the ESI section S5. Our goal is to explore what factors are of importance
and how close the model predictions can describe the experimental results.

We analyze the relatively small changes of the $g$-factors with energy, which
arise from the temperature variations. Therefore, we consider them in the
linear regime. For that the results of Ref. 24 may be compiled in the
following trends:

$\frac{\partial g_{e}}{\partial E_{g}}=\zeta_{e}^{b}\,,\;\;\;\frac{\partial
g_{e}}{\partial E_{e}}=\zeta_{e}^{qc}\,,\;\;\;\frac{\partial g_{h}}{\partial
E_{h}}\approx\frac{\partial g_{h}}{\partial E_{g}}=\zeta_{h}\,.$ (4)

Here $E_{g}$ is the bulk band gap, $E_{e,h}$ are the energies of quantum
confinement of electrons and holes. Atomistic calculations predict that
$\zeta_{e}^{qc}\gg\zeta_{e}^{b}$, which is in good agreement with experimental
data. Moreover, this result is qualitatively reproduced in k$\cdot$p
calculations [24].

Figure 5: (a) Electron $g$-factors calculated in the empirical tight-binding
model following Ref. 24 for CsPbI3 NCs at different temperatures, see text for
details. (b) Black and green open dots show the electron $g$-factors measured
for sample #3 for the temperatures of 6 K and 120 K, respectively. Solid black
and dashed green lines show the $g$-factors calculated in the empirical tight-
binding model, see panel (a).

The temperature variation changes both the band gap and the quantum
confinement energy. The bulk band gap change is linear with temperature [34].
The quantum confinement energy varies due to temperature expansion of the
lattice and, therefore, the NC size and also due to the temperature variation
of the carrier effective mass. The temperature expansion coefficient for
CsPbI3 is almost independent of the crystal phase. The temperature dependence
of the effective masses is not well analyzed in the literature. In a simple
two-band model, the effective mass of electrons and holes should increase with
increase of the bulk band gap [34].

The hole $g$-factor as function of energy is expected to follow the bulk
trend, as we showed in Ref. 24. Quantum confinement does not change this
result qualitatively. In accordance with the theory predictions, the
temperature shifts the energies to larger values and the $g$-factors slightly
increase, see Figures 3a and 4b. This prediction is supported by the measured
data.

For electrons, however, the situation is more complicated. We calculate the
expected change of the electron $g$-factor with temperature increase,
following the procedure from Ref. 24 with temperature-dependent values of the
lattice constant and band gap. The variation of the band gap was accounted by
a change of the tight-binding parameter $E_{pc}$ fitted to reproduce the
experimental value $\frac{\partial E_{g}}{\partial T}=3.1\times 10^{-4}$ eVK-1
from Ref. 34, which closely match the temperature shifts that we measured in
the studied NCs, see ESI Figure S2. We also consider the (almost negligible)
change of the lattice constant $\frac{\partial a_{0}}{\partial T}=3.39\times
10^{-5}$ $a_{0}$ K-1 [35]. In Figure 5a we show the calculated electron
$g$-factors for different temperatures. From the atomistic calculations, it
follows that the temperature variation results in a band gap change, which is
reflected by a shift of the $g$-factor values following the universal
dependence [22].

In Figure 5b we compare the calculated variation of the $g$-factor with
experimental data for temperatures of 6 K and 120 K. One can see, that at
$T=6$ K the calculations and experiment are much closer to each other than at
120 K. In theory, the main effect of a temperature increase is the change of
the $g$-factor by the band gap energy, which roughly follows a universal
dependence, with small changes of the slope of the electron $g$-factor as
function of transition energy. In contrast, the experimental data demonstrate
a strong renormalization of the electron $g$-factor with temperature (larger
than expected from the theory) and a much smaller slope of the electron
$g$-factor as function of the transition energy at elevated temperatures, see
the ESI Figure S7.

##  IV Conclusions

We have studied the spin dynamics of charge carriers in CsPbI3 perovskite
nanocrystals of different size by means of time-resolved Faraday ellipticity
in the temperature range from 6 K up to 120 K. The spectral dependences of the
electron and hole $g$-factors correspond well to model predictions accounting
for the mixing of the electronic bands with increasing confinement energy,
which is accompanied by a decrease of the NC size. With increasing
temperature, the NC optical transition shifts to higher energies due to
increase of the band gap energy. The $g$-factors are independent of
temperature in the studied range when they are measured at the same energy. An
increase in temperature shifts the spectral dependences of the $g$-factors to
higher energies, qualitatively following the model predictions. Namely, the
electron $g$-factor decreases and the hole $g$-factor increases with growing
energy as result of an increase in temperature. It is interesting and
important to check further whether this trend holds also for higher
temperatures up to room temperature as well as in other lead halide perovskite
NCs. As follows from the comparison, the experimentally observed $g$-factors
show a stronger dependence on energy. The effect of quantum confinement on the
$g$-factor value is strongly renormalized by the temperature variation.
Understanding of these details and responsible mechanisms would allow one to
refine band parameters for the lead halide perovskites and their NCs.

##  V Samples and Methods

###  V.1 Samples

The studied CsPbI3 nanocrystals embedded in fluorophosphate Ba(PO3)2-AlF3
glass were synthesized by rapid cooling of a glass melt enriched with the
components needed for the perovskite crystallization. Details of the method
are given in Refs. 11, 20. The samples of fluorophosphate (FP) glass with the
composition BaI2-doped 35P2O5–35BaO–5AlF3–10Ga2O3–10PbF2–5Cs2O (mol. %) doped
with BaI2 were synthesized using the melt-quench technique. The glass
synthesis was performed in a closed glassy carbon crucible at the temperature
of $T=1050^{\circ}$C. About 50 g of the batch was melted in the crucible for
30 minutes, then the glass melt was cast on a glassy carbon plate and pressed
to form a plate with a thickness of about 2 mm. Samples with a diameter of 5
cm were annealed at the temperature of $50^{\circ}$C below
$T_{g}=400^{\circ}$C to remove residual stresses. The CsPbI3 perovskite NCs
were formed from the glass melt during the quenching. The glasses obtained in
this way are doped with CsPbI3 NCs. The dimensions of the NCs in the initial
glass were regulated by the concentration of iodide and the rate of cooling of
the melt without heat treatment above $T_{g}$. Three samples were investigated
in this paper, which we label #1, #2 and #3. Their technology codes are EK31,
EK7 and EK8, respectively. They differ in the NC sizes, which is reflected by
relative spectral shifts of their optical spectra. Note, that samples from the
same synthesis (same codes) were investigated in Ref. 24, even though their
optical spectra slightly differ due to spatial inhomogeneity.

The change of the NC size was achieved by changing the concentration of iodine
in the melt. Due to the high volatility of iodine compounds and the low
viscosity of the glass-forming fluorophosphate melt at elevated temperatures,
an increase in the synthesis time leads to a gradual decrease in the iodine
concentration in the equilibrium melt. Thus, it is possible to completely
preserve the original composition and change only the concentration of iodine
due to a smooth change in the synthesis duration. Glasses with NC sizes in
range $8-16$ nm were synthesized using different synthesis time. Glasses with
the photoluminescence lines centered at 1.801, 1.808 and 1.809 eV (room
temperature measurements) were synthesized within 40, 35, and 30 min,
respectively. NC sizes are evaluated from spectral shift of exciton line in
absorption spectra.

