# EVIMO2: An Event Camera Dataset for Motion Segmentation, Optical Flow,
Structure from Motion, and Visual Inertial Odometry in Indoor Scenes with
Monocular or Stereo Algorithms

Levi Burner , Anton Mitrokhin, Cornelia Fermüller, Yiannis Aloimonos
Perception and Robotics Group, University of Maryland Institute for Advanced
Computer Studies
University of Maryland, College Park
Levi Burner is the corresponding author. The support of the NSF under awards
DGE-1632976 and OISE 2020624 is gratefully acknowledged.

###### Abstract

A new event camera dataset, EVIMO2, is introduced that improves on the popular
EVIMO dataset by providing more data, from better cameras, in more complex
scenarios. As with its predecessor, EVIMO2 provides labels in the form of per-
pixel ground truth depth and segmentation as well as camera and object poses.
All sequences use data from physical cameras and many sequences feature
multiple independently moving objects. Typically, such labeled data is
unavailable in physical event camera datasets. Thus, EVIMO2 will serve as a
challenging benchmark for existing algorithms and rich training set for the
development of new algorithms. In particular, EVIMO2 is suited for supporting
research in motion and object segmentation, optical flow, structure from
motion, and visual (inertial) odometry in both monocular or stereo
configurations.

EVIMO2 consists of 41 minutes of data from three 640$\times$480 event cameras,
one 2080$\times$1552 classical color camera, inertial measurements from two
six axis inertial measurement units, and millimeter accurate object poses from
a Vicon motion capture system. The dataset’s 173 sequences are arranged into
three categories. 3.75 minutes of independently moving household objects,
22.55 minutes of static scenes, and 14.85 minutes of basic motions in shallow
scenes. Some sequences were recorded in low-light conditions where
conventional cameras fail. Depth and segmentation are provided at 60 Hz for
the event cameras and 30 Hz for the classical camera. The masks can be
regenerated using open-source code up to rates as high as 200 Hz.

This technical report briefly describes EVIMO2. The full documentation is
available online111https://better-flow.github.io/evimo. Videos of individual
sequences can be sampled on the download page222https://better-
flow.github.io/evimo/download_evimo_2.html.

Figure 1: EVIMO2’s data can be used to generate high quality ground truth
optical flow. Due to Vicon’s accurate tracking of independently moving objects
at 200 Hz, the flow can be estimated using finite difference over small
temporal intervals (10 milliseconds in the figure). The flow is dense as shown
in the color field. Sparse arrows indicate magnitude.

##  1 Introduction

(a) Flea3 classical

(b) Samsung DVS Gen 3

(c) Left Prophessee Gen3

(d) Right Prophesee Gen 3

Figure 2: In each subframe, the left side visualizes the raw data (events or
brightness) with the segmentation mask overlayed as a color. Average event
time in a pixel is color coded. The right side of each frame shows the ground
truth depth. There are black pixels where depth or segmentation are
unavailable.

EVIMO2 is an event camera dataset featuring per-pixel ground truth depth,
segmentation, and object pose for the footage from three event cameras and one
classical color camera. It generally improves on its predecessor EVIMO [24] by
using better cameras, featuring longer recordings, providing different
categories of recordings, and using an improved data format. The recordings
feature indoor, dynamic scenes with multiple independently moving objects. In
contrast to existing simulated [17, 26, 21, 30, 10, 15, 27, 16, 34, 11, 33,
28] and physical [7, 14, 25, 2, 38, 36, 37, 12, 32, 38, 1, 22, 35, 3, 26, 4,
38, 5, 9, 8, 20, 19, 13, 31, 18, 29] event camera datasets, EVIMO2 focuses on
close range indoor situations with fast moving objects and provides per-pixel
ground truth depth and segmentation as well as poses for the physical cameras
and all objects.

While pixel accurate semantic segmentation is sometimes available in simulated
event camera datasets [21, 34, 11, 33, 28] it is not usually available for
physical event camera datasets with the exception being EVIMO2’s direct
predecessor [24]. On the other hand, while depth and semantic bounding boxes
are sometimes present in physical event camera datasets from LIDAR, structured
light sensors, and hand labeling [23, 2, 38, 12, 20, 19, 29], EVIMO2’s depth
and segmentation comes from 3D scans of household objects combined with Vicon
pose measurements. This results in more detailed ground truth depth maps and
optical flow fields than could previously be achieved, even in datasets
designed for optical flow estimation [32, 1, 3, 26].

Due to the availability of ground truth pose of the camera, EVIMO2 is suitable
for indoor Visual Inertial Odometry (VIO) and Simultaneous Localization and
Mapping (SLAM) research. It differs from existing VIO and SLAM event camera
datasets by focusing on close-up indoor scenes containing moving household
objects. In contrast, existing datasets focus on indoor scenes without moving
objects [3, 35, 26, 5, 9, 18], outdoor or driving/flying scenes [4, 38, 8, 13,
31, 18], or extreme environments [20, 19, 23]. Further, EVIMO2 provides poses
for the camera and all objects which opens up exciting opportunities for
object pose estimation.

EVIMO2’s ground truth depth and segmentation masks are obtained using the same
methods as its predecessor EVIMO [24]. The 3D scans of the objects are
projected into the camera’s field-of-view using poses measured by a Vicon
motion capture system. 60 Hz ground truth is available for event cameras and
30 Hz ground truth for the classical camera. Higher framerate ground truth can
be generated using the released open-source tools. Due to the detailed 3D
object scans and the high quality tracking of the Vicon system, exceptional
ground-truth optical flow can be generated for scenes with fast moving objects
undergoing complex motions as shown in Figure 1 and 3.

##  2 Methods

EVIMO2 features sensor data from three event cameras, one classical camera,
two six-axis inertial measurement units, and poses from a Vicon motion capture
system. Two of the event cameras are Prophesee Gen3 VGA 640$\times$480 sensors
with a 71 degree diagonal field-of-view
333https://docs.prophesee.ai/stable/hw/evk/gen3.html. They are arranged in a
binocular stereo configuration with a baseline of approximately 22 cm. The
third event camera is a Samsung DVS Gen3 with 640$\times$480 resolution and 75
degree diagonal FOV
444https://rpg.ifi.uzh.ch/docs/CVPR19workshop/CVPRW19_Eric_Ryu_Samsung.pdf.
The classical camera is a Flea 3 camera featuring the Sony IMX036 sensor which
recorded at 2080$\times$1552 and 30 Hz while fitted with a lens that achieved
a 64 degree diagonal FOV 555https://www.flir.com/products/flea3-usb3/. The
inertial measurements are supplied by the Prophesee camera’s internal MPU9250
IMUs which are sampled at 1 kHz
666https://support.prophesee.ai/portal/en/kb/articles/imu-data-format-in-
metavision-software.

