# Simple Methods to Represent Shapes with Sample Spheres

Li-Yi Wei  Adobe Research ,  Arjun V Anand  Adobe Inc. ,  Shally Kumar  Adobe
Inc. and  Tarun Beri  Adobe Inc.

(2020)

###### Abstract.

Representing complex shapes with simple primitives in high accuracy is
important for a variety of applications in computer graphics and geometry
processing. Existing solutions may produce suboptimal samples or are complex
to implement. We present methods to approximate given shapes with user-tunable
number of spheres to balance between accuracy and simplicity: touching
medial/scale-axis polar balls and k-means smallest enclosing circles. Our
methods are easy to implement, run efficiently, and can approach quality
similar to manual construction.

element, sampling

††journalyear: 2020††copyright: acmlicensed††conference: SIGGRAPH Asia 2020
Technical Communications; December 4–13, 2020; Virtual Event, Republic of
Korea††booktitle: SIGGRAPH Asia 2020 Technical Communications (SA ’20
Technical Communications), December 4–13, 2020, Virtual Event, Republic of
Korea††price: 15.00††doi: 10.1145/3410700.3425424††isbn:
978-1-4503-8080-5/20/11††submissionid: tcom_108††copyright: acmlicensed††ccs:
Computing methodologies Texturing††ccs: Computing methodologies Shape modeling

##  1\. Background

Natural and man-made objects come into a variety of sizes and shapes. These
geometry shapes play a central role in many scientific and engineering
disciplines. Some shapes can be complex and thus challenging to store and
compute. Thus, how to represent general shapes in sufficiently simple and yet
accurate forms is an important research question.

Ideally, we would like to represent a given shape (e.g., a polygonal mesh
model or a vector graphics design) using a small number of simple primitives
(e.g., spheres) that can approximate the desired properties of the original
shapes with sufficient accuracy. The definition of the latter is often
application dependent, such as collision handling for physics simulation
[Hubbard, 1996], level of details for progressive rendering [Rusinkiewicz and
Levoy, 2000], shadow computation for global illumination [Wang et al., 2006],
control handles for deformation [Jacobson et al., 2011; Phogat et al., 2019],
or element distribution for design [Hsu et al., 2020].

(a) grass

(b) leaf

(c) semiquaver

(d) treble clef

Figure 1. Element representation. The black dots indicate the sample
positions. The radii of the yellow circles represent the sample weights
$\mathbf{w}_{s}$. The blue lines denote the element graphs. The sampling can
be sparse LABEL:sub@fig:element_representation:grass, dense
LABEL:sub@fig:element_representation:leaf, overlapping
LABEL:sub@fig:element_representation:semiquaver or hybrid
LABEL:sub@fig:element_representation:treble. These samples are manually placed
and connected in [Hsu et al., 2020].  Figure 2. Variety in element sampling.
Depending on the design choice, the same element (purple) can be represented
in different ways, such as 1 sample for a rigid component (left), 2 samples
for a mildly deformable shape (middle), and 8 samples to allow further
deformation among the two branches (right).

Due to the importance of this problem, a variety of solutions have been
proposed to approximate shapes with sample spheres [Wang et al., 2006;
Stolpner et al., 2012; Thiery et al., 2013]. The method in [Hsu et al., 2020]
lets users manually place and connect these spheres to reflect their design
intention as illustrated in Figure 1 and Figure 2 that might not be divinable
by automatic computation. However, it might not be reasonable to require all
users to go through this process as it is more natural for them to directly
design the element shapes without thinking about the underlying
implementation. The Puppet Warp tool in Adobe Illustrator has a machine-
learning-based method for automatic pin placement [Phogat et al., 2019], but
it often produces undesirable outcomes, such missing symmetries and leaving
one part of the shape inconsistent with the other (Figure 6).

##  2\. Method

We provide two automatic methods to sample given shapes with spheres. Both
methods are very simple to implement, run fast, and can well represent the
original shapes with a few spheres, whose numbers can be tuned by the users or
the rest of the authoring system as a quality-performance trade off:

$\xi\leftarrow$ user specified spacing ratio $\in[0\;\infty)$

compute MAT/SAT $\mathbf{T}$ of a given element $e$ // [Amenta et al., 2001;
Steenkamp, 2019]

sample set $\mathbf{S}\leftarrow\emptyset$

for  _each polar ball $b\in\mathbf{T}$ in decreasing radius_ do

$include\leftarrow true$ for  _each $b^{\prime}\in\mathbf{S}$_ do

if  _$\left\lVert b_{center}-b^{\prime}_{center}\right\rVert
<\xi\left(b_{radius}+b^{\prime}_{radius}\right)$_ then

$include\leftarrow false$

end if

end for

if  _$include$_ then

$\mathbf{S}\leftarrow\mathbf{S}+b$

end if

end for

return _$\mathbf{S}$_

Algorithm 1:  Element sampling via touching polar balls from a MAT/SAT tree.

(a) semi breve (b) minim (c) semi quaver (d) treble clef

Figure 3. Automatic element sampling results via touching MAT/SAT polar balls.
Users can set an upper limit on the overlap among sample spheres $\xi$ (for
performance reason), and our method can optimize the number of samples, their
locations, and weights. In this example, $\xi=0.8$.

$\mathbf{P}(e)\leftarrow$ uniform point sets from a given element $e$

$B\leftarrow$ user-specified number of samples for $e$

sample set $\mathbf{S}\leftarrow$ random $B$ centers from $\mathbf{P}(e)$ with
$0$ radii

while  _not enough iterations_ do

// compute nearest ball center for each point for  _each $b\in\mathbf{S}$_ do

$\mathbf{P}(b)\leftarrow\emptyset$

end for

for  _each $p\in\mathbf{P}(e)$_ do

$b^{\prime}=argmin_{b}\left\lVert p-b_{center}\right\rVert,\;\forall
b\in\mathbf{S}$ $\mathbf{P}(b^{\prime})\leftarrow\mathbf{P}(b^{\prime})+p$

end for

// compute smallest enclosing circles for  _each $b\in\mathbf{S}$_ do

$b\leftarrow SmallestEnclosingCircle\left(\mathbf{P}(b)\right)$

end for

end while

return _$\mathbf{S}$_

Algorithm 2:  Element sampling via k-means smallest enclosing circles.

(a) semi breve (b) minim (c) semi quaver (d) treble clef

Figure 4. Automatic element sampling results via k-means smallest enclosing
circle. Users can set an upper limit on the number of samples per element (for
performance reason) $B$, and our method can optimize the number of samples,
their locations, and weights for each element. In this example, $B=8$. Notice
the number of samples for different elements are automatically chosen
depending on their shape complexity.

(a) semi breve (b) minim (c) semi quaver (d) treble clef

Figure 5. K-means smallest enclosing circle with at least two samples per
element.