###  V.2 Time-resolved Faraday ellipticity

To study the coherent spin dynamics of carriers we use a time-resolved pump-
probe technique with detection of the Faraday ellipticity (TRFE) [29, 32].
Spin oriented electrons and holes are generated by circularly polarized pump
pulses. The used laser system (Light Conversion) generates pulses of 1.5 ps
duration with a spectral width of about 1 meV at a repetition rate of 25 kHz
(repetition period 40 ${\mu}$s). The laser photon energy is tuned in the
spectral range of $1.65-1.85$ eV in order to resonantly excite NCs of various
sizes at various temperatures. The laser beam is split into the pump and probe
beams with the same photon energies. The time delay between the pump and probe
pulses is controlled by a mechanical delay line. The pump beam is modulated
with an electro-optical modulator between ${\sigma}^{+}$ and ${\sigma}^{-}$
circular polarization at a frequency of 26 kHz. The probe beam is linearly
polarized. The Faraday ellipticity of the probe beam, which is proportional to
the carrier spin polarization, is measured as function of the delay between
the pump and probe pulses using a balanced photodetector connected to a lock-
in amplifier synchronized with the modulator. Both pump and probe beams have
power of 0.5 mW and spot sizes of about 100$~{}\mu$m. For the time-resolved
measurements the samples are placed in a helium-flow optical cryostat and the
temperature is varied in the range of $6-300$ K. Magnetic field up to 430 mT
is applied using an electromagnet perpendicularly to the laser beam (Voigt
geometry, $\textbf{B}\perp\textbf{k}$).

## Conflicts of interest

There are no conflicts of interest to declare.

## Data availability

The data supporting this article have been included as part of the
Supplementary Information.

## Acknowledgments

Research performed at the P. N. Lebedev Physical Institute was financially
supported by the Ministry of Science and Higher Education of the Russian
Federation, Contract No. 075-15-2021-598. E.V.K. and M.S.K. acknowledge the
Saint-Petersburg State University (Grant No. 122040800257-5).

AUTHOR INFORMATION

Corresponding Authors
Sergey R. Meliakov, Email<EMAIL_ADDRESS>
Dmitri R. Yakovlev, Email<EMAIL_ADDRESS>

ORCID
Sergey R. Meliakov 0000-0003-3277-9357
Evgeny A. Zhukov: 0000-0003-0695-0093
Vasily V. Belykh: 0000-0002-0032-748X
Mikhail O. Nestoklon: 0000-0002-0454-342X
Elena V. Kolobkova: 0000-0002-0134-8434
Maria S. Kuznetsova: 0000-0003-3836-1250
Manfred Bayer: 0000-0002-0893-5949
Dmitri R. Yakovlev: 0000-0001-7349-2745

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## Supplementary Information

Temperature dependence of the electron and hole Landé $g$-factors in CsPbI3
nanocrystals in a glass matrix

Sergey R. Meliakov, Evgeny A. Zhukov, Vasilii V. Belykh, Mikhail O. Nestoklon,
Elena V. Kolobkova, Maria S. Kuznetsova, Manfred Bayer, Dmitri R. Yakovlev

###  S1 Photoluminescence and absorption spectra of CsPbI3 NCs

The photoluminescence (PL) spectra are excited by a continuous-wave laser
operating at the wavelength of 405 nm (3.06 eV) with a power of 0.5 mW. The
absorption spectra are measured using a Cary 6000i UV Vis-NIR
spectrophotometer. The spectra measured at the temperature of $T=6$ K for the
three studied samples are presented in Figure S1. The shift to higher energies
among the samples results from a decrease of the NC size from sample #1
towards sample #3. The significant width of the PL emission lines and of the
exciton absorptions peaks indicates a significant dispersion of the NC sizes
in each sample. For all samples, a Stokes shift of the PL maximum from the
absorption maximum is observed.

Figure S1: Normalized photoluminescence (blue line) and absorption (red line)
spectra of the studied CsPbI3 NCs: (a) sample #1, (b) sample #2 and (c) sample
#3. $T=6$ K.

Figures S2(a,b,c) show the absorption spectra of the samples under study,
measured for temperatures from 4 K up to 150 K. The edge of the absorption for
all samples shifts to higher energies with increasing temperature. For samples
#2 and #3 the dependence of the shift on temperature, presented in Figures
S2(e,f), was evaluated from the shift of the maximum of the exciton absorption
peak. For sample #1, the exciton peak is significantly broadened so that its
maximum is almost impossible to determine precisely. Therefore, the
temperature shift of the absorption edge was evaluated from the position of
the half-maximum in the absorption edge. It should also be noted that for all
samples a significant shift was observed only at temperatures above $T\simeq
12$ K. At high $T$, the shift increases almost linearly with temperature. This
is demonstrated by the linear fits of the dependences in Figures S2(d,e,f).
The slopes of these lines are: $3.2\times 10^{-4}$ eVK-1 (#1), $3.8\times
10^{-4}$ eVK-1 (#2), and $3.5\times 10^{-4}$ eVK-1 (#3).

Figure S2: Absorption spectra of sample #1 (a), #2 (b), and #3 (c) in the
temperature range of $T=4-150$ K. (d),(e),(f) Dependences of the shift of the
absorption edge on temperature for samples #1, #2 and #3, respectively. Red
lines are linear fits to the measured dependences with parameters given in
text.

###  S2 Spin dynamics in different magnetic fields

Figure S3a plots the FE dynamics in sample #1 at $T=6$ K for the laser photon
energy of 1.722 eV. The spot on the sample on which the measurements are
carried out differs from the spot corresponding to Figure 1. Here, the hole
spin precession is almost invisible. Therefore, it is possible to evaluate
only the electron spin parameters. Figure S3b shows the magnetic field
dependence of the electron Larmor precession frequency. A fit to these data
with eq. (1) gives the electron $g$-factor $g_{\rm e}=2.26$. The magnetic
field dependence of the electron spin dephasing time $T_{\text{2,e}}^{*}$ is
presented in Figure S3c. $T_{\text{2,e}}^{*}$ decreases with growing magnetic
field from about 800 ps at zero field to 200 ps at the maximum magnetic field
of 430 mT. Fitting the data with eq. (3) gives the electron $g$-factor spread
$\Delta g_{\rm e}=0.17$.

Figure S3: Magnetic field dependence of spin dynamics at $T=6$ K for laser
photon energy $E_{\text{L}}=1.722$ eV in sample #1. (a) TRFE dynamics measured
in various magnetic fields from 3 mT up to 430 mT. (b) Magnetic field
dependence of the electron Larmor precession frequency. Line shows a fit with
eq. (1). The slope of the fit corresponds to $g_{\rm e}=2.26$. (c) Magnetic
field dependence of the electron spin dephasing time. Fitting the experimental
data with eq. (3) (line) yields $\Delta g_{\rm e}=0.17$.  Figure S4: Magnetic
field dependence of the spin dynamics in sample #2 measured at $T=6$ K for
$E_{\text{L}}=1.703$ eV. (a) FE dynamics measured in various magnetic fields
from 180 mT up to 430 mT. (b) Dependences of the hole (blue triangles) and
electron (red circles) Larmor precession frequencies on magnetic field. Lines
show fits using eq. (1). The slopes of the fits correspond to $g_{\rm e}=2.29$
and $g_{\rm h}=-0.13$. (c) Magnetic field dependences of the electron (red
circles) and hole (blue triangles) spin dephasing times. Fits to the
experimental data with eq. (3) (lines) yield $\Delta g_{\rm e}=0.36$ and
$\Delta g_{\rm h}=0.10$.

