% File: nlopt/lect10/lect10.tex [pure TeX code]
% Last change: March 10, 2003
%
% Lecture No 10 in the course ``Nonlinear optics'', held January-March,
% 2003, at the Royal Institute of Technology, Stockholm, Sweden.
%
% Copyright (C) 2002-2003, Fredrik Jonsson
%
\input epsf
%
% Read amssym to get the AMS {\Bbb E} font (strikethrough E) and
% the Euler fraktur font.
%
\input amssym
\font\ninerm=cmr9
\font\twelvesc=cmcsc10
%
% Use AMS Euler fraktur style for short-hand notation of Fourier transform
%
\def\fourier{\mathop{\frak F}\nolimits}
\def\Re{\mathop{\rm Re}\nolimits} % real part
\def\Im{\mathop{\rm Im}\nolimits} % imaginary part
\def\Tr{\mathop{\rm Tr}\nolimits} % quantum mechanical trace
\def\sinc{\mathop{\rm sinc}\nolimits} % the sinc(x)=sin(x)/x function
\def\sech{\mathop{\rm sech}\nolimits} % the sech(x)=... function
\def\sgn{\mathop{\rm sgn}\nolimits} % sgn(x)=0, if x<0, sgn(x)=1, otherwise
\def\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm
\topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt
\headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
\else\hfill\fi}
\def\rightheadline{\tenrm{\it Lecture notes #1}
\hfil{\it Nonlinear Optics 5A5513 (2003)}}
\def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)}
\hfil{\it Lecture notes #1}}
\noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps}
\vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}}
\hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1}
\vskip 24pt\noindent}
\def\section #1 {\medskip\goodbreak\noindent{\bf #1}
\par\nobreak\smallskip\noindent}
\def\subsection #1 {\smallskip\goodbreak\noindent{\it #1}
\par\nobreak\smallskip\noindent}
\lecture{10}
In this lecture, we will focus on examples of electromagnetic wave
propagation in nonlinear optical media, by applying the forms of Maxwell's
equations as obtained in the eighth lecture to a set of particular
nonlinear interactions as described by the previously formulated nonlinear
susceptibility formalism.
\medskip
\noindent The outline for this lecture is:
\item{$\bullet$}{What are solitons?}
\item{$\bullet$}{Basics of soliton theory}
\item{$\bullet$}{Spatial and temporal solitons}
\item{$\bullet$}{The mathematical equivalence between spatial and
temporal solitons}
\item{$\bullet$}{The creation of temporal and spatial solitons}
\medskip
\section{What are solitons?}
The first reported observation of solitons was made in 1834 by
John Scott Russell, a Scottish scientist and later famous Victorian
engineer and shipbuilder, while studying water waves in the
Glasgow-Edinburgh channel.
As part of this investigation, he was observing a boat being pulled along,
rapidly, by a pair of horses. For some reason, the horses stopped the boat
rather suddenly, and the stopping of the boat caused a verystrong wave to
be generated. This wave, in fact, a significant hump of water stretching
across the rather narrow canal, rose up at the front of the boat and
proceeded to travel, quite rapidly down the canal. Russell, immediately,
realised that the wave was something very special. It was ``alone'', in
the sense that it sat on the canal with no disturbance to the front or
the rear, nor did it die away until he had followed it for quite a long
way. The word ``alone'' is synonymous with ``solitary'', and Russell soon
referred to his observation as the Great Solitary Wave.
The word ``solitary'' is now routinely used, indeed even the word
``solitary'' tends to be replaced by the more generic word ``soliton''.
Once the physics behind Russell's wave is understood, however, solitons,
of one kind or another, appear to be everywhere but it is interesting that
the underlying causes of soliton generation were not understood by Russell,
and only partially by his contemporaries.
\section{Classes of solitons}
\subsection{Bright temporal envelope solitons}
Pulses of light with a certain shape and energy that can propagate
unchanged over large distances.
This is the class of solitons which we will focus on in this lecture.
\smallskip
\subsection{Dark temporal envelope solitons}
Pulses of ``darkness'' within a continuous wave, where the pulses are
of a certain shape, and possess propagation properties similar to
the bright solitons.