### Touching medial/scale axis polar balls (MAT)

In this method, we sample the medial/scale axis polar balls from a given shape
[Amenta et al., 2001; Steenkamp, 2019], and select a subset in which any two
spheres overlay no more than a user selected threshold $\xi$. In our current
implementation, we set a threshold of overlap between the radii of the two
spheres. For example, if the threshold is $0$, no spheres can intersect. And
if the threshold is $\infty$, all polar balls will be selected. We find it
helpful to select the polar balls in decreasing size so ensure the larger and
thus more representative ones are retained first. The method is summarized in
Algorithm 1. See Figure 3 for examples.

### K-means clustering with smallest enclosing circles (SEC)

In this method, we combine k-means clustering with smallest enclosing circle
[Nayuki, 2018; Wikipedia, 2019] to iterate the 2 steps: find the nearest ball
center for each point on the element, and compute the smallest enclosing
ball/circle from all belonging points (instead of the centroid step in
k-means). The points can be uniformly sampled from the element boundary paths
(including interior holes). The method is summarized in Algorithm 2. Users can
specify a $[min,max]$ range of the number of samples for a given element, and
run our algorithms to decide which number produces the minimal total sample
ball areas. ($min>1$ prevents 1-sample elements that can never deform.) See
Figure 4 and Figure 5 for examples.

As can be seen, the MAT method tends to produce more spheres that all fit
within the original shapes, while the SEC method tends to produce less spheres
that cover the original shapes. Thus, they are more suitable for deformable
and rigid elements, respectively.

### Graph construction

After placing the samples, we can connect them to form element graphs (Figure
1 and Figure 2). For a fully rigid shape, we can connect all pairs of samples
to form a complete graph. To allow (skeletal) deformation, we can connect
along the medial/scale axis for MAT or overlapping/touching sample spheres for
SEC.

##  3\. Refinement

[Phogat et al., 2019]

Our SEC method

Figure 6. Comparison with Adobe Illustrator Puppet Warp automatic pin
placement [Phogat et al., 2019]. The results for [Phogat et al., 2019] were
directly screen-captured and thus have different rendering effects from our
results. Notice that the samples produced via [Phogat et al., 2019] might miss
key parts of the shapes, while our results can provide more balanced coverage.

The SEC method in Section 2 can produce unpredictable results due to random
initialization (e.g., comparing Figure 4b and Figure 5b). We describe a better
initialization method and more principled criteria for choosing the number of
samples.

### Objectives

The desired sampling should satisfy the following goals:

  1. (1)

Sample centers should lie on, or close to, the medial axis of the input shape.

  2. (2)

Overlap among samples should be minimal.

  3. (3)

Samples should occupy minimum external area (i.e., area outside the input
shape), but maximum internal area.

  4. (4)

There should be a low number of adjacent samples. One adjacent sample (for
every sample) is okay, two are good if they lie on opposite sides, more than
two are only desired in ’T’ or ’$+$’ like shapes.

  5. (5)

Samples should symmetrically cover the shape, i.e., samples with left side
outside the shape while right side inside the element are less preferred,
while samples completely inside or equivalently outside on left/right and
top/bottom are more preferred.

Some of the criterion mentioned above might work against each other. Let us
consider the case of a narrow U shaped element versus a wider one. If the
parallel boundary walls of the element are far apart, then having a sample
center at the medial axis would mean large samples. This could increase the
external area they occupy. Thus, point $1$ might work against point $3$. On
the other hand, if we reduce the size of samples (and keep their centers at
medial axis), it could reduce the external area but violate adjacencies. Thus,
we do not directly enforce keeping sample centers at medial axis and rather
achieve that as a side-effect of other criteria. We keep the external area
low, maintain desired adjacency and symmetry to enforce sample centers to be
as close to the medial axis as possible.

### Initialization

The k-means clustering in the SEC method requires a set of points to operate
with. Random initialization can produce unstable results; each time we execute
sampling, a different result is returned. To meet our first objective, we
place points uniformly along the boundary and randomly in the interior of the
input shape, and sort all the points first on y-axis and then on x-axis. From
this sorted point set, we pick up uniformly separated points (like rejection
sampling for dart throwing [Yuksel, 2015]) for initialization. This always
produces stable results and has the tendency to converge faster.

### Optimizing sample count

Let $D_{A}{}$ be the area of the input shape and $r_{min}{}$ be the radius
$\mathbf{w}_{s}$ of the smallest sample that we would like to create. If
$D_{A}{}$ is less than the area of the smallest possible sample $S_{A_{min}}$
(i.e., $\pi{}r_{min}{}^{2}$), we treat that as a special case. We can either
rescale the shape or create one tight fitting sample (ignoring $r_{min}{}$)
located at the center of the shape. Otherwise, we use the following algorithm
to determine the number and locations of samples.

We first find the minimum and maximum possible number of samples. Minimum
count ($N_{min}$) is default to 1 and maximum count ($N_{max}$) is computed as
$\lfloor\frac{D_{A}{}}{S_{A_{min}}}\rfloor$. We linearly scan all
possibilities from $N_{min}$ to $N_{max}$ and find the number of samples that
gives us the minimum cost, defined in Equation 1. At every step (in the search
space $N_{min}$ to $N_{max}$), we perform the SEC method (Algorithm 2), and
calculate the following energy value to identify the minimum that indicates
the desired sample count:

(1) $\displaystyle
C=\mathbf{w}_{o}{}M_{o}+\mathbf{w}_{e}{}M_{e}+\mathbf{w}_{a}{}M_{a}+\mathbf{w}_{y}{}M_{y}$

, where the subscripts $o$, $e$, $a$ and $y$ represent overlap, exterior area,
adjacency, and asymmetry, $\mathbf{w}$ and $M$ represent weight and measure of
the subscripted entities. We normalize all measures on an equivalent scale
$[0\;1]$ to avoid results being skewed towards any term.

$M_{o}$ is computed as total overlapping area of all samples divided by total
area of all samples. Since all samples are circles, this is straight forward
to compute. Note that if two samples overlap, we count the overlapping area in
both samples.

$M_{e}$ is computed as the ratio of total area of samples not inside the shape
to the total area of samples. Note that we do not deal with overlaps here. For
every sample, we just compute how much of it is outside the shape. This
requires circle and triangle intersection evaluations which we perform via
triangle tessellation of the input shape.