Figure S4 plots similar data for sample #2, measured at $E_{\text{L}}=1.703$
eV. Here we observe both electron and hole spin precession. The slopes of the
Larmor precession frequency dependences on magnetic field give $g_{\rm
e}=2.29$ and $g_{\rm h}=-0.13$ (the sign of the hole $g$-factor is taken
according to Figure 1e). The decrease of the electron and hole spin dephasing
times with growing magnetic field corresponds to $\Delta g_{\rm e}=0.36$ and
$\Delta g_{\rm h}=0.10$, respectively.

###  S3 Sign of the hole $g$-factor

Figure S5 plots the dependence of the absolute value of the hole $g$-factor on
the laser photon energy $E_{\text{L}}$ (the corresponding spectral dependence
of $g_{\rm h}$ is shown in Figure 1e). $|g_{\rm h}|$ behaves in a nonmonotonic
way with growing energy: it decreases in the spectral range from 1.69 eV to
1.71 eV, is close to zero at $E_{\text{L}}\approx 1.71$ eV, and increases in
the range from 1.71 eV to 1.78 eV. According to Ref. [Nestoklon, et al., Nano
Lett. 2023, 23, 8218] $g_{\rm h}$ should monotonically change in this spectral
range with increasing confinement energy in CsPbI3 NCs, starting from negative
values and rising toward positive values. Thus, we suggest that $g_{\rm h}<0$
at $E_{\text{L}}<1.71$ eV, $g_{\rm h}$ crosses zero at $E_{\text{L}}\approx
1.71$ eV, and $g_{\rm h}>0$ at $E_{\text{L}}>1.71$ eV. We use this hypothesis
for presenting data in the main text, e.g., in Figure 1e.

Figure S5: Spectral dependence of the absolute value of the hole $g$-factor in
samples #1 (green triangles), #2 (red open triangles), and #3 (black
triangles). Error bars correspond to the spreads of the $g$-factor
distribution $\Delta g_{\rm h}\approx 0.07$ for samples #1 and #3 and $\Delta
g_{\rm h}\approx 0.10$ for sample #2.

###  S4 Spin dynamics at temperatures up to 120 K

Figure S6 shows the temperature dependence of the spin dynamics in sample #1.
The Voigt magnetic field is equal to 430 mT. These measurements are performed
on a sample spot different from that used for the measurements in Figure 2.
The laser energy $E_{\text{L}}$ is adjusted to the maximum of the signal for
each temperature. Figure S6a presents the FE dynamics in sample #1 at various
temperatures from 9 K to 120 K. We do not observe hole spin oscillations up to
temperature of about 50 K. Figures S6b and S6c show the temperature
dependences of the electron and hole $g$-factors. Figure S6d shows
corresponding laser photon energies $E_{\text{L}}$. The $g$-factor behavior
with growing temperature is in agreement with the results presented in the
main text.

Figure S6: Temperature dependence of the spin dynamics in sample #1 measured
in the Voigt magnetic field of 430 mT. (a) TRFE traces at various temperatures
from 9 K to 120 K. They are shifted vertically for clarity. (b,c) Temperature
dependences of the electron and hole $g$-factors. (d) Laser photon energies at
which the spin dynamics at various temperatures are measured. (e) Temperature
dependence of the spin dephasing times $T_{2}^{*}$ for electrons (red circles)
and holes (blue triangles). (f) Temperature dependence of the FE amplitudes
for electrons (red circles) and holes (blue triangles).

###  S5 Theory

In this section we estimate the change of the effective band gap, masses, and
$g$-factors in CsPbI3 nanocrystals as function of temperature. We start from
the bulk material parameters. In Ref. [34], a linear change of the band gap as
function of temperature is observed, with the slope for bulk CsPbI3 given by
$\frac{\partial E_{g}}{\partial T}\approx 3.1\times 10^{-4}$ eVK-1. For the
analysis, it is more convenient to use this value divided by the band gap

$\xi=\frac{1}{E_{g}}\frac{\partial E_{g}}{\partial T}\approx 1.8\times
10^{-4}\,{\text{K}}^{-1}\,.$ (S1)

We will also take into account the linear extension of the lattice with
temperature

$\lambda=\frac{1}{a_{0}}\frac{\partial a_{0}}{\partial T}\,,$ (S2)

for CsPbI3 $\lambda\approx 3.39\times 10^{-5}$ K-1 [35].

As mentioned in Ref. [34], the change of the band gap leads to renormalization
of the effective masses of the charge carriers, which may be estimated
assuming that this change is small. In the k$\cdot$p method, the effective
masses of the carriers are [22]:

$\displaystyle\frac{m_{0}}{m_{e}}$
$\displaystyle=1+\frac{2}{3}\frac{p^{2}}{m_{0}E_{g}}\,,$ (S3a)
$\displaystyle\frac{m_{0}}{m_{h}}$
$\displaystyle=-1+\frac{2}{3}\frac{p^{2}}{m_{0}E_{g}}\frac{3E_{g}}{E_{g}+\Delta}\,,$
(S3b)

where $\Delta$ is the spin-orbit splitting of the conduction band. Assuming
all changes to be small, we may estimate the linear slopes of the mass changes
by differentiating Eqs. (S3), giving the result

$\displaystyle\frac{1}{m_{e}}\frac{\partial m_{e}}{\partial T}$
$\displaystyle=\left(1-\frac{m_{e}}{m_{0}}\right)\frac{1}{E_{g}}\frac{\partial
E_{g}}{\partial T}\,,$ (S4a) $\displaystyle\frac{1}{m_{h}}\frac{\partial
m_{h}}{\partial T}$
$\displaystyle=\left(1-\frac{m_{h}}{m_{0}}\right)\left(1-\frac{2\Delta}{E_{g}+\Delta}\right)\frac{1}{E_{g}}\frac{\partial
E_{g}}{\partial T}\,.$ (S4b)

Using the actual parameters of CsPbI3, where $m_{e,h}\ll m_{0}$ and
$\Delta\approx E_{g}$, for the estimations it is safe to approximate like

$\frac{1}{m_{e}}\frac{\partial m_{e}}{\partial
T}\approx\frac{1}{E_{g}}\frac{\partial E_{g}}{\partial
T}\,,\;\;\;\frac{1}{m_{h}}\frac{\partial m_{h}}{\partial T}\approx 0\,.$ (S5)

Note that in Ref. [34] the authors neglect the contribution to the hole mass
from the spin-split electron band and thus severely overestimate its change.

Next, we need to estimate the change of the effective band gap. It may be
evaluated as

$\frac{1}{E_{qc}}\frac{\partial E_{qc}}{\partial
T}=-\left[\frac{1}{m_{e,h}}\frac{\partial m_{e,h}}{\partial
T}+2\frac{1}{L}\frac{\partial L}{\partial
T}\right]=-\left[\xi+2\lambda\right]\,,$ (S6)

where $L$ is the nanocrystal edge length. Since $\lambda\ll\xi$, the change of
the slope of the effective band gap as function of temperature due to
confinement is of the order of $E_{qc}/E_{g}$, which is always small, even for
the smallest NCs.