\smallskip
\subsection{Spatial solitons}
Continuous wave beams or pulses, with a transverse extent of the beam that
via the refractive index change due to optical Kerr-effect can compensate
for the diffraction of the beam. The optically induced change of refractive
index works as an effective waveguide for the light.
\smallskip
\section{The normalized nonlinear Schr\"{o}dinger equation for
temporal solitons}
The starting point for the analysis of temporal solitons is the time-dependent
wave equation for the spatial envelopes of the electromagnetic fields in
optical Kerr-media, here for simplicity taken for linearly polarized light
in isotropic media,
$$
\Big(i{{\partial}\over{\partial z}}+i{{1}\over{v_{\rm g}}}
{{\partial}\over{\partial t}}
-{{\beta}\over{2}}{{\partial^2}\over{\partial t^2}}\Big)
{\bf A}_{\omega}(z,t)
=-{{\omega n_2}\over{c}}|{\bf A}_{\omega}(z,t)|^2
{\bf A}_{\omega}(z,t),\eqno{(1)}
$$
where, as previously, $v_{\rm g}=(dk/d\omega)^{-1}$ is the linear group
velocity, and where we introduced the notation
$$
\beta={{d^2 k}\over{d\omega^2}}\Big|_{\omega_{\sigma}}
$$
for the second order linear dispersion of the medium, and (in analogy
with Butcher and Cotter Eq.~(6.63)),
$$
n_2=({{3}/{8n_0}})\chi^{(3)}_{xxxx}
$$
for the intensity-dependent refractive index $n=n_0+n_2|{\bf E}_{\omega}|^2$.
Since we here are considering wave propagation in isotropic media,
with linearly polarized light (for which no polarization state cross-talk
occur), the wave equation~(1) is conveniently taken in a scalar form as
$$
\Big(i{{\partial}\over{\partial z}}+i{{1}\over{v_{\rm g}}}
{{\partial}\over{\partial t}}
-{{\beta}\over{2}}{{\partial^2}\over{\partial t^2}}\Big) A_{\omega}(z,t)
=-{{\omega n_2}\over{c}}|A_{\omega}(z,t)|^2 A_{\omega}(z,t).\eqno{(2)}
$$
Equation (2) consists of three terms that interact. The first two terms
contain first order derivatives of the envelope, and these terms can
be seen as the homogeneous part of a wave equation for the envelope,
giving travelling wave solutions that depend on the other two terms,
which rather act like source terms.
The third term contains a second order derivative of the envelope,
and this terms is also linearly dependent on the dispersion $\beta$
of the medium, that is to say, the change of the group velocity
of the medium with respect to the angular frequency $\omega$ of the
light. This term is generally responsible for smearing out a short pulse as
it traverses a dispersive medium.
Finally, the fourth term is a nonlinear source term, which depending on
the sign of $n_2$ will concentrate higher frequency components either at
the leading or trailing edge of the pulse, as soon will be shown.
\subsection{The effect of dispersion}
The {\sl group velocity dispersion} $d v_{\rm g}/d\omega$ is related to the
introduced dispersion parameter $\beta\equiv d^2k/d\omega^2$ as
$$
{{dv_{\rm g}}\over{d\omega}}
={{d}\over{d\omega}}\left[\left(
{{d k(\omega)}\over{d\omega}}\right)^{-1}\right]
=-\underbrace{\left({{d k(\omega)}\over{d\omega}}\right)^{-2}}_{
\equiv v^2_{\rm g}}
\underbrace{{{d^2 k(\omega)}\over{d\omega^2}}}_{\equiv\beta}
=-v^2_{\rm g}\beta,
$$
and hence the sign of the group velocity dispersion is the opposite
of the sign of the dispersion parameter $\beta$.