$M_{a}$ is computed by finding the number of samples that are overlapping or
adjoining. If a sample has radius $r_{1}$ and another has radius $r_{2}$ and
the separation between their centers is less than $r_{1}+r_{2}$, we count that
as an adjacency. We also use some epsilon tolerance for stable results. The
extreme case is that every sample is adjacent to every other sample. Thus, for
$N$ samples, the maximum possible number of adjacencies $A_{max}$ is given by
$N{}(N-1)$ and for a linearly connected sampling (like in a leaf shape in
Figure 1b), the desired number of adjacencies $A_{min}$ is given by
$2{}(N-1)$. Thus, we can compute $M_{a}$ as following:

(2) $\displaystyle M_{a}=\frac{\lvert A-A_{min}\rvert}{A_{max}}$

, where $A$ is the total count of adjacencies among all samples. Note that if
two samples are adjacent, we count adjacency from both sides. Since it will be
rare to have $A_{max}$ adjacencies in a practical setup, there will be a
higher tendency to have $M_{a}$ less than $M_{o}$, $M_{e}$ and $M_{y}$. Thus,
to counter this, we do two things: use $3{}(N-1)$ as a practical upper bound
for $A_{max}$, and set $\mathbf{w}_{a}$ higher than $\mathbf{w}_{o}$,
$\mathbf{w}_{e}$ and $\mathbf{w}_{y}$. Also note that $M_{a}$ is mostly
immaterial (being zero) when number of samples is $2$. Thus, in that scenario,
we ignore $M_{a}$ and evaluate cost $C$ by considering the other three
measures only.

$M_{y}$ is computed by dividing every sample into four quadrants and measuring
how much asymmetric coverage is exhibited by opposite quadrants (quadrant 1
versus 3 and quadrant 2 versus 4; quadrants are numbered anti-clockwise from
x-axis). We find the number of interior pixels $p$ in each of the four
quadrants and use the following formula to find $M_{y}$:

(3) $\displaystyle M_{y}=\frac{0.5{}\lvert p_{1}-p_{3}\rvert+0.5{}\lvert
p_{2}-p_{4}\rvert}{Q}$

, where $Q$ represents area of a quadrant; $p_{1}$, $p_{2}$, $p_{3}$ and
$p_{4}$ represent interior pixels (i.e. inside the element) in quadrants 1, 2,
3 and 4.

### Band normalization

Different measures ($M_{o}$, $M_{e}$, $M_{a}$ and $M_{y}$) can have different
range of values. Specifically, $M_{a}$ tends to have lower absolute values
than others. On the other hand, the weights ($\mathbf{w}_{o}$,
$\mathbf{w}_{e}$, $\mathbf{w}_{a}$ and $\mathbf{w}_{y}$) associated with these
measures are defined to establish their relative importance and they are going
to behave correctly only when the values of measures are on an equivalent
scale. Thus, we band normalize all the measures before computing the final
cost. For band normalization, we divide each measure by its average value
across all runs from $N_{min}$ to $N_{max}$. We perform band normalization
separately from normalizing all measures to $[0\;1]$ range. However, it is
possible to combine both these in a single step.

Figure 6 compares [Phogat et al., 2019] with our refined SEC method.

##  4\. Concluding Remarks

We describe simple methods to represent given shapes with sample spheres. We
have evaluated our methods for 2D vector graphics applications including
element synthesis [Hsu et al., 2020] and shape warping [Phogat et al., 2019],
and plan to explore applications in 3D modeling. We also plan to perform more
comprehensive analysis and comparisons against other methods.

###### Acknowledgements.

We would like to thank the anonymous reviewers for their valuable feedback.

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# Nonlinear Faraday effect and spin noise in rare-earth activated crystals

A. N. Kamenskii  Experimentelle Physik 2, Technische Universität Dortmund,
44221 Dortmund, Germany  G. G. Kozlov  Spin Optics Laboratory, St. Petersburg
State University, Peterhof, 198504 St. Petersburg, Russia  E. I. Baibekov
Kazan Federal University, 420008 Kazan, Russia  B. Z. Malkin  Kazan Federal
University, 420008 Kazan, Russia  M. Bayer  Experimentelle Physik 2,
Technische Universität Dortmund, 44221 Dortmund, Germany  Ioffe Institute,
Russian Academy of Sciences, 194021 St. Petersburg, Russia  A. Greilich
Experimentelle Physik 2, Technische Universität Dortmund, 44221 Dortmund,
Germany  V. S. Zapasskii  Spin Optics Laboratory, St. Petersburg State
University, Peterhof, 198504 St. Petersburg, Russia

(August 27, 2024)

###### Abstract

The spin-noise spectroscopy (SNS) method implies high efficiency of conversion
of the spin-system magnetization to the Faraday rotation angle. Generally,
this efficiency cannot be estimated using the characteristics of the regular
magneto-optical activity of a paramagnet. However, it may be drastically
enhanced in systems with strong inhomogeneous broadening of the optical
transitions. This enhancement leads to the giant spin-noise gain effect and
previously allowed one to apply the SNS to rare-earth-activated crystals. We
show that the nonlinear resonant Faraday effect can be used to measure the
homogeneous width of the inhomogeneously broadened transition and, thus, to
estimate the applicability of the SNS to this type of paramagnet. We present
the theoretical description of the effect and perform measurements on
intraconfigurational ($4f$-$4f$) transitions of the trivalent rare-earth ions
of neodymium and ytterbium in fluorite-based crystals. The proposed
experimental approach establishes new links between the effects of nonlinear
optics and spin-noise characteristics of crystals with paramagnetic impurities
and offers new ways of research in the physics of impurity crystals.

## Introduction

The method of detecting magnetic resonances in the noise of magneto-optical
activity – the so-called spin-noise spectroscopy (SNS) – has been intensely
developed during the last 15 years [1, 2, 3, 4]. Having been primarily
demonstrated on an atomic system [5], this technique has gained extensive use
since its successful application to semiconductors [6]. In spite of its
apparently lower sensitivity of detecting stochastic (rather than regular)
signals, SNS proved to be highly efficient not only as a conceptually new
method of radiospectroscopy, but also as a peculiar method of optical
spectroscopy, with its abilities substantially exceeding those of conventional
linear spectroscopy [4].

Until recently, all attempts to apply SNS with its unique potentialities to
dielectrics with paramagnetic impurities had failed [7] because of the
smallness of the specific Faraday rotation (Faraday-rotation cross section
[8]) in these systems. This result can be explained by the fact that the
allowed transitions, in activated crystals, are usually strongly broadened,
while spectrally narrow lines are mostly weak. Both these factors affect
unfavorably the Faraday rotation (FR) per unit spin density and the FR noise
power.

In Ref. [9], it was shown, however, that the value of the FR noise (spin-
noise) power detected under conditions of resonant probing is determined not
only by the FR per unit spin density, but also strongly depends on the
homogeneous linewidth, which is hidden inside the inhomogeneously broadened
profile and cannot be revealed in linear optical spectroscopy. By neglecting
this fact, one may significantly underestimate the noise signal for the
following reason. When the inhomogeneously broadened system is probed by a
monochromatic laser beam in the absorption region, the polarization noise
signal is contributed only by the ions of the sample whose resonant
frequencies coincide with that of the laser light to within the homogeneous
width $\gamma$. The fraction of these ions with respect to their total number
in the beam can be estimated as $\gamma/\Gamma$, (here, $\Gamma$ is the
inhomogenous linewidth of the optical transition) and may be extremely small.
Under these conditions, the relative spin fluctuations of these ions may
strongly exceed those of all the ions in the laser beam. It was found that the
gain factor that determines the “enhancement” of the FR noise power is equal
to the ratio of the inhomogeneous linewidth to the homogeneous one and may
cover several orders of magnitude.