$\frac{\partial E_{qc}}{\partial T}\bigg{/}\frac{\partial E_{g}}{\partial
T}\sim-\frac{E_{qc}}{E_{g}}\,.$ (S7)

Now we proceed to the $g$-factor value change with temperature. In the
k$\cdot$p method, its value is given by [24]

$\displaystyle g_{h}(E_{g},E_{h})$
$\displaystyle=2-\frac{4}{3}\frac{p^{2}w_{h}}{m_{0}}\left[\frac{1}{E_{g}+E_{h}}-\frac{1}{E_{g}+E_{h}+\Delta}\right]\,,$
(S8a) $\displaystyle g_{e}(E_{g},E_{e})$
$\displaystyle=-\frac{2}{3}+\frac{4}{3}\frac{p^{2}}{m_{0}}\frac{w_{e}}{E_{g}+E_{e}}+\Delta
g_{\mathrm{rem}}-40\frac{m_{0}}{m_{e}}\frac{E_{e}}{\Delta}\,,$ (S8b)

where $\Delta g_{\mathrm{rem}}$ is the correction from the remote bands and
$w_{h}$, $w_{e}$ account for the confinement-induced band mixing, see details
in [24]. In equations above, in the electron $g$-factor we used a rough
estimate for the term arising from the mixing with the spin-orbit split-off
electron band, see Supporting Information of Ref. [24].

From Eq. (S8a) it follows that

$\frac{\partial g_{h}}{\partial E_{g}}=\frac{\partial g_{h}}{\partial E_{h}}$
(S9)

and the slope of the hole $g$-factor dependence on energy does not depend on
the reason of the energy change, including the change as function of
temperature.

For the electron $g$-factor the situation is more complicated. From Eq. (S8b)
it follows that

$\displaystyle\zeta_{e}^{b}\equiv\frac{\partial g_{e}}{\partial E_{g}}$
$\displaystyle=-\frac{4}{3}\frac{p^{2}w_{e}}{m_{0}}\frac{1}{(E_{g}+E_{h})^{2}}\,,$
(S10a) $\displaystyle\zeta_{e}^{qc}\equiv\frac{\partial g_{e}}{\partial
E_{e}}$
$\displaystyle=-\frac{4}{3}\frac{p^{2}w_{e}}{m_{0}}\frac{1}{(E_{g}+E_{h})^{2}}-40\frac{m_{0}}{m_{e}}\frac{1}{\Delta}\,.$
(S10b)

The second term in Eq. (S10b) is responsible for the deviation of the
$g$-factors as function of quantum confinement energy in Fig. 2 of Ref. [24]
from the universal dependence found in Ref. [22].

Next, we want to understand the evolution of the electron $g$-factor as
function of the effective band gap when this change is due to a temperature
change. The electron quantum confinement energy may be estimated as

$E_{e}=\frac{\hbar^{2}}{2m_{e}}\frac{3\pi^{2}}{L^{2}}$ (S11)

and

$\frac{\partial E_{e}}{\partial T}=-E_{e}\left[\xi+2\lambda\right]\,.$ (S12)

Let us assume a small change of the temperature ${\rm d}T$, resulting in a
change of the electron $g$-factor

${\rm d}g_{e}=\left[\frac{\partial g_{e}}{\partial E_{g}}\frac{\partial
E_{g}}{\partial T}+\frac{\partial g_{e}}{\partial E_{e}}\frac{\partial
E_{e}}{\partial T}\right]{\rm d}T\,,$ (S13)

and the peak energy change

${\rm d}E(T)=\frac{\partial E_{g}}{\partial T}{\rm d}T+\frac{\partial
E_{qc}}{\partial T}{\rm d}T\,.$ (S14)

The value of interest for us is

$\frac{{\rm d}g_{e}}{{\rm
d}E(T)}\approx\zeta_{e}^{b}\left(1+\frac{E_{qc}}{E_{g}}\left(1+2\frac{\lambda}{\xi}\right)\right)-\zeta_{e}^{qc}\frac{E_{e}}{E_{g}}\left(1+2\frac{\lambda}{\xi}\right)\,,$
(S15)

where we used $E_{e},\;E_{qc}\ll E_{g}$.

Note that for CsPbI3 NCs the “quantum confinement” contribution to the
$g$-factor change relative to bulk is large: $\zeta_{e}^{qc}\sim-10$ eV-1,
while $\zeta_{e}^{b}\sim-2$ eV-1. This means that the renormalization of the
electron $g$-factor as a function of energy should be seen already when
quantum confinement is small compared to the bulk band gap value. However, the
change is expected to be small even for small nanocrystals and the bulk trend
should be reproduced, in contrast to the experimental data.

To highlight the difference between the experimental data and the expected
trend, we also analyze the change of the $g$-factor as function of the peak
energy for different temperatures. For a particular sample, this value is
perfectly fitted by a linear function from which $\frac{\partial
g_{e}}{\partial E_{g}}$ may be extracted from the experimental data, see Fig.
S7. As can be seen, this value changes from $-9.2$ eV-1 to $-4.2$ eV-1 when
the temperature changes from 6 K to 120 K.

Figure S7: Black squares show the experimentally measured $\frac{\partial
g_{e}}{\partial E_{g}}$ for different temperatures. Red circles show the same
value estimated from ETB calculations which take into account only the band
gap change and the lattice extension. It is clear that the experimental trend
is not reproduced in the atomistic calculations.

To check the theory expectations, we use the empirical tight-binding (ETB)
method following Ref. 24. To account for the temperature change, we adjust the
parameter $E_{pc}$ to reproduce the band gap change. We also change the
lattice constant. The bulk values of the effective masses and $g$-factors
change qualitatively following Eqs. (S5) and the results of Ref. [22]
respectively. However, the value of the $g$-factor derivative with respect to
$E_{g}$ changes only marginally, see Fig. S7. The striking difference between
theoretical analysis and experimental results show that an important
ingredient is missing in the theoretical analysis.


Further author information: Send correspondence to Alexandre E. Adler, E-mail:
<EMAIL_ADDRESS>
# Modeling sidelobe response for ground-based mm-wavelength telescopes with
the geometrical theory of diffraction

Alexandre E. Adler  The Oskar Klein Centre, Department of Physics, Stockholm
University, SE-106 91 Stockholm, Sweden  Jon E. Gudmundsson  The Oskar Klein
Centre, Department of Physics, Stockholm University, SE-106 91 Stockholm,
Sweden

###### Abstract

Accurate optical modeling is important for the design and characterisation of
current and next-generation experiments studying the Cosmic Microwave
Background (CMB). Geometrical Optics (GO) cannot model diffractive effects. In
this work, we discuss two methods that incorporate diffraction, Physical
Optics (PO) and the Geometrical Theory of Diffraction (GTD). We simulate the
optical response of a ground-based two-lens refractor design shielded by a
ground screen with time-reversed simulations. In particular, we use GTD to
determine the interplay between the design of the refractor’s forebaffle and
the sidelobes caused by interaction with the ground screen.

###### keywords:

Telescopes, Microwave radiation, Polarization, Optical systematics, Physical
optics, GTD, Baffling

##  1 INTRODUCTION

The temperature and polarization anisotropies of the Cosmic Microwave
Background (CMB) have been studied for now nearly thirty years with ever-
increasing precision by space [1, 2, 3], balloon [4, 5, 6], and ground-
based[7, 8] millimetre telescopes. Temperature anisotropies are variations of
a few tens of $\mu K_{\mathrm{CMB}}$, polarization anisotropies are an order
of magnitude weaker. Increasingly precise knowledge of the CMB’s anisotropies
has produced some of the most stringent constraints on our $\Lambda$CDM
cosmological model. This has been experimentally enabled by concurrent
advances in cryogenics, electronics, detector design and optics. As
instruments become more sensitive, the impact of various systematic effects
grows. Improving the accuracy of our optical models represents one key aspect
of the overall effort to control systematic effects.