In order to get a qualitative picture of the effect of linear dispersion,
let us consider the effect of the sign of $\beta$:
\medskip
\item{$\bullet$}{$\beta>0$:
For this case, the group velocity dispersion is negative, since
$${{dv_{\rm g}}\over{d\omega}}=-v^2_{\rm g}\beta < 0.$$}
\item{$~$}{
This implies that the {\sl group velocity decreases} with an increasing
angular frequency $\omega$. In other words, the ``blue'' frequency components
of the pulse travel slower than the ``red'' components. Considering the
effects on the pulse as it propagates, the leading edge of the pulse will
after some distance contain a higher concentration of low (``red'')
frequencies, while the trailing edge rather will contain a higher
concentration of high (``blue'') frequencies. This effect is illustrated
in Fig.~1.}
\bigskip
\centerline{\epsfxsize=50mm\epsfbox{nonchirp.eps}
\qquad
$\buildrel{\rm propagation}\over\longmapsto$
\qquad
\epsfxsize=50mm\epsfbox{poschirp.eps}}
\medskip
\centerline{Figure 1. Pulse propagation in a linearly dispersive medium
with $\beta>0$.}
\medskip
\item{$~$}{
Whenever ``red'' frequency components travel faster than ``blue'' components,
we usually associate this with so-called {\sl normal dispersion}.}
\medskip
\item{$\bullet$}{$\beta<0$:
For this case, the group velocity dispersion is instead positive, since now
$${{dv_{\rm g}}\over{d\omega}}=-v^2_{\rm g}\beta > 0.$$}
\item{$~$}{
This implies that the {\sl group velocity increases} with an increasing
angular frequency $\omega$. In other words, the ``blue'' frequency components
of the pulse now travel {\sl faster} than the ``red'' components. Considering
the effects on the pulse as it propagates, the leading edge of the pulse will
after some distance hence contain a higher concentration of high (``blue'')
frequencies, while the trailing edge rather will contain a higher
concentration of low (``red'') frequencies. This effect, being the inverse
of the one described for a negative group velocity dispersion, is illustrated
in Fig.~2.}
\bigskip
\centerline{\epsfxsize=50mm\epsfbox{nonchirp.eps}
\qquad
$\buildrel{\rm propagation}\over\longmapsto$
\qquad
\epsfxsize=50mm\epsfbox{negchirp.eps}}
\medskip
\centerline{Figure 2. Pulse propagation in a linearly dispersive medium
with $\beta<0$.}
\medskip
\item{$~$}{
Whenever ``blue'' frequency components travel faster than ``red'' components,
we usually associate this with so-called {\sl anomalous dispersion}.}
\medskip
\noindent
Notice that depending on the distribution of the frequency components
of the pulse as it enters a dispersive medium, the pulse may for some
propagation distance actually undergo {\sl pulse compression}.
For $\beta>0$, this occurs if the leading edge of the pulse contain
a higher concentration of ``blue'' frequencies, while for $\beta<0$,
this occurs if the leading edge of the pulse instead contain a higher
concentration of ``red'' frequencies.
\medskip
\subsection{The effect of a nonlinear refractive index}
Having sorted out the effects of the sign of $\beta$ on the pulse
propagation, we will now focus on the effects of a nonlinear, optical
field dependent refractive index of the medium.
In order to extract the effect of the nonlinear refractive index, we
will here go to the very definition of the instantaneous angular frequency
of the light from its real-valued electric field,
$$
{\bf E}({\rm r},t)=\Re[{\rm E}_{\omega}({\bf r},t)\exp(-i\omega t)].
$$
For light propagating in a medium where the refractive index depend
on the intensity as
$$
n(t)=n_0+n_2 I(t),
$$
the spatial envelope will typically be described by an effective
propagation constant (see lecture notes as handed out
during lecture nine)
$$
k_{\rm eff}(\omega,I(t))=(\omega/c)(n_0+n_2 I(t)),
$$
and the local, instantaneous angular frequency becomes
$$
\eqalign{
\omega_{\rm loc}
&=-{{d}\over{dt}}\bigg\{{\rm phase\ of\ the\ light}\bigg\}\cr
&=-{{d}\over{dt}}\left[k_{\rm eff}(\omega,I(t))-\omega t\right]\cr
&=-{{d}\over{dt}}\left[{{\omega}\over{c}}(n_0(\omega)+n_2(\omega)I(t))
\right]+\omega\cr
&=\omega-{{\omega n_2(\omega)}\over{c}}{{dI(t)}\over{dt}}.\cr
}
$$
The typical behaviour of the instantaneous angular frequancy
$\omega_{\rm loc}(t)$ on a typical pulse shape is shown in Fig.~3, for the
case of $n_2>0$ and a Gaussian pulse.