In Ref. [10], this idea was successfully applied to crystals activated by
rare-earth (RE) ions, with parity-forbidden intraconfigurational ($f$-$f$)
transitions, whose inhomogeneous linewidths (usually lying in the range of
several GHz) may exceed their homogeneous width by many orders of magnitude.
This giant spin-noise gain effect has allowed us to observe, up to that point
unreported, magnetic resonances in the FR noise spectrum of an impurity
crystal. At the same time, it was found, that among the chosen RE ions and
chosen $f$-$f$ transitions, only some of them appeared to be amenable for
detection of the ground-state magnetic resonance in the FR noise spectrum,
while others, showing similar linear magneto-optics, did not reveal any
noticeable FR noise [10]. For further development of the SNS of impurity
crystals, it is required to find out in more detail the laws of formation of
the spin noise signal. As follows from our previous experimental results [9,
10], reliable information about the applicability of SNS to an object cannot
be obtained from its linear optical or magneto-optical properties. Therefore,
we plan to establish a relation between the magnitude of the spin-noise power
of a paramagnet and its nonlinear magneto-optical characteristics.

In this paper, we study resonant spectra of the nonlinear FR of RE-activated
crystals in the range of the $f$-$f$ transitions and show that the diamagnetic
contribution to the nonlinear FR spectrum (controlled only by the magnetic
splitting of the energy levels) strongly depends on the probe beam intensity
and dramatically differs for lines with different types of broadening. Thus,
we have found that high-resolution spectroscopy of nonlinear FR can be used to
measure, in a single-beam configuration, the homogeneous widths of
inhomogeneously broadened transitions and thus to distinguish the $f$-$f$
transitions capable to reveal the giant spin-noise gain effect.

The paper is organized as follows: after the general motivation in Sec. I we
provide the theoretical background of the expected effect in Sec. II; Sec. III
describes the experimental setup and studied samples, while in Sec. IV we
discuss the results of the measurements on different transition of RE ions;
Sec. V concludes the paper.

##  I General considerations and motivation

The FR spectra of paramagnetic ions in dielectric media are known to be
controlled by two main contributions – diamagnetic and paramagnetic [11, 12].
The first one is determined by the magnetic splitting of optical transitions,
while the second is related to their intensity difference resulting from the
Boltzmann distribution of populations over magnetic sublevels of the ground
state. Correspondingly, the diamagnetic contribution is temperature-
independent, while the paramagnetic one is proportional to the spin-system
magnetization and, in the high-temperature limit, obeys Curie’s law. Without
entering into details of the FR spectra, we note that the diamagnetic
contribution is described by the derivative of the line’s dispersion curve
and, therefore, increases with decreasing linewidth.

Figure 1: (a) and (b) absorption ($D$) and refraction ($n$) spectra of an
isolated inhomogeneously broadened optical transition; (b) for the transition
with a spectral hole burnt by monochromatic laser light. (c) and (d)
corresponding spectra of diamagnetic FR ($\phi$). The spectra of the
transition with a spectral hole are supposed to be measured with an additional
light source. $\Delta\omega$ is the relative detuning from the resonance.

Along with the temperature and spectral properties of these two contributions,
they can be distinguished by their inertial characteristics: the diamagnetic
contribution responds to magnetic field variations practically
instantaneously, while the paramagnetic contribution does it with some delay
that is needed to establish thermal equilibrium of the populations over
magnetic sublevels of the ground state. Thus, in the response to a magnetic
field modulated at sufficiently high frequency (exceeding the ground-state
spin-relaxation rate), the paramagnetic contribution can be strongly
suppressed even at low temperatures [13].

The behavior of the absorption and FR spectra of inhomogeneously broadened
lines under conditions of strong resonant excitation has been previously
studied for optical transitions in atomic (gaseous) systems, where the
inhomogeneous broadening is associated with the Doppler effect [14, 15]. In
these systems, under usual experimental conditions, the magnetic splitting of
the energy levels appears to be smaller than the thermal energy
$k_{\text{B}}T$ ($k_{\text{B}}$ being the Boltzmann constant) by many orders
of magnitude, and, as a result, the paramagnetic contribution to the Faraday
effect may be neglected. In this case, the pure diamagnetic FR spectrum of an
isolated optical transition looks as shown in Fig. 1(c).

Under conditions of strong resonant laser excitation of an inhomogeneously
broadened transition, the absorption line profile (see Fig. 1(a)) becomes
distorted by the hole-burning effect (Fig. 1(b)), and an additional spectral
contribution (contribution of the ’hole’) to the FR spectrum arises. As
compared with the FR spectrum in the linear regime, this contribution has the
opposite sign, is spectrally narrower, and, accordingly, is greater in
magnitude (Fig. 1(d)). The ratio of the amplitudes of the two components
(’broad’ and ’narrow’) to the FR spectrum, as can be expected, should
correlate with the ratio of the inhomogeneous to homogeneous widths, which, as
seen from the figure, can be measured in a single-beam (rather than pump-
probe) configuration with the probe beam tuned to the line center.

As was shown in a number of previous publications [16, 17] and noted in our
recent work [10], this ratio, in the $f$-$f$ transitions of RE ions in
crystals, may reach $6-8$ orders of magnitude, and, correspondingly, the
nonlinear Faraday effect in these systems may exceed its linear counterpart
also by many orders of magnitude. It is important to note that, in the single-
beam measurements of the nonlinear Faraday effect, the wavelength of the laser
beam always coincides with the hole center, where the nonlinear FR angle is
the greatest. Thus, we can conclude that, with increasing intensity of the
probe laser light, the resonant FR in the center of the $f$-$f$ transition
will change its sign and increase in magnitude by many orders of magnitude.

##  II Theoretical background

This section presents a semi-quantitative consideration of the nonlinear
Faraday effect observed for resonant probing of an inhomogeneously broadened
spectral line. Similar phenomena in gas systems, with inhomogeneous broadening
caused by the Doppler effect, were described in Refs. [14, 15]. Here, we
consider optical transitions of paramagnetic ions in a crystal lattice, with a
static inhomogeneous broadening associated with spatial fluctuations of the
local crystal fields. The results of this treatment are further applied to RE-
activated crystals. For consistency of their relation, we first consider the
general characteristics of the FR spectra in the linear regime and, then,
specific features of the nonlinear FR from inhomogeneously broadened
transitions of ensembles of paramagnetic atoms (ions).