Optical systems can be modeled using a wide range of techniques. A common and
relatively computationally inexpensive approach is known as geometrical optics
(GO), or more commonly as ray tracing. This method can be used to quickly
identify key properties of common optical systems. However, geometrical optics
does not account for diffraction effects, and therefore this approach is
unable to capture some features in the far-field response of these telescopes.

Current and upcoming experiments searching for faint hypothesized primordial
B-modes in the polarization of the CMB require exquisite control of instrument
non-idealities. These CMB polarimeters typically employ some kind of optical
baffling to shield the primary optics from unwanted sources. Unfortunately,
diffraction caused by such baffling elements can redistribute optical power to
wide angles, thereby solving one problem but introducing another. In order to
properly quantify such trade-offs, we need optical simulations that account
for the wavelike properties of light; this includes a commonly used method
known as physical optics (PO). Some baffling elements however, such as
satellite V-grooves or the screens used by ground-based telescopes, tend to be
electrically large, thereby hindering the use of physical optics simulations
in a reasonable computation time.

In this work, we combine physical optics simulations with a modified ray-
tracing method known as the Geometrical Theory of Diffraction (GTD) [9, 10,
11] to predict the far field beam response of a fiducial ground-based
telescope. Using a Python API that communicates with a commercial software
named TICRA Tools (formerly GRASP) [12], we generate an ensemble of PO and GTD
simulations.111See https://www.ticra.com/ Similar approaches are described in
Refs. 13, 14, and 15.

This proceeding is organised as follows: Section 2 introduces the mathematical
and physical framework for PO and GTD. Section 3 describes our instrument
design and the parameters for our baffling elements for GTD simulations of
diffraction by a ground screen. The simulation results are then presented and
discussed in Section 4. We conclude in Section 5.

##  2 BASICS OF PHYSICAL OPTICS AND GTD

###  2.1 Physical optics

Physical Optics is a method that allows one to sequentially propagate electric
fields scattered on surfaces with arbitrary geometries and material
properties. An incoming electric field on that surface will induce surface
currents, which will themselves induce a radiated electric field. The
scattered electric field is then the sum of the incoming field and the
radiated one. Physical optics approximates the current induced by the incoming
field at a point on a curved surface with the current induced at the same
point on an infinite flat plane tangent to the curved surface:

${\bf J}_{\textrm{e}}=2{\bf\hat{n}}\times{\bf H}_{\mathrm{inc}},$ (1)

where ${\bf J}_{\textrm{e}}$ is the induced current, ${\bf\hat{n}}$ is the
surface normal, and ${\bf H}_{\textrm{inc}}$ is the magnetic field of the
incoming radiation. For regions that are not illuminated by the source, the
induced currents are assumed to be zero. By suitably sampling the surface, it
is thus possible to establish the field scattered by the object at arbitrary
locations. The spacing of the integration points needs to be smaller than the
wavelength of the illuminating radiation. In some cases, assumptions of
azimuthal symmetry can reduce the number of integration points, however this
simplification is not valid for many systems observing at mm-wavelengths. When
objects become electrically large, PO simulations become computationally
intensive: in those cases, we require alternative methods to make modeling
tractable.

###  2.2 The geometrical theory of diffraction: expanding from the Sommerfeld
solution

In geometrical optics, rays travel along straight lines until they either
reflect or refract. Snell’s law and the Fresnel equations describe the rays
that result from these interactions. In addition to the laws of GO, GTD
introduces rules to create diffracted fields in shadow regions: for instance
in the shadow formed when an aperture truncates a bundle of rays. GTD creates
extra rays to bridge over the discontinuity in that boundary region.
Furthermore, these diffracted rays are generated only by rays incident on
discontinuities of the object, the so-called edge rays that delineate the
boundary between light and shadow in GO. The amplitude of the field for the
diffracted rays is proportional to the amplitude of the diffracted ray at the
edge multiplied by a diffraction coefficient
$D(\mathbf{n}_{1},\mathbf{n}_{2})$ where $\mathbf{n}_{1}$ and $\mathbf{n}_{2}$
are the unit vectors of the incident and diffracted rays. The best known
example of GTD is the Sommerfeld solution for diffraction on a reflecting
half-plane.

We define a reflective half-plane along the $\varphi=-\pi/2$ line in polar
coordinates $(r,\varphi)$, and propagate a plane wave with wavenumber $k$ so
that the wave front is incident on the half-plane reflector with angle
$\alpha$. The angle conventions are shown in the Fig. 1. The diffracted field
amplitude at a distance $r>\\!\\!>k^{-1}$ from the edge is then:

$u_{\alpha}(r,\varphi)=u_{0}e^{i(kr+\frac{\pi}{4})}\frac{1}{2\sqrt{2\pi
kr}}f_{s}(\alpha,\varphi)=\frac{u_{0}e^{i(kr+\frac{\pi}{4})}}{2\sqrt{2\pi
r}}D_{s}(k,\alpha,\varphi)\textrm{,}$ (2)

where $u_{0}$ the amplitude of the field at the edge of the screen,
$D_{s}=k^{-1/2}f_{s}$ is the diffraction coefficient, and $f_{s}$ contains the
angular dependence:

$f_{s}(\alpha,\varphi)=-\frac{1}{\cos{\frac{\varphi-\alpha}{2}}}\mp\frac{1}{\cos{\frac{\varphi+\alpha}{2}}}\>\textrm{.}$
(3)

The second term in the above equation is negative (positive) if the electric
field is parallel (perpendicular) to the edge of the semi-infinite plane. The
solution is not applicable when $\varphi=\pi\pm\alpha$, as this creates
unphysical caustics in the field. This expression, derived by Sommerfeld [16],
accurately describes diffraction on sharp edges. Similar expressions have been
worked out by Keller et al. for diffraction on rounded edges or wedges[9, 10,
11].

The module that provides GTD simulation capabilities in TICRA Tools is called
GRASP. Unfortunately, GRASP provides a version of the GTD approach that only
includes Sommerfeld’s solution. We are therefore currently unable to use its
GTD implementation to model diffraction on rounded edges. As expected,
however, the semi-analytical expressions derived for diffraction on rounded
edges generally predicts diffraction patterns that have lower amplitude
compared to ones found for diffraction on a sharp edge by Sommerfeld. Since
GRASP only implements Sommerfeld’s solution, we expect that our results
represent upper limits compared to experimental configurations that implement
rounded edges. We will end this section by describing how the GRASP GTD
procedure works, to explicit its computational edge over PO.

The GTD method implemented by GRASP starts the set of points in space on which
the output will be computed. From these points, rays are propagated towards
the light source and any specified surface. Rays that encounter a surface get
reflected, either towards the source directly or towards another reflector in
the GTD system. Rays that encounter an edge will generate extra rays following
the rules of GTD. The total number of reflections and diffraction steps
allowed for a given input ray is an adjustable parameter that represents the
maximum order of the simulation. Once the paths from source to output are
established, the field can be computed by adding up the rays, much like in GO.
The obvious advantage over PO comes from not having to compute the equivalent
currents in the objects on which reflections and diffractions occur, using the
computationally easier formulations of GTD. This is beneficial for
electrically large objects such as the surface of a ground screen. The GTD
implementation in GRASP allows surfaces to block ray paths, but ignores
partial refraction. In that respect, PO analysis is superior.