###  2.1 Objects

The dataset features over 20 real world objects including a large table that
forms the main backdrop of many sequences as illustrated by Figure 2. The
other objects are small household items such as toy blocks, remote control
cars, and small drones. Silver motion capture markers are attached to each
object to enable tracking by the Vicon motion capture system.

The 3D Models for the objects were obtained using an Artec Space Spider
scanner777https://www.artec3d.com/portable-3d-scanners/artec-spider. This
results in high quality meshes, which were then edited using the open-source
Meshlab software to remove the Vicon markers [6].

###  2.2 Ground Truth

Ground truth depth and segmentation masks are calculated using Vicon pose
estimates and 3D scans of small objects and a large table. This makes accurate
ground truth depth available for the objects and the table, which typically
cover most of the field of view as exemplified by Figure 2.

Due to the small size of the objects used, and the sensitivity of the Vicon
motion capture system to occlusion of object’s markers, there are occasional
temporal gaps in ground truth availability. As a result, users must take care
to respect that the ground truth data is not evenly spaced in time.

###  2.3 Sensor Availability

Not all sensors are available in all sequences. A complete camera availability
matrix is available on the EVIMO2 downloads page. The left Prophesee camera is
available in 155 sequences. The right Prophesee camera is available in 124
sequences. The Samsung camera is available in 154 sequences. The Flea3 camera
is available in 139 sequences (the Flea3 data is excluded from several of the
low-light sequences because it did not produce usable data).

###  2.4 Sequence Categories

The sequences are organized in three distinct categories. “Independently
Moving Objects” (IMO), “Structure from Motion” (SfM), and “Simple Motion in
Planar Scenes” (Sanity). All categories have a smaller set of sequences
designated “low light”. In low light sequences the ambient light levels in the
recording room were greatly reduced which makes the events much “noisier”.
Video previews of a prototypical member of each category are available on
EVIMO2’s download page.

For IMO and SfM sequences, sequences have been split into distinct “train” and
“test” sets of approximately equal length. The sanity sequences are split into
“checkerboard”, “depth variation”, “sliding”, and “tabletop” subsets where
basic translations and rotations are performed in almost entirely planar
scenes.

Table 1 summarizes the number and duration of the sequences in each category.

###  2.5 Optical Flow

Due to the availability of ground truth depth and per object pose, EVIMO2 is
particularly suited for evaluation of optical flow methods. An example of the
quality of the ground truth optical flow is shown in Figure 1. There, the
ground truth flow is approximated by finite-differencing object positions
within a time interval of 10 milliseconds. This small time interval is made
possible due to the Vicon’s 200 Hz update rate. The result is accurate ground-
truth optical flow even when multiple objects are flipping through the air in
different directions as illustrated by Figure 1. Further, the high-quality 3D
object models allow fine details to be captured in the flow field as shown in
Figure 3.

Figure 3: EVIMO2’s 3D models allow fine details in the optical flow fields to be captured such as the differences in velocity of the features on a toy car as shown above. Category | Sequences | Minutes
---|---|---
| Train | Test | Train | Test
IMO | 21 | 8 | 2.74 | 0.51
IMO LL | 3 | 2 | 0.36 | 0.14
SfM | 52 | 10 | 16.17 | 2.45
SfM LL | 5 | 6 | 2.56 | 1.37
Sanity | N/A | 31 | N/A | 6.96
Sanity LL | N/A | 35 | N/A | 7.89
Totals | 81 | 92 | 21.83 | 19.32
Table 1: Number of sequences and duration of sequences for each sequence
category in EVIMO2. Abbreviations: Independent Moving Objects (IMO), Structure
from Motion (SfM), Simple Motion in Planar Scenes (Sanity), Low Light (LL).

##  3 Conclusion

EVIMO2 is an event camera dataset featuring three event cameras, one classical
camera, two IMUs, per-pixel ground truth depth, segmentation, and object
poses. Typically, ground segmentation and object poses are only available in
simulated datasets. Further, EVIMO2 contains numerous sequences with small,
independent, fast moving objects, which present a challenge for optical flow,
visual odometry, and SLAM methods. Thus, EVIMO2 provides a challenging
benchmark for existing methods and rich set of sequences for training new
ones.

##  4 Acknowledgements

The authors thank Tobi Delbruck for assisting in testing and debugging the
ground truth optical flow generation tool during the writing of [22]. The
authors also thank Suteerth Vishnu for assisting with collection and curation
of the data.

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# Transmission of electromagnetic waves through nonlinear optical layers

L. M. Hincapie-Zuluaga, J. D. Mazo-Vásquez, C. A. Betancur-Silvera, and E.
Reyes-Gómez  Instituto de Física, Universidad de Antioquia UdeA, Calle 70 No.
52-21, Medellín, Colombia

###### Abstract

The excitation of soliton states in optical layers exhibiting Kerr
nonlinearities is theoretically investigated. The optical transmission
coefficient is obtained as a function of the intensity of an incident light
beam. The incident electromagnetic wave is considered to be monochromatic and
linearly polarized in the transverse electric configuration. In materials
exhibiting self-defocusing nonlinearity, soliton excitations are not observed
if the product of the nonlinear dielectric coefficient of the slab and the
absolute square of the incident electric-field amplitude is below a well-
defined value. At such a limit, a soliton excitation with a position-
independent electric field amplitude is observed within the nonlinear layer,
regardless of the frequency value of the incident wave.