\bigskip
\centerline{\epsfxsize=50mm\epsfbox{nonlin1.eps}
\qquad
\epsfxsize=50mm\epsfbox{nonlin2.eps}
\qquad
\epsfxsize=50mm\epsfbox{nonlin3.eps}}
\medskip
\centerline{Figure 3. Effect of a intensity dependent refractive index
$n=n_0+n_2I(t)$ on frequency content of the pulse.}
\medskip
As seen in the figure, the leading edge of the pulse has a slight decrease
in angular frequency, while the trailing edge has a slight increase.
This means that in the presence of an intensity dependent refractive index,
for $n_2>0$, the pulse will have a concentration of ``red'' frequencies at
the leading edge, while the trailing edge will have a concentration of
``blue'' frequencies. This is illustrated in Fig.~4.
\vfill\eject
\centerline{\epsfxsize=90mm\epsfbox{poschirp.eps}}
\centerline{Figure 4. Typical frequency chirp of an optical pulse in
a nonlinear medium with $n_2>0$.}
\medskip
If instead $n_2<0$, i.~e.~for an intensity dependent refractive index that
decrease with an increasing intensity, the roles of the ``red'' and ``blue''
edges of the pulse are reversed.
\subsection{The basic idea behind temporal solitons}
As seen from Figs.~2 and 4, the effect of anomalous dispersion
(with $\beta<0$) and the effect of a nonlinear, intensity dependent
refractive index (with $n_2>0$) are opposite of each other.
When combined, that is to say, considering pulse propagation in a medium
which simultaneously possesses anomalous dispersion and $n_2>0$,
these effects can combine, {\sl giving a pulse that can propagate without
altering its shape}. This is the basic pronciple of the {\sl temporal soliton}.
\subsection{Normalization of the nonlinear Schr\"{o}dinger equation}
Equation~(2) can now
be cast into a normalized form, the so-called {\sl nonlinear Schr\"{o}dinger
equation}, by applying the change of variables\footnote{${}^1$}{Please note
that there is a printing error in Butcher and Cotter's book in the section
that deals with the normalization of the nonlinear Schr\"odinger equation.
In the first line of Eq.~(7.55), there is an ambiguity of the denominator,
as well as an erroneous dispersion term, and the equation
$$u=\tau\sqrt{n_2\omega/c|d^2 k/d\omega^2|^2}\widehat{E}$$
should be replaced by
$$u=\tau\sqrt{n_2\omega/(c|d^2 k/d\omega^2|)}\widehat{E}.$$
(The other lines of Eq.~(7.55) in Butcher and Cotter are correct.)}
$$
u=\tau\sqrt{{{n_2\omega}\over{c|\beta|}}} A_{\omega},\qquad
s=(t-z/v_{\rm g})/\tau,\qquad
\zeta=|\beta|z/\tau^2,
$$
where $\tau$ is some characteristic time of the evolution of the pulse,
usually taken as the pulse duration time, which gives the normalized form
$$
\Big(i{{\partial}\over{\partial\zeta}}
-{{1}\over{2}}\sgn(\beta)
{{\partial^2}\over{\partial s^2}}\Big)u(\zeta,s)
+|u(\zeta,s)|^2 u(\zeta,s)=0.\eqno{(3)}
$$
This normalized equation has many interesting properties, and for some
cases even analytical solutions exist, as we will see in the following
sections. Before actually solving the equation, however, we will consider
another mechanism for the generation of solitons.
Before leaving the temporal pulse propagation, a few remarks on the signs
of the dispersion term $\beta$ and the nonlinear refractive index $n_2$
should be made.
Whenever $\beta>0$, the group velocity dispersion
$$
{{d v_{\rm g}}\over{d\omega}}
\equiv{{d}\over{d\omega}}
\left[\left({{dk}\over{d\omega}}\right)^{-1}\right]
=-\left({{dk}\over{d\omega}}\right)^{-2}{{d^2 k}\over{d\omega^2}}
=-v^2_{\rm g}{{d^2 k}\over{d\omega^2}}
$$
will be negative, and the pulse will experience what we call a
{\sl normal} dispersion, for which the refractive index of the
medium decrease with an increasing wavelength of the light.