###  II.1 Homogeneous broadening, linear FR

As is known, the FR angle $\phi$ is expressed in terms of the refractive
indices $n_{\pm}(\omega)$ of the medium for the circularly polarized waves
$\sigma_{\pm}$ as:

$\phi={\omega l\over c}\bigg{[}n_{+}(\omega)-n_{-}(\omega)\bigg{]},$ (1)

where $\omega$ is the optical frequency of the probe beam, $l$ is the sample
length, and $c$ is the speed of light. The contribution of the impurity ions
to the refractive index $n_{+}(n_{-})$ is determined by the transitions
between the energy states, with the difference $\Delta M$ of the projections
of angular momentum on the light propagation direction equal to $+1$ ($-1$)
(Fig. 2). In zero magnetic field, $B=0$, the values of the relative population
difference $p_{\pm}$ (the difference between the diagonal elements of the
density matrix of the impurity system), as well as the frequencies
$\omega_{\pm}$ for the transitions with $\Delta M=+1$ and $\Delta M=-1$ are
the same (for zero magnetic field, we denote $p_{\pm}\equiv p$ and
$\omega_{\pm}\equiv\omega_{0}$), and the FR vanishes ($\phi=0$). When the
magnetic field $B$ is turned on, the energy levels of the impurity centers
undergo the Zeeman splitting, which for the states with $|\Delta M|=1$ is
equal to the Larmor frequency $\omega_{L}\equiv g\mu_{\text{B}}B/\hbar$ (here,
$g$ is the factor describing the magnetic splitting of a given ion,
$\mu_{\text{B}}$ is the Bohr magneton, and $\hbar$ is the reduced Planck
constant). Therefore, the frequencies and populations for the transitions with
$\Delta M=+1$ and $\Delta M=-1$ become different and can be presented in the
form:

$\omega_{\pm}=\omega_{0}\pm\omega_{L},\hskip
14.22636ptp_{\pm}=p\pm\hbar\omega_{L}/2k_{\text{B}}T,$ (2)

where $k_{\text{B}}$ is the Boltzmann constant, and $T$ is the temperature.
(Eq. (2) is valid for $\hbar\omega_{L}/2k_{\text{B}}T<1$). In the case of
absence of the inhomogeneous broadening, when the frequency $\omega_{0}$ is
the same for all impurity centers, we can represent the above refractive
indices $n_{\pm}$ in the form:

$n_{\pm}(\omega)={2\pi\mathrm{d}^{2}\over\hbar}\hskip 2.84526ptNp_{\pm}\hskip
2.84526ptf(\omega_{\pm}-\omega)=\\\ ={2\pi\mathrm{d}^{2}\over\hbar}\hskip
2.84526ptN[p\pm\hbar\omega_{L}/2k_{\text{B}}T]f(\omega_{0}\pm\omega_{L}-\omega).$
(3)

Here, $\mathrm{d}$ is the dipole moment of the optical transitions
$\sigma_{\pm}$, $N$ is the concentration of the impurity centers, the function
$f$ describes the frequency dependence of the refractive indices
$n_{\pm}(\omega)$, and for the homogeneously broadened optical transition has
the form:

$f(\nu)={\nu\over\nu^{2}+\gamma^{2}},$ (4)

where $\gamma$ is the width of the homogeneously broadened transition. In Eq.
(3), we do not take into account the background part of the refractive index
not associated with the considered transition of the impurity ion.

Figure 2: The simplest energy-level diagram of a Kramers impurity ion in
magnetic field. $E_{1}$ and $E_{2}$ are the ground and excited states, and
$\pm 1/2$ indicate the spin levels split by the longitudinal magnetic field.
Under real experimental conditions, the magnetic splittings of the ground and
excited states are much smaller than the total width of the transition, and
the line profile, under strong resonant pumping, looks like curve $D$ in Fig.
1(a) or 1(b) depending on whether the inhomogeneous width predominates
homogeneous or not.

At low magnetic fields, where $\omega_{L}<\gamma,k_{\text{B}}T$, the
dependence of the FR angle on the magnetic field (or on the Larmor frequency
$\omega_{L}$) is linear and can be found using Eq. (1):

${\phi\over\omega_{L}}={\omega l\over c}{d\over
d\omega_{L}}\bigg{[}n_{+}(\omega)-n_{-}(\omega)\bigg{]}\bigg{|}_{\omega_{L}=0}.$
(5)

Substituting here $n_{\pm}(\omega)$ from Eq. (3), we have:

$\displaystyle\phi$ $\displaystyle=\phi_{C}-\phi_{A},$ $\displaystyle\phi_{C}$
$\displaystyle\equiv{V\hbar\over 2k_{\text{B}}T}f(\omega_{0}-\omega),\hskip
14.22636pt\phi_{A}\equiv Vp{\partial\over\partial\omega}f(\omega_{0}-\omega),$
$\displaystyle V$ $\displaystyle\equiv{4\pi\mathrm{d}^{2}N\over\hbar}\hskip
2.84526pt{\omega_{L}\omega l\over c}.$ (6)

The contribution $\phi_{C}$, usually referred to as paramagnetic or C-term
[11] is associated with variations of the impurity energy-level populations in
the magnetic field. As seen from Eq. (6), the $C$-term decreases with
increasing temperature. Dynamics of this contribution under variations of the
magnetic field is controlled by the population relaxation time $T_{1}$, so
that, upon modulation of the magnetic field at frequencies exceeding
$1/T_{1}$, the contribution $\phi_{C}$ is being suppressed. Thus, at
sufficiently high temperatures ($k_{\text{B}}T\gg\hbar\omega_{L}$) and under
conditions of sufficiently high frequency of the magnetic field modulation,
the contribution of the $C$-term to the detected FR signal can be neglected.

The contribution $\phi_{A}$, usually referred to as diamagnetic or A-term
[11], is related to the shift of the impurity energy levels in the applied
magnetic field. This contribution, which is temperature-independent and
practically inertialess, will be most important for interpretation of our
experimental data. Below, we consider the behavior of this contribution for
inhomogeneously broadened transitions at high intensities of the probe beam,
where the effects of optical nonlinearity become essential.

###  II.2 Inhomogeneous broadening, nonlinear FR

Let the line of the inhomogeneously broadened transition be centered at the
frequency $\bar{\omega}$ and described by the shape function ${\cal P}(\nu)>0$
$(\int{\cal P}(\nu)d\nu=1$), so that $N{\cal
P}(\omega_{0}-\bar{\omega})d\omega_{0}$ is the concentration of the impurity
centers with the transition frequency lying within the range
$[\omega_{0},\omega_{0}+d\omega_{0}]$. With the inhomogeneous broadening
introduced in this way, the diamagnetic contribution to the FR is given by the
relation:

$\phi_{A}(\omega)=V\int d\omega_{0}{\cal
P}(\omega_{0}-\bar{\omega})p(\omega_{0}){\partial\over\partial\omega}f(\omega_{0}-\omega).$
(7)

Equations (3) and (7) show that the contribution $\phi_{A}(\omega)$ is
proportional to the derivative of the refractive index $n_{\pm}$ (at
${\omega_{L}=0}$).