###### pacs:

41.20.Jb; 42.65.Tg; 42.70.Gi

The propagation of electromagnetic waves through nonlinear materials has been
extensively investigated over the last decades. The nonlinear response of a
medium, i.e., the dependence of the system’s optical properties on the
intensity of the incident electromagnetic field, leads to the existence of
physical phenomena that are not observed in conventional linear materials.
Different mechanisms could give rise to nonlinear optical properties in
various materials. For example, it has been shown that thermal effects and
induced polarization could be responsible for third-order optical nonlinearity
in high-purity silica samples containing Cu nanoparticles Torres-Torres et al.
(2008). Changes in molecular orientation in organic solvents and solutions
could lead to the nonlinear behavior of the refractive index Bundulis et al.
(2020), etc. Of particular interest turns out to be the excitation of
transparency states in low-absorptive nonlinear materials when the intensity
of the incident electromagnetic radiation is appropriately tuned. In such
cases, the system is observed to switch from states of low transparency to
states of maximum transmission, a situation that can be understood from the
concept of soliton Lomdahl (Spring 1984). A change in the incident
electromagnetic-field intensity can cause soliton excitations, propagating
through the nonlinear medium with a maximum transmission. This phenomenon,
known as transmission-switching phenomenon, has been extensively studied in
both nonlinear thin films Chen and Mills (1987a); Peschel et al. (1988); Leung
(1989) and multilayer structures built by alternating linear and nonlinear
materials along a predefined growth axis Trutschel and Lederer (1988); Chen
and Mills (1987b). In this particular case, soliton excitation was reported
for frequency values in the vicinity of the Bragg gaps Chen and Mills (1987c).
Moreover, when the linear material composing the periodic structure exhibits
left-handed material properties, soliton excitations associated with both the
zero-n gap Hegde and Winful (2005) and the longitudinal plasmon-polariton gap
Cavalcanti et al. (2014) can also be observed.

Of great relevance for the present study are the works by Chen and Mills Chen
and Mills (1987a) and Leung Leung (1989), who were able to develop exact
analytical approaches to theoretically investigate the tunneling effect of
electromagnetic waves through non-absorptive Kerr-like nonlinear layers. They
also investigated the transmission-switching phenomenon in both self-focusing
and self-defocusing materials. Despite the considerable amount of literature
on the formation of solitons in nonlinear layers, a detailed review of the
works mentioned above has led us to understand some transmission properties of
nonlinear systems that, at the time, were not clearly explained. The present
work aims to investigate such properties. In particular, we will show that,
for soliton excitations to occur in layers with a self-defocusing Kerr
nonlinearity, the product of the nonlinear dielectric coefficient of the slab
and the absolute square of the incident electric-field amplitude must exceed a
specific minimum value. This finding opens up the possibility of predicting
intensity regions where the formation of solitons would take place and other
ones where such effects would not be observed at all, and could be of
relevance in optical-computing investigations Corbelli et al. (2013); Silva et
al. (2021).

Figure 1: (Color online) Pictorial view of an electromagnetic wave incident on
a nonlinear layer, with the angle of incidence $\theta$ and electric-field
intensity $\vec{\mathbf{E}}$. The electromagnetic response of the nonlinear
layer is characterized by the magnetic permeability $\mu$ and the electric
permittivity $\epsilon$. The corresponding parameters in the surrounding
linear medium are $\mu_{0}$ and $\varepsilon_{0}$, respectively. The incident
electromagnetic wave is assumed to be polarized in the transverse electric
configuration. The plane of incidence is the $xz$ plane containing the wave
vector $\vec{\mathbf{k}}$, and the electric field is oriented along the $y$
direction.

We consider an electromagnetic wave of frequency $\omega=2\pi\nu$ incident on
a slab of thickness $d$, electric permittivity $\epsilon$ and magnetic
permeability $\mu$, grown along the $z$ direction as shown in Fig. 1. We
suppose that the slab is composed of a non-absorptive material, that is, both
$\epsilon$ and $\mu$ are real parameters. The incidence angle, denoted as
$\theta$, is measured in the surrounding medium where the electric
permittivity and magnetic permeability are given by $\varepsilon_{0}$ and
$\mu_{0}$, respectively. The wave is assumed to be linearly polarized
according to the transverse electric (TE) configuration, i.e., the electric-
field intensity of the incident wave is perpendicular to the plane of
incidence. The electric-field component of the electromagnetic wave is then
given by $\mbox{$\vec{\mathbf{E}}$}(\mbox{$\vec{\mathbf{r}}$},t)={\cal
E}(z)e^{i(qx-\omega t)}\mbox{$\vec{\mathbf{e}}$}_{y}$, where
$q=\frac{\omega}{c}\sqrt{\varepsilon_{0}\mu_{0}}\sin(\theta)$ is the
projection of the wave vector $\vec{\mathbf{k}}$ along the $x$ direction (cf.
Fig. 1) and $\mbox{$\vec{\mathbf{e}}$}_{y}$ is the unit vector along the $y$
direction. The electric-field amplitude of the wave satisfies the equation
Cavalcanti et al. (2014)

$-\frac{d}{dz}\frac{1}{\bar{\mu}(z)}\frac{d}{dz}{\cal
E}(z)=\left[\frac{\omega^{2}}{c^{2}}\bar{\varepsilon}(z)-\frac{\omega^{2}}{c^{2}\bar{\mu}(z)}\mathrm{sin^{2}}(\theta)\right]{\cal
E}(z),$ (1)

where $\bar{\varepsilon}(z)$ and $\bar{\mu}(z)$ are the position-dependent
electric permittivity and magnetic permeability, respectively, along the $z$
growth axis. One may notice that $\bar{\varepsilon}(z)=\varepsilon_{0}$ and
$\bar{\mu}(z)=\mu_{0}$ in the surrounding medium, whereas
$\bar{\varepsilon}(z)=\epsilon$ and $\bar{\mu}(z)=\mu$ within the slab.

We assume that the surrounding medium is linear and non-dispersive, whereas
the optical material composing the slab exhibits a Kerr-type nonlinear
behavior. The nonlinear electric permittivity is then related to the third-
order susceptibility tensor, resulting from the third-order expansion of the
nonlinear polarization in series of powers of the electric field. Here we
suppose that the optical material is isotropic. Therefore, as the incident
electromagnetic wave is considered to be linearly polarized, the electric
permittivity of the slab may be given by Sutherland (2003)
$\epsilon=\varepsilon+\alpha|{\cal E}|^{2}$, where $\varepsilon$ is the
dielectric background constant of the slab. The parameter $\alpha$ is the so-
called nonlinear dielectric coefficient which is a real-scalar number
proportional to the real part of the $\chi^{(3)}_{yyyy}$ component of the
third-order susceptibility tensor Sutherland (2003); Kivshar and Agrawal
(2003) of the nonlinear layer. The term $\alpha|{\cal E}|^{2}$ is the
nonlinear Kerr contribution to the electric permittivity. It is important to
note that the sign of $\alpha$ is determined by the sign of
$\chi^{(3)}_{yyyy}$ Kivshar and Agrawal (2003). The case $\alpha>0$
corresponds to self-focusing materials, whereas $\alpha<0$ corresponds to
self-defocusing nonlinearities. We want to emphasize that our mathematical
derivation below is consistent with any value of the sign of $\alpha$.