Figure 1: Modulus of the relative diffracted amplitude following Equation 2
for a range of $(r,\varphi)$. The incoming plane wave is assumed to have
incidence $\alpha=\pi/6$. The angle conventions for $\alpha$ and $\varphi$ are
labeled in blue and red respectively. Left (right) panel is for the electric
field parallel (perpendicular) to the plane of incidence. Note the caustics in
each panel, as the solution diverges when $\varphi\pm\alpha=\pi$.

##  3 INSTRUMENT SETUP

###  3.1 Telescope design

We study a simple two-lens silicon refractor telescope, first described in Ref
17, that could be used to observe the degree-scale CMB anisotropies over a
range of frequencies. A simple ray tracing diagram is shown in Fig. 2(a). The
design employs two roughly $310\text{\text{-}}\mathrm{mm}$ diameter silicon
lenses with a maximum sag of about $16\text{\,}\mathrm{mm}$ on the primary
lens. The system has an average effective $f$-number spanning 1.5–1.7 and a
telecentricity angle not exceeding $None$ over the entire field. This design
supports a relatively wide diffraction-limited field of view (DLFOV) of
approximately $None$ up to roughly $350\text{\,}\mathrm{GHz}$. This
corresponds to an active focal plane area with a diameter of approximately
$250\text{\,}\mathrm{mm}$ ($None$/mm plate scale) and more than 2500 physical
pixels, assuming a $6\text{\text{-}}\mathrm{mm}$ pitch size. By employing
dichroic bolometers with two polarization directions, this telescope could
have over 5,000 bolometer channels.

For the purpose of studying diffraction effects as they relate to CMB
polarimeters, our analysis is focused on the performance at
$90\text{\,}\mathrm{GHz}$. However, the API that we have developed for the
purpose of this study can be easily extended to other frequencies used to map
the polarization of the CMB.

###  3.2 Unshielded telescope beams

We generate a linearly polarized Gaussian pattern at $90\text{\,}\mathrm{GHz}$
on the focal plane, with a far-field beam FWHM of $None$. This corresponds to
a conservative $-18\text{\,}\mathrm{dB}$ edge taper on the cold stop, which
should produce relatively low amplitude sidelobe response. The pixel is
polarized such that the electric field is oscillating horizontally relative to
gravity, corresponding to the right panel in Figure 1. We do not consider the
other polarization in this proceeding.

The source can be translated on the focal plane and the maximum field location
(the edge of the focal plane), corresponding to roughly $None$ field angle, is
approximately $125\text{\,}\mathrm{mm}$. The field from the Gaussian source is
propagated through the secondary and the primary lenses using PO. The
equivalent currents on the primary can then be used as a source for PO
analysis of the forebaffle. In a real telescope, passive elements such as
filters and the vacuum window could scatter light. We do not simulate these
effects directly, instead we approximate any scattering by adding one or
several unpolarized Lambertian sources on the sky side of the primary lens, at
the aperture of the forebaffle. Testing indicated that the number of sources
made very little difference for our simulations, as long as their total
intensity remained constant. Therefore, all simulations use a single
unpolarized Lambertian source with optical power that corresponds to
$0.5\text{\,}\mathrm{\char 37\relax}$ of the detector horn illuminating the
secondary lens.

(a) Lenses and Gaussian feedhorn, rays traced to $None$ from axis.

(b) Lenses, and Gaussian feedhorn, Lambertian source and forebaffle

(c) Lenses, source horn, Lambertian source, forebaffle with a curved rim

Figure 2: Parts of the system used for the PO analysis

###  3.3 Baffling the telescope

As our telescope design has a wide field of view, we need to be mindful of
ground pickup, illumination from the Moon and the Sun as well as other
unwanted sources of radiation. To suppress it, our telescope includes a
conical forebaffle mounted beyond the vacuum window and a fixed reflective
ground screen several meters away from the optics. The forebaffle is a
truncated cone of length $h_{\mathrm{fb}}$ and apex half-angle
$\alpha_{\mathrm{fb}}$. The circular aperture on the primary side is situated
$10\text{\,}\mathrm{mm}$ in front of the primary lens and has a radius of
$170\text{\,}\mathrm{mm}$. The inner side of the forebaffle can be either
perfectly reflective or absorbing. We also have the option of a
$5\text{\text{-}}\mathrm{mm}$ thick layer of dielectric meant to mimic
ECCOSORB HR-10[15], with a relative electric permittivity
$\varepsilon_{r}=3.54$ and loss tangent $\delta=0.057$.

To mitigate possible diffraction effects at the outer rim of the
forebaffle[15], a curved flare can be added, whose radius is $n\lambda$, with
$\lambda$ corresponding to the wavelength of our source. The ground screen is,
at its base, a cylinder $7\text{\,}\mathrm{m}$ in radius. At a height $z_{t}$,
the wall starts angling outward at an angle $\beta$ to the vertical. It
reaches a maximum height of $z_{\mathrm{apex}}$. We assume the ground level to
be at the bottom of the ground screen, and the ground within the screen to be
fully absorbing. The telescope rotates in elevation (and azimuth) around the
centre of the secondary lens, which is $1.5\text{\,}\mathrm{m}$ above the
ground.

Figure 3: Schematic view of the telescope and ground screen with relevant
angles distances labeled. The thick solid black line is the ground screen, the
trapezoid represents the forebaffle, the blue elements are the lenses and the
source pixel is the red cone. The purple cone represents a pixel at the edge
of the field of view with a centroid shifted by $None$ relative to the
principal ray. The elevation spherical cut is depicted here inside the ground
screen for ease of view; it is in reality in the far field. The field of view
is indicated by the orange dotted cone.

###  3.4 From the forebaffle to the far-field

The GRASP GTD simulation framework does not allow us to use the lens or the
forebaffle as a direct source. Therefore, we sample the total field generated
by the main beam, the Lambertian source, and the possible scattering by the
forebaffle using a near field grid placed a few centimetres beyond the
aperture of the forebaffle. This grid is a pattern of 24 equally spaced radial
segments, each comprised of 150 uniformly spaced points, extending to
$5\text{\,}\mathrm{cm}$ beyond the edge of the forebaffle’s rim. We chose that
density of points after testing several other configurations: under-sampling
the grid would result in unphysical results. We allow the GTD rays to interact
once with the ground screen between that source and their far-field
destination, an elevation cut spanning from 0 to 135 degrees over the horizon.

We do not want the forebaffle to obstruct our telescope’s field of view, but
we also want it to suppress sidelobes and illumination by bright sources.
These two goals pull the constraints on the forebaffle design in different
directions. We set $z_{t}=$3.64\text{\,}\mathrm{m}$$, $\beta=$$$ and
$z_{\mathrm{apex}}=$5.64\text{\,}\mathrm{m}$$. In this case, the ground
screen’s edge is at $None$ elevation as seen from the center of the secondary
lens, and at $None$ elevation when seen from the center of the primary lens
when the telescope’s elevation is $None$. It disappears from the primary’s
$None$ field of view at a telescope elevation of $None$. We model fully
reflective forebaffles with $\alpha_{\mathrm{fb}}$ either 16, 18 or $None$,
and three forebaffle lengths: 60, 80 or $100\text{\,}\mathrm{cm}$. There is no
curved rim on these forebaffle models. The flare’s radius of curvature is
either 5, 15 or 25 times the wavelength of our main source,
$\lambda\approx$3.3\text{\,}\mathrm{mm}$$. When a flare is implemented, we
extend the forebaffle until its surface is nearly normal to the optical axis.