At low light intensity, when the impurity system is practically not perturbed
by the light, the population difference $p(\omega_{0})$ for all spectral
fractions remains the same ($p(\omega_{0})=p_{eq}$), as well as the total
population of the lowest states of the impurity centers. With increasing
intensity, the population difference for the resonant and near-resonant
spectral fractions may decrease, giving rise to the so-called hole-burning
effect and to the nonlinear Faraday effect.

Let us estimate, first, the value of the FR in the linear regime. We assume
here that the function ${\cal P}(\nu)$ is essentially nonzero within the range
$\nu\in[-\Gamma,\Gamma]$, where ${\cal P}(\nu)\sim 1/[2\Gamma]\equiv\bar{\cal
P}$. In this case, the quantity $\Gamma$ characterizes the inhomogeneous
width, which is assumed to be much greater than $\gamma$. Denoting the FR at
the center of the inhomogeneously broadened line ($\bar{\omega}$), in the
absence of saturation, by $\phi_{A0}$ and using the accepted simplifications,
we obtain for this quantity (setting $\omega=\bar{\omega}$ in Eq. (7)) the
following estimate:

$\phi_{A0}\approx-Vp_{eq}\bar{\cal
P}\int_{\bar{\omega}-\Gamma}^{\bar{\omega}+\Gamma}{\partial\over\partial\omega_{0}}f(\omega_{0}-\bar{\omega})\hskip
2.84526ptd\omega_{0}=-{Vp_{eq}\bar{\cal P}\over 2\Gamma}.$ (8)

Let us make now a similar estimate for the FR $\phi_{A1}$ under the condition
of optical saturation. As noted above, in this case, the function
$p(\omega_{0})$ appears to be distorted by the ’hole’ at the frequency
$\omega_{0}\approx\bar{\omega}$. We will characterize this hole by a bell-
shaped function ${\cal L}(\nu)$, which reaches its maximum at $\nu=0$, has a
width of $\tilde{\gamma}>\gamma$, and is, in magnitude, restricted to $0<{\cal
L}(\nu)<1$. Using this function, we can represent the population distribution
$p(\omega_{0})$ in the form:

$p(\omega_{0})=p_{eq}[1-{\cal L}(\omega_{0}-\bar{\omega})].$ (9)

As follows both from intuitive and from more rigorous theoretical
considerations, the function ${\cal L}(\nu)$, describing the hole, should meet
certain requirements. The depth of the hole ${\cal L}(0)$ and its width
$\tilde{\gamma}$ should depend on the probe beam intensity $I$, so that
$\lim_{I\rightarrow 0}{\cal L}(0)=0$ and $\lim_{I\rightarrow
0}\tilde{\gamma}=\gamma$. With increasing intensity, the amplitude ${\cal
L}(0)$ should increase and tend to unity as $I\rightarrow\infty$. Regarding
the hole width $\tilde{\gamma}$, at low and moderate intensities, when ${\cal
L}(0)\ll 1$, it should be $\sim\gamma$, and start to increase with further
growth of the intensity.

Substituting Eq. (9) into Eq. (7) and taking into account that
$\Gamma\gg\tilde{\gamma}$ (i.e., the width of the burnt hole is always smaller
than the inhomogeneous width), we can write for the FR angle $\phi_{A1}$
measured under these conditions the following relations:

$\displaystyle\phi_{A1}$ $\displaystyle=\phi_{A0}+$ $\displaystyle+Vp_{eq}\int
d\omega_{0}{\cal P}(\omega_{0}-\bar{\omega}){\cal
L}(\omega_{0}-\bar{\omega}){\partial\over\partial\omega_{0}}f(\omega_{0}-\bar{\omega})\approx$
$\displaystyle\approx\phi_{A0}+Vp_{eq}{\cal P}(0)\int d\omega_{0}{\cal
L}(\omega_{0}-\bar{\omega}){\partial\over\partial\omega_{0}}f(\omega_{0}-\bar{\omega}).$
(10)

Now, taking into account that the function ${\cal L}(\omega_{0}-\bar{\omega})$
is essentially different from zero at
$\omega_{0}\in[\bar{\omega}-\tilde{\gamma},\bar{\omega}+\tilde{\gamma}]$
(where it is $\sim{\cal L}(0)$), and also the fact that ${\cal
P}(0)\sim\bar{\cal P}$, we obtain, for $\phi_{A1}$, the following estimate:

$\displaystyle\phi_{A1}$ $\displaystyle\approx\phi_{A0}+Vp_{eq}\bar{\cal
P}{\cal
L}(0)\int_{\bar{\omega}-\tilde{\gamma}}^{\bar{\omega}+\tilde{\gamma}}d\omega_{0}{\partial\over\partial\omega_{0}}f(\omega_{0}-\bar{\omega})\approx$
$\displaystyle\approx Vp_{eq}\bar{\cal P}\bigg{[}{{\cal L}(0)\over
2\tilde{\gamma}}-{1\over 2\Gamma}\bigg{]}.$ (11)

In accordance with the properties of the quantities ${\cal L}(0)$ and
$\tilde{\gamma}$ described above, with increasing intensity of the probe, the
first fraction in the square brackets first increases (since ${\cal L}(0)$
grows), reaches its maximum at a certain intensity $I=I_{c}$ and then slowly
decreases (when ${\cal L}(0)\approx 1$ and $\tilde{\gamma}$ continues to
grow). From Eq. (11) one can see that, at sufficiently high intensities, the
first term becomes greater than the second, and $\phi_{A1}$ changes its sign.

The relationship (11) allows us to propose a way to estimate the ratio
$\Gamma/\gamma$ (inhomogeneous width to homogeneous width), which is, as noted
in Ref. [10], an important parameter for assessing the applicability of SNS to
a particular impurity system. Indeed, when the probe beam intensity is equal
to $I_{c}$, the value of ${\cal L}(0)$ is already close to unity, while the
hole width $\tilde{\gamma}$ is yet close to the homogeneous width $\gamma$.
Now, as can be seen from Eqs. (11) and (8), the easily measurable ratio
$R\equiv\phi_{A1}/\phi_{A0}$ can be estimated as:

$R|_{I=I_{c}}={\phi_{A1}|_{I=I_{c}}\over\phi_{A0}}\sim{\Gamma\over\gamma}.$
(12)

Thus, we see that the large value of the ratio of the inhomogeneous width of a
transition to its homogeneous width provides not only the “giant SN gain
effect” [10], but may also give rise to a “giant nonlinear FR”.