The optical parameters that describe the system depend, in general, on the
wave frequency. For instance, it has been shown in recent experimental work
that simultaneous changes in wave frequency and slab thickness lead to
switching between self-focusing and self-defocusing regimes in certain
nonlinear materials under femtosecond and picosecond laser pulses Torres-
Torres et al. (2013). However, in other experimental situations, considering
the optical parameters independent of the wave frequency can be a good
approximation Peschel et al. (1988). For the sake of simplicity, we have taken
$\varepsilon_{0}$, $\mu_{0}$, $\varepsilon$, $\mu$, and $\alpha$ as frequency-
independent parameters, an approximation that might be valid in specific
experimental situations.

One may propose

${\cal E}(z)=E_{i}\begin{cases}e^{iQ_{0}z}+re^{-iQ_{0}z}&$if
$z<0\cr\phi(z)&$if $0<z<d\cr te^{iQ_{0}(z-d)}&$if $z>d\end{cases}$ (2)

as the solution of Eq. (1), where $E_{i}$ is the amplitude of the incident
electric field, $Q_{0}=\frac{\omega}{c}p_{0}$, and
$p_{0}=\sqrt{\varepsilon_{0}\mu_{0}}\cos{(\theta)}$. By taking into account
the continuity of ${\cal E}$ and $\frac{1}{\bar{\mu}}\frac{d}{dz}{\cal E}$ at
the interfaces, one may find the reflection ($R=rr^{*}$) and transmission
($T=tt^{*}$) coefficients corresponding to the slab. By defining
$\zeta=\frac{\omega}{c}z$, Eq. (1) becomes

$\frac{d^{2}}{d\zeta^{2}}\phi(\zeta)+\left(\varepsilon\mu-\varepsilon_{0}\mu_{0}\mathrm{sin^{2}}\theta\right)\phi(\zeta)+a\mu|\phi(\zeta)|^{2}\phi(\zeta)=0$
(3)

within the region $0<\zeta<\zeta_{d}$, where $\zeta_{d}=\frac{\omega}{c}d$. We
refer to $a=\alpha|E_{i}|^{2}$ as the normalized input intensity Peschel et
al. (1988) since such parameter is proportional to the intensity of the
incident wave but could be either positive or negative, depending on the sign
of the nonlinear dielectric coefficient corresponding to the material layer.
We propose $\phi=f\mathrm{e}^{i\varphi}$, where $f$ and $\varphi$ are real
functions of $\zeta$. Accordingly, Eq. (3) leads to

$\frac{d}{d\zeta}\left(f^{2}\frac{d\varphi}{d\zeta}\right)=0,$ (4)

and

$\frac{d^{2}f}{d\zeta^{2}}-f\left(\frac{d\varphi}{d\zeta}\right)^{2}+p^{2}f+a\mu
f^{3}=0,$ (5)

where $p^{2}=\varepsilon\mu-\varepsilon_{0}\mu_{0}\sin^{2}(\theta)$. The first
integrals of Eqs. (4) and (5) are

$\frac{d\varphi}{d\zeta}=\frac{A}{f^{2}}$ (6)

and

$\left(\frac{1}{2}\frac{dF}{d\zeta}\right)^{2}+A^{2}+p^{2}F^{2}+\frac{a\mu}{2}F^{3}-BF=0,$
(7)

respectively, where $A$ and $B$ are integration constants and $F=f^{2}$. We
calculate the integration constants $A$ and $B$ by taking into account the
boundary conditions that the electric-field amplitude fulfills at the slab
interfaces. From the continuity of $\phi$ and
$\frac{1}{\mu}\frac{d\phi}{d\zeta}$ at the interface $\zeta=0$ one finds

$\begin{cases}1-r=\frac{1}{i\eta}\left[\dot{f}_{0}\mathrm{e}^{i\varphi_{0}}+i\dot{\varphi}_{0}f_{0}\mathrm{e}^{i\varphi_{0}}\right]\cr
1+r=f_{0}\mathrm{e}^{i\varphi_{0}},\end{cases}$ (8)

where $\eta=p_{0}\frac{\mu}{\mu_{0}}$, $f_{0}=f(0),\varphi_{0}=\varphi(0)$,
$\dot{f}_{0}=\frac{df}{d\zeta}|_{\zeta=0}$, and
$\dot{\varphi}_{0}=\frac{d\varphi}{d\zeta}|_{\zeta=0}$. One may note, from the
above equations, that

$4\eta^{2}=\dot{f}_{0}^{2}+f_{0}^{2}(\eta+\dot{\varphi}_{0})^{2}.$ (9)

Similarly, at the interface $\zeta=\zeta_{d}$ one obtains

$\begin{cases}t=f_{d}\mathrm{e}^{i\varphi_{d}}\cr
t=\frac{1}{i\eta}\left[\dot{f}_{d}\mathrm{e}^{i\varphi_{d}}+if_{d}\dot{\varphi}_{d}\mathrm{e}^{i\varphi_{d}}\right],\end{cases}$
(10)

where $f_{d}=f(\zeta_{d}),\varphi_{d}=\varphi(\zeta_{d})$,
$\dot{f}_{d}=\frac{df}{d\zeta}|_{\zeta=\zeta_{d}}$, and
$\dot{\varphi}_{d}=\frac{d\varphi}{d\zeta}|_{\zeta=\zeta_{d}}$. Notice that
the transmission coefficient is given by $T=tt^{*}=f_{d}^{2}=F_{d}$. From the
set of Eqs. (10) one may see that $\dot{\varphi}_{d}=\eta$ and
$\dot{f_{d}}=0$. By evaluating Eq. (6) at the interface $\zeta=\zeta_{d}$ one
obtains $A=\eta\,F_{d}$. Moreover, from Eq. (7) evaluated at $\zeta=\zeta_{d}$
one has the constant $B=F_{d}\left[\eta^{2}+p^{2}+\frac{a\mu}{2}F_{d}\right]$
as a function of both $F_{d}$ and $a$. Finally, we use the above results to
expand Eq. (9). In the process we have taken into account that
$\dot{\varphi}_{0}=\frac{A}{F_{0}}$ [cf. Eq. (6)], where $F_{0}=f_{0}^{2}$.
The result is