This is the ``common'' way dispersion enters in optical processes, where
the pulse is broadened as it traverses the medium.
\section{Spatial solitons}
As a light beam with some limited spatial extent in the transverse direction
enter an optical Kerr media, the intensity variation across the beam will
via the intensity dependent refractive index $n=n_0+n_2 I$ form a lensing
through the medium.
Depending on the sign of the coefficient $n_2$ (the ``nonlinear refractive
index''), the beam will either experience a defocusing lensing effect
(if $n_2<0$) or a focusing lensing effect (if $n_2>0$); in the latter
case the beam itself will create a self-induced waveguide in the medium
(see Fig.~5).
\bigskip
\centerline{\epsfxsize=130mm\epsfbox{../images/selfocus/selfocus.1}}
\medskip
\centerline{Figure 5. An illustration of the effect of self-focusing.}
\medskip
As being the most important case for beams with maximum intensity in
the middle of the beam (as we usually encounter them in most situations),
we will focus on the case $n_2>0$. For this case, highly intense beams
may cause such a strong focusing that the beam eventually break up again,
due to strong diffraction effects for very narrow beams, or even due
to material damage in the nonlinear crystal.
For some situations, however, there exist stationary solutions to the
spatial light distribution that exactly balance between the self-focusing
and the diffraction of the beam. We can picture this as a balance between
two lensing effects, with the first one due to self-focusing, with an
effective focal length $f_{\rm foc}$ (see Fig.~6), and the second one
due to diffraction, with an effective focal length of $f_{\rm defoc}$
(see Fig.~7).
\vfill\eject
\centerline{\epsfxsize=130mm\epsfbox{../images/equilens/equilens.1}}
\medskip
\centerline{Figure 6. Self-focusing seen as an effective lensing of the
optical beam.}
\medskip
\centerline{\epsfxsize=130mm\epsfbox{../images/defolens/defolens.1}}
\medskip
\centerline{Figure 7. Diffraction seen as an effective defocusing of the
optical beam.}
\medskip
\noindent
Whenever these effects balance each other, we in this picture have the
effective focal length $f_{\rm foc}+f_{\rm defoc}=0$.
In the electromagnetic wave picture, the propagation of an optical
continuous wave in optical Kerr-media is governed by the wave equation
$$
\eqalign{
\nabla\times\nabla\times{\bf E}_{\omega}({\bf r})
-k^2{\bf E}_{\omega}({\bf r})
&=\mu_0\omega^2{\bf P}^{({\rm NL})}_{\omega}({\bf r})\cr
&={{3}\over{4}}{{\omega^2}\over{c^2}}\chi^{(3)}_{xxxx}
|{\bf E}_{\omega}({\bf r})|^2{\bf E}_{\omega}({\bf r}),\cr
}\eqno{(4)}
$$
with $k=\omega n_0/c$, using notations as previously introduced in this
course. For simplicity we will from now on consider the spatial extent of
the beam in only one transverse Cartesian coordinate~$x$.
By introducing the spatial envelope ${\bf A}_{\omega}(x,z)$ according to
$$
{\bf E}_{\omega}({\bf r})={\bf A}_{\omega}(x,z)\exp(ikz),
$$
and using the slowly varying envelope approximation in the direction of
propagation $z$, the wave equation~(4) takes the form
$$
i{{\partial{\bf A}_{\omega}(x,z)}\over{\partial z}}
+{{1}\over{2k}}
{{\partial^2{\bf A}_{\omega}(x,z)}\over{\partial x^2}}
=-{{\omega n_2}\over{c}}
|{\bf A}_{\omega}(x,z)|^2{\bf A}_{\omega}(x,z).\eqno{(5)}
$$
Notice the strong similarity between this equation for continuous wave
propagation and the equation~(3) for the envelope of a infinite plane wave
pulse. The only significant difference, apart from the physical dimensions
of the involved parameters, is that here nu additional first order
derivative with respect to $x$ is present.
In all other respects, Eqs.~(3) and (5) are identical, if we interchange
the roles of the time $t$ in Eq.~(3) with the transverse spatial coordinate
$x$ in Eq.~(5).