A more rigorous quantitative consideration, which we do not present here,
confirms the above reasoning and leads to the following formula for the ratio
$R$ introduced above:

$\displaystyle R$ $\displaystyle={\Gamma^{2}+\gamma^{2}\over\Gamma\gamma}{\pi
s\over 2\sqrt{s+1}\bigg{(}\sqrt{s+1}+1\bigg{)}^{2}}-1,$ $\displaystyle s$
$\displaystyle\equiv{\Omega_{\text{R}}^{2}\over\gamma\gamma^{exc}}.$ (13)

Here, $\Omega_{\text{R}}\sim\sqrt{I}$ and $\gamma^{exc}$ are the Rabi
frequency and the excited-state decay rate, respectively, of the impurity-ion
transition. An example of the dependence $R(s)$ is presented in Fig. 3.
Qualitatively, the behavior of the nonlinear FR described by this formula is
well correlated with our expectations: with increasing intensity of the probe
beam, the FR, first, decreases, then changes its sign, before it strongly
increases in magnitude.

Figure 3: Dependence of the resonant nonlinear FR (in units of linear FR) on
the probe beam intensity (in units of the saturation factor $s$) for three
values of the ratio $\Gamma/\gamma$. The inset in the center shows inversion
of the FR sign at low light intensities.

The value $s\sim I$ is usually referred to as the saturation factor. The
second fraction in Eq. (13) reaches its maximum (0.27) at $s=s_{c}=4.81$. The
probe beam intensity (the light power density) corresponding to this value of
the saturation factor was denoted above by $I_{c}$.

Now, one can easily see that, at $\Gamma\gg\gamma$, Eq. (13) yields a result
similar to (but more accurate than) the estimate (12):

$R|_{I=I_{c}}=0.27{\Gamma\over\gamma}.$ (14)

This equation can be used to estimate the homogeneous width of the optical
transition from the ”FR gain factor” and to predict the efficiency of
application of the SNS technique to this particular system.

##  III Methods and Samples

To perform the experiments suggested above with a solid-state paramagnet, it
is not enough to measure the FR at high intensity of the probe beam. One also
has to get rid of the paramagnetic contribution, which usually predominates at
low temperatures and may substantially distort the results of the
measurements. In addition, the spectral width of the probe beam should be
smaller than the homogeneous width of the transition under study.

Figure 4: A sketch of the experimental setup for measuring the FR angle. The
laser beam power was controlled by the $\lambda/2$ plate and Glan-Taylor (GT)
prism. Longitudinal magnetic field was created by a pair of coils.
Polarimetric detector consisted of $\lambda/2$ plate, Wollaston prism (WP),
and a pair of photo diodes. The FR angle was obtained by measuring the
difference signal $I_{1}-I_{2}$ normalised by $I_{1}$.

The schematics of the experimental setup is presented in Fig. 4. As light
source we used a tunable Ti:sapphire continuous wave ring laser (Coherent
MBR110) with the linewidth of around 40 kHz. The linearly polarized laser beam
of 2 mm diameter was focused at the sample using the 60 mm focal-length lens.
The rotation of the polarization plane of the transmitted beam was analyzed by
the half-wave plate followed by the Wollaston prism and balanced photodiodes
(Newport Nirvana 2007). Using additionally the single diode output, we were
able to evaluate the absolute value of the rotation angle. The sample was
mounted inside the continuous flow cryostat on a cold finger at the
temperature of $\sim 6$ K. To measure the diamagnetic contribution to the FR,
we applied a longitudinal oscillating magnetic field using an electro magnet,
with its frequency ($f_{m}=333$ Hz) substantially exceeding the spin-lattice
relaxation rate of the ground-state spin system. This frequency was then used
as a reference for a lock-in amplifier to detect the signal.

To be able to compare the results of magneto-optical and spin-noise
measurements, we used here the same crystals that were studied in our previous
work on SNS [10]. These include a SrF2 crystal coactivated with Nd3+ (0.5 mol
%) and Yb3+ (0.15 mol %), and a crystal of CaF2:Nd3+ (0.1 mol %).

##  IV Results of the measurements

###  IV.1 Nonlinear FR spectra

The measurements of the FR spectra were performed on several lines of the
$f$-$f$ transitions of Nd3+ and Yb3+ ions in the CaF2 and SrF2 crystals. As
expected, the behavior of the FR spectra strongly differed at different
transitions. Some of them had a complicated structure which did not allow us
to apply our simplified model. Still, there were several well isolated lines
with a pronounced dependence of their FR spectra on the probe beam intensity.
As an example, Fig. 5(a) shows the evolution of the FR spectrum of the line at
862.68 nm of Nd3+ in CaF2 with increasing probe beam intensity. This
transition of the tetragonal center is characterized by the longest lifetime
of the excited state ($\sim 1.5$ ms) [18]. Several FR spectra at the lowest
light intensities (presumably in the linear regime) are shown in the inset. As
we can see, the observed behavior perfectly agrees with the predictions of our
model: the spectrum of the linear FR (at low light intensity) approximately
corresponds to the derivative of the line’s refractive index. Then, with
increasing light intensity, the FR, in the line center, inverts its sign and
strongly increases in magnitude. In this particular case, the greatest
nonlinear FR exceeded the linear FR by nearly two orders of magnitude.

This is, however, not a common type of intensity-related behavior of the FR
spectrum. Figure 5(b) shows another example of this behavior, when the FR
spectrum virtually does not vary with the light intensity. In accordance with
Ref. [18], this transition belongs to the Nd3+ M-center (pair center),
characterized by fast cross-relaxation between the levels of the Nd ions
comprising the center, which drastically shortens the excited-state lifetime
(to 92 $\mu$s). Under these conditions, the line appears to be broadened
homogeneously, and no spectral hole is produced.

Figure 5: Two patterns of behavior of the FR spectrum vs probe light intensity
for inhomogeneously (a) and homogeneously (b) broadened transitions measured
for the CaF2-Nd (0.1%) crystal. In the first case, panel (a), the FR, as a
function of laser intensity, changes its sign at the resonance and grows in
magnitude (see also the inset for lower powers), while in the second case,
panel (b), it remains the same in a wide range of laser intensities. The
spectra are shifted for clarity, whereby the zero level is marked by the
dashed horizontal line. Note the scale difference between panels (a) in radian
and (b) in milliradian.

###  IV.2 Measuring the homogeneous widths

As follows from the above treatment, the homogeneous width of the transition
can be found by measuring the factor of enhancement of the nonlinear FR ($R$).
Figure 6 shows the light-intensity dependencies of the resonant FR measured on
several transitions of the RE ions in the studied crystals. The general
pattern of these dependences, for all lines, correlates well with the results
of our treatment presented in Fig. 3.