$4\eta^{2}-F_{d}\left[3\eta^{2}+p^{2}+\frac{a\mu}{2}F_{d}\right]+F_{0}\left[p^{2}-\eta^{2}+\frac{a\mu}{2}F_{0}\right]=0.$
(11)

Figure 2: (Color online) Transmission coefficient as a function of
$\tilde{\lambda}=\frac{\alpha}{\varepsilon}|E_{i}|^{2}$. Lines correspond to
the results obtained from Eq. (11), whereas symbols correspond to theoretical
results reported in Ref. Chen and Mills (1987a). Calculations were performed
for $\varepsilon_{0}=\mu_{0}=1$, $\mu=1$, and $\alpha>0$. The dashed line and
the squares correspond to $\varepsilon=4$, whereas the circles and the solid
line correspond to $\varepsilon=100$. In each case, the thickness of the slab
was taken as $d=0.4\lambda_{f}$, where
$\lambda_{f}=\frac{2\pi}{\omega}\frac{c}{\sqrt{\varepsilon\mu}}$.

The above equation may be used to obtain the transmission coefficient as a
function of $a=\alpha|E_{i}|^{2}$. The procedure consists of providing a value
of $F_{d}$ such that $0<F_{d}\leq 1$, that is, the transmission coefficient
$T$. The electric-field intensity at the $\zeta=0$ interface can be calculated
from Eq. (7) as a function of both $F_{d}$ and $a$ (see below). Eq. (11) then
results in a transcendental equation for $a$, which leads to the possible
values of $a$ for a given value of $T=F_{d}$ that guarantee the fulfillment of
the boundary conditions of the electric field at the interfaces of the slab.
We want to emphasize that this procedure is slightly different but completely
equivalent to those described in Refs. Chen and Mills (1987a) and Leung
(1989). In this regard, we have compared in Fig. 2 the present results
obtained by solving Eq. (11) with the calculations reported by Chen and Mills
Chen and Mills (1987a). According to the definitions given in Ref. Chen and
Mills (1987a), the transmission coefficient was obtained as a function of
$\tilde{\lambda}=\frac{\alpha}{\varepsilon}|E_{i}|^{2}$ for
$\varepsilon_{0}=\mu_{0}=1$ , $\mu=1$, and $\alpha>0$. The dashed line and the
squares in Fig. 2 correspond to a layer with $\varepsilon=4$, whereas the
circles and the solid line correspond to a different layer with
$\varepsilon=100$. In each case, as in Ref. Chen and Mills (1987a), we chose
the thickness of the slab as $d=0.4\lambda_{f}$, where
$\lambda_{f}=\frac{2\pi}{\omega}\frac{c}{\sqrt{\varepsilon\mu}}$. Therefore
$\zeta_{d}=0.4\frac{2\pi}{\sqrt{\varepsilon\mu}}$ is independent of wave
frequency. It is apparent from Fig. 2 that the present theoretical results are
in good agreement with calculations by Chen and Mills Chen and Mills (1987a).

If we take $F_{d}=1$, then the left-hand side of Eq. (11) vanishes for
$F_{0}=1$. In other words, Eq. (11) is satisfied identically by simultaneously
choosing $F_{d}=F_{0}=1$. In this situation, the incident electromagnetic
field transmits completely through the slab. The complete transparency of the
slab is due to the excitation of solitons within the nonlinear layer, that is,
to the existence of electromagnetic states exhibiting unit transmission and
maximum amplitude at the interfaces. From symmetry considerations, it can be
seen that the function $F=F(\zeta)$ corresponding to a soliton state is an
even function with respect to the center of the slab. The condition
$F_{d}=F_{0}=1$ may be assumed as a necessary (but not sufficient) condition
for the existence of solitons within the nonlinear layer.

The square $F$ of the electric-field amplitude may be obtained by integrating
Eq. (7), i.e,

$2(\zeta-\zeta_{d})=\int_{F(\zeta_{d})}^{F(\zeta)}\frac{dF}{\sqrt{-\frac{a\mu}{2}}\sqrt{P(F)}},$
(12)

where $P(F)=(F-F_{1})(F-F_{2})(F-F_{3})$. The three roots of $P$ are given by

$F_{1}=-\frac{1}{2}\left[\sqrt{\left(\frac{2p^{2}}{a\mu}+F_{d}\right)^{2}+\frac{8\eta^{2}}{a\mu}F_{d}}+\frac{2p^{2}}{a\mu}+F_{d}\right],$
(13)
$F_{2}=\frac{1}{2}\left[\sqrt{\left(\frac{2p^{2}}{a\mu}+F_{d}\right)^{2}+\frac{8\eta^{2}}{a\mu}F_{d}}-\frac{2p^{2}}{a\mu}-F_{d}\right],$
(14)

and

$F_{3}=F_{d}.$ (15)

The solutions of Eq. (12) depend on the properties of the roots $F_{1}$,
$F_{2}$, and $F_{3}$. As mentioned above, we have assumed that both
$\varepsilon$ and $\mu$ are not frequency-dependent functions. Also, we have
taken $\varepsilon>\mu>\varepsilon_{0}=\mu_{0}=1$. For computational purposes
we have taken $\varepsilon=11.680$ and $\mu=1.036$. If the hypothetical
material layer we are considering exhibits a self-focusing nonlinearity
($a>0$) then the three roots of $P$ are real and $F_{1}<F_{2}<F_{3}$. In
contrast, in the case of a self-defocusing nonlinearity ($a<0$) several
situations of interest can occur. Let us define the quantities $\xi_{1}$,
$\xi_{2}$, and $\xi_{3}$ such that