While the sign of the dispersion parameter $\beta$ occurring in Eq.~(3)
has significance for the compression or broadening of the pulse, no
such sign option appear in Eq.~(5) for the spatial envelope of the
continuous wave beam.
This follows naturally, since the spatial broadening mechanism
(in contrary to the temporal compression or broadening of the pulse)
is due to diffraction, a non-reversible process which in nature always
tend to broaden a collimated light beam.
As with Eq.~(3) for the temporal pulse propagation, we may now for
the continuous wave case cast Eq.~(5) into a normalized form, by
applying the change of variables
$$
u=L\sqrt{{{n_2\omega k}\over{c}}} A_{\omega},\qquad
s=x/L,\qquad
\zeta=z/(k L^2),
$$
where $L$ is some characteristic length of the evolution of the beam,
usually taken as the transverse beam width, which gives the normalized form
$$
\Big(i{{\partial}\over{\partial\zeta}}
+{{1}\over{2}}{{\partial^2}\over{\partial s^2}}\Big)u(\zeta,s)
+|u(\zeta,s)|^2 u(\zeta,s)=0.\eqno{(6)}
$$
\section{Mathematical equivalence between temporal and spatial solitons}
As seen in the above derivation of the normalized forms of the equations
governing wave propagation of temporal and spatial solitons, they are
described by exactly the same normalized nonlinear Schr\"odinger equation.
The only difference between the two cases are the ways the normalization
is being carried out.
In the interpretation of the solutions to the nonlinear Schr\"odinger
equation, the $s$ variable could for the temporal solitons be taken
as a normalized time variable, while for the spatial solitons, the $s$
variable could instead be taken as a normalized transverse coordinate.
\section{Soliton solutions}
The nonlinear Schr\"odinger equations given by Eqs.~(3) and (6) possess
infinitely many solutions, of which only a few are possible to obtain
analytically. In the regime where $dv_{\rm g}/d\omega>0$ (i.~e.~for
which $\beta<0$), an exact temporal soliton solution to Eq.~(3) is
though obtained when the pulse $u(\zeta,s)$ has the initial shape
$$
u(0,s)=N\sech(s),
$$
where $N\ge 1$ is an integer number. Depending on the value of $N$,
solitons of different order can be formed, and the so-called ``fundamental
soliton'' is given for $N=1$. For higher values of $N$, the solitons
are hence called ``higher order solitons''.
The first analytical solution to the nonlinear Schr\"odinger
equation is given for $N=1$ as\footnote{${}^2$}{Please note that there is
a printing error in Butcher and Cotter's {\sl The Elements of Nonlinear
Optics} in their expression for this solution, on page 241, row 30,
where their erroneous equation ``$u(\zeta,s)=\sech(s)\exp(-i\zeta/2)$''
should be replaced by the proper one, {\sl without} the minus sign in
the exponential.}
$$
u(\zeta,s)=\sech(s)\exp(i\zeta/2).
$$
The shape of this fundamental solution is shown in Fig.~8.
\vfill\eject
\centerline{\epsfxsize=70mm\epsfbox{fund3d.eps}\qquad
\epsfxsize=70mm\epsfbox{fund2d.eps}}
\medskip
\centerline{Figure 8. The fundamental bright soliton solution to the NLSE.}
\medskip
\noindent
For higher order solitons, the behaviour is usually not stable
with respect to the normalized distance $\zeta$, but rather of an
oscillatory nature, as shown in Fig.~10.1 of the handed out material.
(Figure 10.1 is copied from Govind P.~Agrawal {\sl Fiber-Optic
Communication systems} (Wiley, New York, 1997).)
This figure shows the fundamental soliton together with the third
order ($N=3$) soliton, and one can see that there is a continuous,
oscillatory energy transfer in the $s$-direction of the pulse.
(See also Butcher and Cotter's Fig.~7.8 on page 242, where the $N=4$
soliton is shown.)
The solutions so far discussed belong to a class called ``bright solitons''.