In the case of ground-based experiments, scan-synchronous noise caused by
sidelobe coupling to warm objects such as the ground, mountains, or near-by
structures can impact data analysis. In general, sidelobe response can pick up
strong signal from point sources and the Galactic plane and create a residual
in maps of regions thought to be relatively clean. As the amplitude of the
sidelobe profile obviously impacts the detector signal, reducing the level of
the sidelobe response is a key design goal of most if not all experiments
observing the CMB. As an example, we note that a $-50\text{\,}\mathrm{dB}$
sidelobe coupling to a 100 square degree object at $300\text{\,}\mathrm{K}$
should cause $\sim$0.1\text{\,}\mathrm{pW}$$ loading on a
$90\text{\text{-}}\mathrm{GHz}$ single-moded $25\text{\,}\mathrm{\char
37\relax}$ frequency bandwidth detector coupled to a
$30\text{\text{-}}\mathrm{cm}$ aperture. This is roughly 5% of the loading
caused by Earth’s atmosphere assuming $1\text{\text{-}}\mathrm{mm}$
precipitable water vapor (PWV) and a $None$ elevation angle. Reducing the
sidelobe amplitude down to -60 dB, $-10\text{\,}\mathrm{dBi}$ assuming a
$50\text{\,}\mathrm{dBi}$ forward gain, reduces that signal amplitude by an
order of magnitude. This is quite significant given that atmospheric loading
tends to dominate loading on ground-based detectors.

##  4 Results

###  4.1 Near-field

Figures 4 and 5 show the near-field beam power patterns for a selection of
forebaffle configurations, normalized to the maximum of the co-polar beam. All
the models depicted are based on either a short forebaffle
$\\{z_{\mathrm{fb}}=$60\text{\,}\mathrm{cm}$,\alpha_{\mathrm{fb}}=$$\\}$ or a
longer, narrower one
$\\{z_{\mathrm{fb}}=$10\text{\,}\mathrm{cm}$,\alpha_{\mathrm{fb}}=$$\\}$. In
Figure 4, we examine the effect of an added curved rim on these two models. We
expect that curving the rim is equivalent to apodizing the aperture of the
forebaffle, smoothing the transition between the beam and the surrounding
space. We plot a cut on our grid: since our optical system has rotational
symmetry around the forebaffle’s axis, a single cut should represent the
physics of the co-polar beam well. We see the apodization happen clearly for
the co-polar beam of the short forebaffle, where the model with the straight
rim has a steep $\sim$15\text{\,}\mathrm{dB}$$i drop around
$38\text{\,}\mathrm{cm}$ from the beam’s centre. That drop gets gradually
attenuated in the models with curved rims: while it is still visible when
$r_{\mathrm{rim}}=5\lambda$, it becomes far less pronounced for
$r_{\mathrm{rim}}=15\lambda$ or $25\lambda$. For the longer, narrower
forebaffle the apodization is not immediately apparent. It is just about
visible at the very edge of the cut, at about $50\text{\,}\mathrm{cm}$ for the
straight-rim configuration. This might be due to the size of our grid: it
extends $5\text{\,}\mathrm{cm}$ beyond the rim in every simulation, and
$5\text{\,}\mathrm{cm}$ represents a greater fraction of the aperture size for
the short forebaffle than for the long one.

Figure 4: A cut of the near-field beam power after the edge of the forebaffle
for the two geometries: on the left,
$(z_{\mathrm{fb}}=$60\text{\,}\mathrm{cm}$,\alpha_{\mathrm{fb}}=$$)$ and on
the right, $(z_{\mathrm{fb}}=1m,\alpha_{\mathrm{fb}}=$$)$. The cut’s size
differs from simulation to simulation, since it is defined relative to the
forebaffle’s aperture size and the curved rim increases it. Power is
normalized so the maximum is $0\text{\,}\mathrm{dB}$. Figure 5: The components
of the near-field that serve as input for a flare-less forebaffle, with
different material properties in each row. The first three columns are for a
forebaffle with a $z_{\mathrm{fb}}=60$cm and $\alpha_{\mathrm{fb}}=$$$. The
last three columns are for a forebaffle with a $z_{\mathrm{fb}}=1$m and
$\alpha_{\mathrm{fb}}=$$$. The distances on the x- and y-axes are in meters.

In Figure 5, we go back to the straight-rim versions of the long and short
forebaffles. We now plot all three components of the squared electric field
following the Ludwig-III polarization convention[18]. We see the cross-polar
component is weaker when the forebaffle is coated with the dielectric
described in Section 3.3. When the forebaffle is modelled as purely absorbing,
that component is nearly uniformly $-45\text{\,}\mathrm{dB}$ for the two
models. Conversely, a quadrupolar pattern visible in the co-polar component of
the reflective forebaffles gets suppressed: for the absorbing forebaffles, the
co-polar beam is very nearly rotationally symmetric.

###  4.2 Far-field

Figure 6: Elevation cut of the far-field beam predicted by GTD simulations
that use tabulated fields at the forebaffle opening as input. The telescope is
at $None$ elevation. Color lightness indicates the different values of
$\alpha_{\mathrm{fb}}$. Different colors are for different forebaffle lengths,
$z_{\mathrm{fb}}$. In the top row, the dashed black line indicates the
elevation of the ground screen’s rim as measured from the centre of the
primary lens, corresponding to $None$. The dotted black line is the edge of
the penumbra for the
$\\{z_{\mathrm{fb}}=$1\text{\,}\mathrm{m}$;\alpha_{\mathrm{fb}}=$$\\}$
forebaffle. The bottom row focuses on the angular region in the ground
screen’s shadow and penumbra. Left column: Stokes I component. Right column:
Stokes U component.

Since the primary lens is finite in size, the angle from the surface of the
primary to the tip of the ground screen has a finite range. We define the
larger of those two angles as the penumbra and the smaller as the shadow. In
some cases, the forebaffle will shield parts of the primary lens from the tip
of the ground screen. In this case, the angular range of the penumbra is
reduced in size. For long forebaffle configurations and/or high telescope
elevation angles, no part of the primary lens will see the tip of the ground
screen. In that case, the shadow is cast by the forebaffle.

We set the telescope’s elevation to $None$, and use the near-field patterns we
described in Section 4.1 as input for our GTD simulations. Sampling the
elevation cut defined in Section 3.4 will help us evaluate the sidelobe
amplitude for different optical configurations. We are aware that the GTD
method implemented by GRASP creates caustics at the location of the main beam
and we ignore this part of the simulation result. Instead, we use the results
of a physical optics model without the ground screen in a $\pm$$$ region
around the main beam. In all results, we normalize power by the forward gain
of a Gaussian beam with a FWHM set by the theoretical diffraction limit of the
$30\text{\text{-}}\mathrm{cm}$ aperture stop; roughly
$50\text{\,}\mathrm{dBi}$ at $90\text{\,}\mathrm{GHz}$.

Figure 7: Elevation cuts of the far-field beam for the
$\\{h_{\mathrm{fb}}=$100\text{\,}\mathrm{cm}$;\alpha_{\mathrm{fb}}=$$\\}$ and
$\\{h_{\mathrm{fb}}=$60\text{\,}\mathrm{cm}$;\alpha_{\mathrm{fb}}=$$\\}$
forebaffles with different flared rim configurations represented by different
shades of the same color. The telescope is set at $None$ elevation. Left
column: Stokes I component. Right column: Stokes U component.