$\xi_{1}=-2\left(p^{2}+2\eta^{2}\right)-4\eta\sqrt{p^{2}+\eta^{2}},$ (16)
$\xi_{2}=-2\left(p^{2}+2\eta^{2}\right)+4\eta\sqrt{p^{2}+\eta^{2}},$ (17)

and

$\xi_{3}=\eta^{2}-p^{2},$ (18)

respectively. One may note that $\xi_{1}<\xi_{2}<\xi_{3}<0$. The roots $F_{1}$
and $F_{2}$ are complex numbers whenever $\frac{\xi_{1}}{\mu
F_{d}}<a<\frac{\xi_{2}}{\mu F_{d}}$. If $\frac{\xi_{2}}{\mu F_{d}}<a<0$ or
$a<\frac{\xi_{1}}{\mu F_{d}}$ then the three roots of $P$ are real. In the
interval $\frac{\xi_{3}}{\mu F_{d}}<a<0$ one has $F_{1}<F_{3}<F_{2}$, whereas
if $\frac{\xi_{2}}{\mu F_{d}}<a<\frac{\xi_{3}}{\mu F_{d}}$ or
$a<\frac{\xi_{1}}{\mu F_{d}}$ then $F_{1}<F_{2}<F_{3}$. In the above intervals
Eq. (12) may be expressed in terms of the incomplete elliptic integral of the
first kind Abramowitz and Stegun (1964). Accordingly,

$F(\zeta)=F_{2}+\left(F_{d}-F_{2}\right)\mathrm{cn}^{2}\left[\kappa_{f}\left(\zeta-\zeta_{d}\right),m_{f}\right]$
(19)

if $a>0$,

$F(\zeta)=\frac{F_{1}+\sigma
F_{2}\mathrm{cn}^{2}\left[\kappa_{d1}\left(\zeta-\zeta_{d}\right),m_{d1}\right]}{1+\sigma\mathrm{cn}^{2}\left[\kappa_{d1}\left(\zeta-\zeta_{d}\right),m_{d1}\right]}$
(20)

within the interval $\frac{\xi_{3}}{\mu F_{d}}<a<0$,

$F(\zeta)=\frac{F_{d}-F_{2}\mathrm{sn}^{2}\left[\kappa_{d2}\left(\zeta-\zeta_{d}\right),m_{d2}\right]}{1-\mathrm{sn}^{2}\left[\kappa_{d2}\left(\zeta-\zeta_{d}\right),m_{d2}\right]}$
(21)

if $\frac{\xi_{2}}{\mu F_{d}}<a<\frac{\xi_{3}}{\mu F_{d}}$ or
$a<\frac{\xi_{1}}{\mu F_{d}}$, and

$F(\zeta)=F_{d}+\Lambda\frac{1-\mathrm{cn}\left[\kappa_{d3}\left(\zeta-\zeta_{d}\right),m_{d3}\right]}{1+\mathrm{cn}\left[\kappa_{d3}\left(\zeta-\zeta_{d}\right),m_{d3}\right]}$
(22)

if $\frac{\xi_{1}}{\mu F_{d}}<a<\frac{\xi_{2}}{\mu F_{d}}$. In the above
expressions one has
$\kappa_{f}=\sqrt{\frac{a\mu}{2}\left(F_{d}-F_{1}\right)}$,
$m_{f}=\frac{F_{d}-F_{2}}{F_{d}-F_{1}}$,
$\kappa_{d1}=\sqrt{\frac{|a|\mu}{2}\left(F_{2}-F_{1}\right)}$,
$m_{d1}=\frac{F_{d}-F_{1}}{F_{2}-F_{1}}$,
$\sigma=\frac{F_{d}-F_{1}}{F_{2}-F_{d}}$,
$\kappa_{d2}=\sqrt{\frac{|a|\mu}{2}\left(F_{d}-F_{1}\right)}$,
$m_{d2}=\frac{F_{2}-F_{1}}{F_{d}-F_{1}}$,
$\Lambda=\sqrt{\frac{2}{|a|\mu}F_{d}\left(\eta^{2}-p^{2}-a\mu F_{d}\right)}$,
$\kappa_{d3}=\sqrt[4]{8|a|\mu F_{d}\left(\eta^{2}-p^{2}-a\mu F_{d}\right)}$,
and $m_{d3}=\frac{1}{2}-\frac{1}{8}\frac{\delta}{\Lambda}$, where
$\delta=\frac{4p^{2}}{a\mu}+6F_{d}$. The behavior of the electromagnetic field
around the points $a=\frac{\xi_{i}}{\mu F_{d}}$ ($i=1$, 2, 3) can be studied
separately. For example, from Eqs. (20) and (21) one obtains

$F(\zeta)\xrightarrow[a\rightarrow\frac{\xi_{3}}{\mu F_{d}}]{}F_{d}.$ (23)

Similar analyzes can be performed in the cases $a\rightarrow\frac{\xi_{1}}{\mu
F_{d}}$ and $a\rightarrow\frac{\xi_{2}}{\mu F_{d}}$ (results are not shown
here).