Figure 6: Dependencies of the resonant FR for the studied transitions on the probe beam power density for SrF2:Yb (0.15%) in the panel (a) and (b), for SrF2:Nd (0.5%) in the panel (b), and for CaF2:Nd (0.1%) in the panel (c). $\lambda$ is the probe wavelength at the transition. The FR enhancement factor $R=|\text{FR}_{\text{max}}/\text{FR}_{\text{min}}|$, see panel (a). $T=6$ K, maximal amplitude of the oscillating magnetic field $B_{\sim}=0.3$ mT, $f_{m}=333$ Hz. The green lines are guides for the eyes and are connecting the data points. Table 1: Evaluation of the homogeneous width for some transitions of Nd3+ and Yb3+ in the studied crystals. $\lambda$ is the probe wavelength at the transition, $R$ is the FR enhancement factor, $\Gamma$ is the measured spectral width of the transition, and $\gamma$ is the homogeneous width of the transition, calculated using Eq. (14). Crystal |  $\lambda$, nm | $R$ |  $\Gamma$, GHz |  $\gamma$, MHz
---|---|---|---|---
SrF2-Yb (0.15%) | 965.44 | 13 | 2.68 | 44
SrF2-Yb (0.15%) | 965.15 | 28 | 4.0 | 31
SrF2-Nd (0.5%) | 863.365 | 134 | 26.8 | 54
CaF2-Nd (0.1%) | 862.68 | 108 | 15.2 | 38
CaF2-Nd (0.1%) | 863.415 | 70 | 12.8 | 50

The results of the calculations of the homogeneous linewidths of the studied
transitions obtained using our experimental data and Eq. (14) are presented in
Table 1. These results show that the quantity $\gamma$ is of the same order
for all the transitions that revealed a pronounced nonlinear Faraday effect
and lies in the range of a few tens of MHz. To make sure that the estimates of
the homogeneous linewidths obtained from the measurements of the nonlinear FR
are correct, we performed independent measurements of this quantity. The used
method was based on the fact that the nonlinear FR measured in our experiments
should depend linearly on the applied oscillating magnetic field only as long
as the Zeeman splitting modulation remains smaller than the width of the hole
burnt by the laser light in the inhomogeneously broadened line. Thus, the
width of the hole can be estimated by measuring the dependence of the FR
signal on the applied field amplitude.

Results of such measurements made on the 862.68 nm line of Nd3+ in CaF2 are
shown in Fig. 7. These measurements were performed at the light power density
of around 0.3 kW/cm2, falling into the range of the initial linear dependence
of the nonlinear FR on the probe beam intensity (Fig. 6), where the hole width
is supposed to be not broadened by the light. It is also important that the
amplitude of the oscillating field used in the measurements presented in Fig.
6, as one can see, belongs to the linear part of the plot in Fig. 7, which
means that the approximation of smallness of the field amplitude, in these
measurements, was satisfied.

As seen from Fig. 7, the presented dependence starts to deviate from a linear
dependence at around 0.5 mT, which, for the mean $g$ factor 3 yields the value
$g\mu_{\text{B}}B/h\approx 21$ MHz, correlating well with the measured
homogeneous widths ($h$ is the Planck constant), see Table 1.

Figure 7: Dependence of the FR signal on the amplitude of the applied
oscillating magnetic field for the transition at 862.68 nm of Nd3+ in CaF2.
$T=6$ K, $f_{m}=333$ Hz. Red line is a linear fit. Vertical dashed line at 0.5
mT shows the point where the data start to deviate from the linear dependence.

###  IV.3 Discussion

Now, we can check whether the results of the FR-based measurements of the
homogeneous widths agree with the theoretical estimates of this value. To
evaluate the homogeneous width associated with the magnetic dipole-dipole
interaction of the paramagnetic ions, we can use the following formula [19]:

$\gamma={\tilde{k}C\tilde{g}^{2}\mu_{\text{B}}^{2}\over\hbar},$ (15)

where $\tilde{g}$ is the characteristic $g$ factor of the impurity ion,
$\tilde{k}\sim 2.53$ is a numerical factor, governed by the relative values
and signs of two pairs of $g$ factors of the ground and excited states of the
ion, and $C$ is the impurity concentration. The estimate of $\gamma$ using Eq.
(15) for the tetragonal Nd3+ center in the CaF2 crystal with $C=2.46\cdot
10^{19}$ cm-3 and $\tilde{g}=4$ yields $\gamma=2\pi\cdot 16$ MHz, which,
taking into account the uncertainties of our measurements and made
approximations, agrees well with the experimental data of $\gamma$ for the
transitions presented in Table 1.

We did not find, in the literature, results of measuring the homogeneous
linewidths of the $f$-$f$ transitions in these particular transitions, whereas
it is known that, for other RE ions and for different $f$-$f$ transitions,
these values may vary from tens of Hz to several GHz [17, 20]. In particular,
in Ref. [21], the homogeneous width of the $f$-$f$ transitions of Nd3+ in CaF2
at 9 K reached 350 MHz. Studies of this kind are mainly aimed at searching for
the smallest linewidths, most interesting from the viewpoint of applications
[22, 23, 24]. For this study, we did not specially choose samples with an
extraordinary small linewidth to observe a strong nonlinear FR, and we believe
that, using other RE-doped crystals and other transitions, the effect of the
enhanced nonlinear FR, being several orders of magnitude stronger, will become
observable.

##  V Conclusions

In this paper, we have demonstrated specific features of the nonlinear
resonant Faraday effect observed on the $f$-$f$ transitions of trivalent rare-
earth ions in crystals. We show that, in transitions broadened essentially
inhomogeneously and under conditions of sufficiently high intensity of the
probe beam, the diamagnetic Faraday rotation may increase by several orders of
magnitude as compared with its linear (unperturbed) value. In other words, we
show that the giant spin-noise gain effect, not revealed in the regular linear
Faraday rotation of the medium, can be revealed as an equally giant regular
nonlinear Faraday effect under conditions of resonant probing. The results of
our magneto-optical measurements confirm the correlation between the magnitude
of the spin noise and the homogeneous width of the transition. The proposed
magneto-optical method of testing optical transitions can be useful not only
for studying the applicability of spin-noise spectroscopy to a particular
system, but also as a single-beam method for measuring the homogeneous width
of the optical transitions of paramagnetic impurities in crystals. Note also
that the giant nonlinear Faraday rotation that may exceed the linear Macaluso-
Corbino effect by many orders of magnitude, can find application in numerous
magneto-optical devices for the laser wavelength stabilization and for
magnetometric purposes [15].

###### Acknowledgements.

We highly appreciate the financial support from the Deutsche
Forschungsgemeinschaft in the frame of the International Collaborative
Research Center TRR 160 (Project A5) and the Russian Foundation for Basic
Research (Grant No. 19-52-12054). The authors from Russian side acknowledge
the Saint Petersburg State University for the research Grant No. 73031758.
E.I.B. acknowledges the support of the Foundation for the Advancement of
Theoretical Physics and Mathematics ”BASIS” and from the subsidy allocated to
Kazan Federal University for the state assignment in the sphere of scientific
activities No. FZSM-2020-0050.

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