We start by examining the impact of the forebaffle’s length and opening angle.
In Figure 6, we plotted the Stokes I and U components of the beam for a $None$
elevation, with the bottom two panels focusing on the ground screen’s shadow.
Surprisingly, we find that lengthening the forebaffle does not reduce the
Stokes I component in the ground screen’s shadow and penumbra. However,
increasing the opening angle typically results in a larger sidelobe response.
Conversely, for high elevation angles the long forebaffle and wide opening
angles actually helps reduce the Stokes I and U sidelobe amplitude (see
70-$None$ region in Figure 6). This behavior is more consistent with our
expectations.

We find that increasing the length of the forebaffle results in a translation
of the drop-off position in the ground screen shadow. This could be explained
by the fact that the forebaffle solid angle subtended from the ground screen
is larger for a long forebaffle. The Stokes U beam profile is larger for the
longer forebaffle configuration. This is caused by reflections on the
forebaffle which are polarization dependent. Finally, we note a secondary
$5\text{\,}\mathrm{dBi}$ peak in the Stokes I beam profiles at a roughly
106-$None$ elevation, which is due to reflection of the beam off the angled
section of the ground screen. The Stokes U beam also shows a peak at this
location, but the amplitude is roughly $-10\text{\,}\mathrm{dBi}$.

Selecting the configuration
$\\{h_{\mathrm{fb}}=$100\text{\,}\mathrm{cm}$;\alpha_{\mathrm{fb}}=$$\\}$,
hereafter the long forebaffle, and
$\\{h_{\mathrm{fb}}=$60\text{\,}\mathrm{cm}$;\alpha_{\mathrm{fb}}=$$\\}$, the
short forebaffle, for further developments, we look at the impact of a flared
rim. Figure 7 shows that the flare adds some Stokes I beam power at elevations
between $None$ and $None$ for the short forebaffle. The difference is less
significant for the long forebaffle. In the Stokes U beam component, we
observe no clear benefit in adding a flare to the model, with a possible
exception of the $25\lambda$ flare for the short forebaffle configuration; in
which the beam response is larger. As in the previous case. the largest
difference between the flare and the no flare case can be seen at 70-$None$
elevation angles. In this region, we note a significant reduction in sidelobe
power for both the Stokes I and U beam profiles when a flare is added.

What happens for off-axis pixels? We now displace our focal plane source by
$12\text{\,}\mathrm{cm}$ relative to the center pixel, in the same plane as
the elevation cut (see purple source in Figure 3). Given the plate scale, this
corresponds to a beam centroid that is offset by $None$ relative to the
boresight. In Figure 8, we compare results for the centered and off-center
sources. We show the Stokes I and U beam profiles for the forebaffle lengths
previously considered, with $\alpha_{\mathrm{fb}}=$$$. As expected, we observe
that changing the position of the source affects the diffraction profile of
the Stokes I beam in the ground screen’s shadow, with a roughly
$\sim$10\text{\,}\mathrm{dBi}$$ higher power for the edge pixel. If some
elevated terrain, such as a structure or a mountain, were to be situated in
that part of the far-field, then the detector response while scanning would be
significantly different for the two pixels. Outside the shadow, we see that
the off-center pixel has a mirror image on the other side of the boresight of
our telescope caused by reflections in the forebaffle. Furthermore, for the
$1\text{\text{-}}\mathrm{m}$ long forebaffle the secondary peak at $None$
elevation has an echo at $None$.

Figure 8: Elevation cut of the far-field for a forebaffle with
$\alpha_{\mathrm{fb}}=$$$. The telescope is at $None$ elevation. Color shade
indicates the different positions of the source on the focal plane. Different
colors are for different $z_{\mathrm{fb}}$.

In Figure 9, the telescope’s elevation is set to $None$. We are interested in
the effect of forebaffle material properties on the sidelobe response, as
previously mentioned in Section 3.3. As expected, the power outside the main
lobe is generally reduced for the dielectric and fully absorbing models
relative to the reflective forebaffle. This is true both in the ground
screen’s shadow and at high elevations. The only exception is for the Stokes U
beam of the dielectric long forebaffle, which is greater under $None$ than for
the reflective model. The height of the secondary peak at $None$-$None$
doesn’t change significantly, confirming that it is mainly due to direct
reflection of the main beam on the ground screen.

Figure 9: Elevation cut of the far-field for the long and short forebaffle
models, respectively in blue and green. Different shades of the color denote
different material properties. The telescope is at $None$ elevation. Figure
10: Elevation cut of the far-field for the long and short forebaffle models
with an off-center pixel, respectively in blue and green. Different shades of
the color denote different material properties. The telescope is at $None$
elevation, indicated by the red dashed line. The off-center detector’s main
lobe nearly clips the ground screen’s edge.

When the telescope is at $None$ elevation, an edge pixel on the telescope will
come close clipping the ground screen. We now repeat our simulations with an
off-center pixel. The results are visible in Figure 10. Like in Figure 8, we
see that the diffracted power in the ground screen’s shadow is greater than
for the center pixel, but the effect is stronger this time with a
$\sim$25\text{\,}\mathrm{dBi}$$ difference in the sidelobe response for the
center and edge pixels. In this scenario, the absorbing configuration no
longer reduces power at high elevations and we also see significant
differences in behavior between the short and long forebaffles. In the ground
screen’s shadow, the Stokes U beam for the reflective long forebaffle is much
weaker than for the two other configurations. There is an additional feature
at $None$ that is only visible for the long forebaffle. The cause of this
feature is currently unknown.

##  5 CONCLUSION

Optical modeling of CMB telescopes can be done through several complementary
methods. In this paper, we introduced a design for a simple ground-based
telescope that could map the cosmic microwave background on degree angular
scales. We have then used physical optics to simulate the near-field of
primary optics and forebaffle of this telescope at $90\text{\,}\mathrm{GHz}$.
Using GTD, we then examined the effect of a ground screen for varying
forebaffle properties. We found that varying the length and opening angle of
the forebaffle had opposite effects at different elevations, with longer and
wider forebaffles increasing the sidelobe response in the shadow and
decreasing it at high elevations. For the design considered, a flared rim on
the forebaffle was not found to reduce the power in the ground’s screen
shadow. However, we found that it reduces the power at high elevations. As
expected, off-center pixels have a different beam profile in the shadow
compared to centered ones. This implies an elevation dependent variation in
scan-synchronous noise for ground-based telescopes. The beam profile of off-
center pixel is also found to have secondary peaks due to reflection off the
forebaffle. Finally, we see that making the forebaffle absorbing reduces
sidelobe power for a centered pixel, but will have varying effects for an off-
center pixel depending on the elevation.

Although the GTD simulation framework provided by GRASP is unable to capture
diffraction on curved edges, we find that this software is still able to
produce useful models of the far-sidelobe response for a generic experiment
designed to map the degree-scale polarization of the CMB. These results are
all reported for a polarization perpendicular to the plane of incidence as
shown in Figure 1. In future work, we will extend and adopt this modeling
framework to real designs of upcoming CMB experiments.

###### Acknowledgements.

We thank Lyman Page for useful discussions at the beginning of this project.
We would also like to express our gratitude to Per Heighwood Nielsen, Senior
Research Engineer at TICRA, for his assistance in understanding the
capabilities and limitations of the software.

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