---
Figure 3: (Color online) The solutions of Eqs. (24)-(25) in the $(\nu,a)$
space ($a=\alpha|E_{i}|^{2}$). Results were obtained for a nonlinear layer of
width $d=5$ mm. The white, light gray, upper gray, black, and lower gray
regions correspond to $a>0$, $\frac{\xi_{3}}{\mu}<a<0$,
$\frac{\xi_{2}}{\mu}<a<\frac{\xi_{3}}{\mu}$,
$\frac{\xi_{1}}{\mu}<a<\frac{\xi_{2}}{\mu}$, and $a<\frac{\xi_{1}}{\mu}$,
respectively. Panels (a) and (b) correspond to normal incidence and oblique
incidence with $\theta=\frac{\pi}{4}$, respectively. Dotted lines correspond
to $a=\frac{\xi_{3}}{\mu}$. Only the solid and dotted lines correspond to
physically realizable soliton states (see the text).
---
Figure 4: (Color online) Square of the electric field amplitude as a function
of the position within a nonlinear layer of thickness $d=5$ mm. Solid and
dashed lines correspond to $a=10$ and $a=-10$ within the intervals $a>0$ and
$\frac{\xi_{3}}{\mu}<a<0$, respectively. Results were obtained for $n=2$ (cf.
the second solid line, from the left to the right, in both Figs. 3(a) and
3(b)), where the frequency values corresponding to $a=10$ and $a=-10$ were
obtained from Eqs. (24)-(25). Panels (a) and (b) correspond to $\theta=0$ and
$\theta=\frac{\pi}{4}$, respectively. The dot-dashed horizontal lines
correspond, in all panels, to the soliton solution obtained at the limit
$a\rightarrow\frac{\xi_{3}}{\mu}$. The same result is obtained regardless of
the value of the wave frequency.
---
Figure 5: (Color online) As in Fig. 4, but for solid, dashed, and dotted lines
corresponding to $a=-11$, $a=-14$ and $a=-44$, respectively, within the
regions $\frac{\xi_{2}}{\mu}<a<\frac{\xi_{3}}{\mu}$,
$\frac{\xi_{1}}{\mu}<a<\frac{\xi_{2}}{\mu}$, and $a<\frac{\xi_{3}}{\mu}$,
respectively. Results were obtained for $n=2$ (cf. the second dashed line,
from the left to the right, in both Figs. 3(a) and 3(b)). The corresponding
frequency value for each value of $a$ was obtained from Eqs. (24)-(25). Figure
6: (Color online) Transmission coefficient as a function of the normalized
input intensity for $\nu=10$ GHz. Results were obtained for a self-defocusing
material with $d=5$ mm. Solid and dashed lines correspond to $\theta=0$ and
$\theta=\frac{\pi}{4}$, respectively. Calculations were obtained by using the
fourth order Runge-Kutta method. Vertical dashed lines in the inset indicate
the position of the first soliton peak
($\alpha|E_{i}|^{2}=\frac{\xi_{3}}{\mu}$) for each value of the incidence
angle.

We can now study the existence of soliton states within the nonlinear layer by
imposing the condition $F_{d}=F_{0}=1$ in Eqs. (19)-(22). From such equations
and by taking into account the properties of the Jacobian elliptic functions
Abramowitz and Stegun (1964), we find that

$\zeta_{d}=2ng(a),$ (24)

where $n$ is a natural number and

$g(a)=\begin{cases}\frac{K(m_{f})}{\kappa_{f}}&$if
$a>0\cr\frac{K(m_{d1})}{\kappa_{d1}}&$if
$\frac{\xi_{3}}{\mu}<a<0\cr\frac{K(m_{d2})}{\kappa_{d2}}&$if
$\frac{\xi_{2}}{\mu}<a<\frac{\xi_{3}}{\mu}\cr
2\frac{K(m_{d3})}{\kappa_{d3}}&$if
$\frac{\xi_{1}}{\mu}<a<\frac{\xi_{2}}{\mu}\cr\frac{K(m_{d2})}{\kappa_{d2}}&$if
$a<\frac{\xi_{1}}{\mu}\cr\end{cases}.$ (25)

Eqs. (24) and (25) define a set of curves in the space $(\nu,a)$ at whose
points the condition $F_{d}=F_{0}=1$ is fulfilled. In the above expression
$K(m)$ is the complete elliptic integral of the first kind of parameter $m$
Abramowitz and Stegun (1964).

Fig. 3 shows the numerical results obtained from Eqs. (24)-(25) for a
nonlinear layer of thickness $d=5$ mm. Calculations were performed for normal
incidence and oblique incidence with $\theta=\frac{\pi}{4}$ (cf. Fig. 3(a) and
Fig. 3(b), respectively). The results obtained for $a<0$ and $a>0$ were
depicted on the same graph. Of course, the hypothetical nonlinear material
considered here can only exhibit a self-defocusing or self-focusing
nonlinearity. Therefore, present results must be understood in the sense of
what would happen if our hypothetical material displayed one or another type
of nonlinearity. In this sense, we highlight in Fig. 3 the different regions
of interest in the $(\nu,a)$ space with different gray scales. The white
region corresponds to $a>0$. The light-gray region corresponds to the case
$\frac{\xi_{3}}{\mu}<a<0$. In the upper gray region we have
$\frac{\xi_{2}}{\mu}<a<\frac{\xi_{3}}{\mu}$, whereas
$\frac{\xi_{1}}{\mu}<a<\frac{\xi_{2}}{\mu}$ in the black region and
$a<\frac{\xi_{1}}{\mu}$ in the lower gray region.

As expected, when evaluating the function $F$ at each point of the solid
curves shown in Fig. 3 for $a>\frac{\xi_{3}}{\mu}$, the result satisfies the
condition $F_{0}=F_{d}=1$. According to Eqs. (19)-(20), the functions
$F=F(\zeta)$ are, in these cases, continuous functions of $\zeta$ (or $z$)
within the slab and are symmetric with respect to the center of the slab. Such
electromagnetic states are soliton states propagating through the slab with
$T=1$. Moreover, in the limit $a\rightarrow\frac{\xi_{3}}{\mu}$ [cf. Eq. (23)
for $F_{d}=1$] an electromagnetic mode is obtained, which is spatially
constant within the slab and does not depend on the wave frequency. As a
consequence of Eq. (23), such solution exhibits unit transmission if
$F_{d}=1$. In other words, the transmission coefficient tends toward one as
$a\rightarrow\frac{\xi_{3}}{\mu}$, which results in a soliton state with the
electric field uniformly distributed within the nonlinear layer regardless of
the value of the wave frequency. Such soliton state was not discussed in
previous works Chen and Mills (1987a); Leung (1989). The corresponding soliton
lines in the $(\nu,a)$ space for $\theta=0$ and $\theta=\frac{\pi}{4}$ are
shown in Figs. 3(a) and 3(b), respectively, as horizontal dotted lines on the
boundaries between the respective light-gray and gray regions. We show in Fig.
4 the square of the electric field amplitude as a function of the position
within the nonlinear layer of width $d=5$ mm. Calculations were performed for
$n=2$ and two different values of the incidence angle. Solid and dashed lines
in Fig. 4 correspond to $a=10$ and $a=-10$, respectively, within the intervals
$a>0$ and $\frac{\xi_{3}}{\mu}<a<0$, respectively, and are associated with the
soliton states predicted by Eq. (25). The dot-dashed horizontal line displayed
in each panel of Figs. 4 correspond to the soliton solution obtained at the
limit $a\rightarrow\frac{\xi_{3}}{\mu}$. In each case, the electric-field
amplitude is spatially constant within the slab, and the same result is
obtained regardless of the value of the wave frequency, as explained above.

In contrast, the evaluation of $F$ at the points of the curves within the
$(\nu,a)$ region such that $a<\frac{\xi_{3}}{\mu}$ (cf. dashed lines in Fig.
3) leads to solutions that are physically inadmissible. In spite of
$F_{d}=F_{0}=1$ in these cases, the $F$ functions obtained from Eq. (21) or
Eq. (22) are not continuous functions within the nonlinear layer. The
singularities of $F$ can be characterized by performing a simple analysis. Let
us suppose, for example, that $\frac{\xi_{2}}{\mu}<a<\frac{\xi_{3}}{\mu}$ or
$a<\frac{\xi_{1}}{\mu}$. The poles of $F$ are obtained at the coordinate
$\zeta$ satisfying the equation
$\kappa_{d2}(\zeta-\zeta_{d})=-(2l-1)K(m_{d2})$, where $l$ is a natural
number. Furthermore, the condition $F_{d}=F_{0}=1$ leads to
$\kappa_{d2}\zeta_{d}=2nK(m_{d2})$ (cf. Eq. (25)). By combining the last two
equations, one has
$\zeta\equiv\zeta_{nl}=\left[2(n-l)+1\right]\frac{K(m_{d2})}{\kappa_{d2}}$,
or, equivalently,

$z_{nl}=\frac{c}{\omega}\zeta_{nl}=\frac{2(n-l)+1}{2n}\,d.$ (26)

Since $0<z_{nl}<d$ one has $l=1$, 2, …, $n$. Therefore, if
$\frac{\xi_{2}}{\mu}<a<\frac{\xi_{3}}{\mu}$ or $a<\frac{\xi_{1}}{\mu}$, then
the $n^{\mathrm{th}}$ electromagnetic mode that satisfies the condition
$F_{d}=F_{0}=1$ will exhibit $n$ poles within the nonlinear layer. The
$z_{nl}$ position of the $l^{\mathrm{th}}$ pole is given by Eq. (26). In
addition, it is possible to see that Eq. (26) is also valid within the region
$\frac{\xi_{1}}{\mu}<a<\frac{\xi_{2}}{\mu}$, where $F$ is evaluated through
Eq. (22). We display in Fig. 5 the existence of divergent solutions for the
amplitude of the electric field as well as the position of their respective
poles within the nonlinear layer of thickness $d=5$ mm. Calculations were
performed for $n=2$, for two different values of the incidence angle, and for
$a=-11$, $a=-14$ and $a=-44$ corresponding to the regions
$\frac{\xi_{2}}{\mu}<a<\frac{\xi_{3}}{\mu}$,
$\frac{\xi_{1}}{\mu}<a<\frac{\xi_{2}}{\mu}$, and $a<\frac{\xi_{3}}{\mu}$,
respectively (cf. solid, dashed and dotted lines in Fig. 5, respectively).

A divergence of $F$ inside the nonlinear layer would imply that the nonlinear
material could store infinite electromagnetic energy, a situation that is not
observed in practice. This fact, together with the loss of continuity of $F$
under the condition $F_{d}=F_{0}=1$, indicates that the existence of soliton
states is prohibited for each value of $a$ such that $a<\frac{\xi_{3}}{\mu}$.
Consequently, the solutions of the equations (24)-(25) corresponding to such
values of the normalized input intensity (see dashed lines in Fig. 3) are not
physically realizable states. Furthermore, we would like to stress that our
study on the properties of the electromagnetic field at the limits
$a\rightarrow\frac{\xi_{1}}{\mu}$ and $a\rightarrow\frac{\xi_{2}}{\mu}$ did
not lead to soliton states in these cases.

The most important conclusion drawn from the present analysis is the existence
of a limiting value of the normalized input intensity
($a=\frac{\xi_{3}}{\mu}$) below which soliton excitations in self-defocusing
materials are not possible. In this regard, only the solid and dotted curves
shown in Fig. 3 represent the points in the $(\nu,a)$ space where soliton
states can exist. We want to stress that the possibility of finding soliton
excitations in a sufficiently wide frequency band essentially depends on the
ordering of the roots $F_{1}$, $F_{2}$, and $F_{3}$ given by Eqs. (13), (14),
and (15), respectively. For nonlinear slabs with optical parameters dependent
on the wave frequency, one could expect different possibilities in arranging
the quantities $F_{1}$, $F_{2}$, and $F_{3}$, even in the case of self-
focusing materials. In such a situation, the analysis of which regions of the
frequency spectrum are allowed for soliton excitations could become a more
intricate matter. Further works could be devoted to investigating this
subject.

The existence of a limit for the excitation of soliton states in the self-
defocusing material layer studied above can be verified by numerically solving
Eq. (1). In Fig. 6 we show the transmission coefficient as a function of $a$
for $\nu=10$ GHz. Calculations were obtained by using the fourth-order Runge-
Kutta method. Solid and dashed lines correspond to $\theta=0$ and
$\theta=\frac{\pi}{4}$, respectively. In each case, the first peak from left
to right corresponds to the soliton state obtained at $a=\frac{\xi_{3}}{\mu}$
(see the vertical dashed lines in the inset of Fig. 6). The second peak
corresponds to a soliton state described by Eqs. (24)-(25). From Fig. 6 it
follows that the transmission coefficient decreases monotonically as $a$
decreases, and no evidence of soliton excitation is observed in the region
$a<\frac{\xi_{3}}{\mu}$.

In summary, we have theoretically investigated the excitation of solitons in
optical layers exhibiting Kerr nonlinearities. We found the transmission
coefficient as a function of the normalized input intensity
$a=\alpha|E_{i}|^{2}$ corresponding to an incident monochromatic
electromagnetic wave, which was assumed to be polarized according to the
transverse electric configuration. Furthermore, the fundamental optical
constants associated with the system were considered independent of the wave
frequency. In the case of self-defocusing materials, the present study
indicates that soliton excitations cannot occur if the parameter $a$ is below
a well-defined limiting value. We have shown that, at that limit, a soliton
excitation can occur regardless of the value of the incident wave frequency.
Present theoretical work extends and complements the previous results by Chen
and Mills Chen and Mills (1987a) and Leung Leung (1989), and we hope that it
will stimulate further experimental research on this topic.

###### Acknowledgements.

The authors would like to thank the Scientific Colombian Agency CODI -
University of Antioquia for partial financial support.

## References

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