The reason for using the term ``bright soliton'' becomes more clear if
we consider another type of solutions to the nonlinear Schrödinger
equation, namely the ``dark'' solitons, given as the solutions
$$
u(\zeta,s)=[\eta\tanh(\eta(s-\kappa\zeta))-i\kappa]\exp(iu^2_0\zeta),
$$
with $u_0$ being the normalized amplitude of the continuous-wave background,
$\phi$ is an internal phase angle in the range $0\le\phi\le\pi/2$, and
$$
\eta=u_0\cos\phi,\qquad\kappa=u_0\sin\phi.
$$
For the dark solitons, one makes a distinction between the ``black''
soliton for $\phi=0$, which drops down to zero intensity in the middle
of the pulse, and the ``grey'' solitons for $\phi\ne 0$, which do not
drop down to zero.
For the black solitons, the solution for $\phi=0$ takes ths simpler form
$$
u(\zeta,s)=u_0\tanh(u_0 s)\exp(iu^2_0\zeta).
$$
The shape of the black fundamental soliton is shown in Fig.~9.
\bigskip
\centerline{\epsfxsize=70mm\epsfbox{dark3d.eps}\qquad
\epsfxsize=70mm\epsfbox{dark2d.eps}}
\medskip
\centerline{Figure 9. The fundamental dark (black) soliton solution
to the NLSE.}
\vfill\eject
Another important difference between the bright and the dark soliton, apart
from their obvious difference in appearances, is that the velocity of a
dark soliton depends on its amplitude, through the internal phase angle
$u^2_0\zeta$. This is not the case for the bright solitons, which propagate
with the same velocity irregardless of the amplitude.
The darks soliton is easily pictured as a dark travelling pulse
in an otherwise continuous level background intensity. The described
dark solitons, however, are equally well applied to spatial solitons
as well, for the case $n_2>0$, where a dark center of the beam causes
a slightly {\sl lower} refractive index than for the illuminated
surroundings, hence generating an effective ``anti-waveguide'' that
compensates for the diffraction experianced by the black center.
\section{General travelling wave solutions}
It should be emphasized that the nonlinear Schr\"odinger equation permits
travelling wave solutions as well. On example of such an exact solution
is given by
$$
u(\zeta,s)=a\sech[a(s-c\zeta/\sqrt{2})]
\exp[ic(s\sqrt{2}-c\zeta)/2+in\zeta]
$$
where $n=(1/2)(a^2+c^2/2)$. That this in fact {\sl is} a solution to the
nonlinear Schr\"odinger equation,
$$
\Big(i{{\partial}\over{\partial\zeta}}
+{{1}\over{2}}
{{\partial^2}\over{\partial s^2}}\Big)u(\zeta,s)
+|u(\zeta,s)|^2 u(\zeta,s)=0,
$$
(here for simplicity taken for the special case $\sgn(\beta)=-1$)
is straightforward to verify by, for example, using the following MapleV
blocks:
\medskip
{\obeyspaces\obeylines\tt
~ restart:
~ assume(s,real);
~ assume(zeta,real);
~ assume(a,real);
~ assume(c,real);
~ n:=(1/2)*(a\^ 2+c\^ 2/2);
~ u(zeta,s):=a*sech(a*(s*sqrt(2)-c*zeta)/sqrt(2))
~ *exp(I*((c/2)*(s*sqrt(2)-c*zeta)+n*zeta));
~ nlse:=I*diff(u(zeta,s),zeta)+(1/2)*diff(u(zeta,s),s\$2)
~ +conjugate(u(zeta,s))*u(zeta,s)\^ 2;
~ simplify(nlse);
}
\medskip
\noindent
For further information regarding travelling wave solutions and higher
order soliton solutions to the nonlinear Schr\"odinger equation, see
P.~G.~Johnson and R.~S.~Drazin, {\sl Solitons: an introduction}
(Cambridge Univrsity Press, Cambridge, 1989).
\section{Soliton interactions}
One can understand the implications of soliton interaction by solving the
NLSE numerically with the input amplitude consisting of a soliton pair
$$
u(0,\tau)=\sech(\tau-q_0)+r\sech[r(\tau+q_0)]\exp(i\vartheta)
$$
with, as previously, $\sech(x)\equiv 1/\cosh(x)$, and
$r$ is the relative amplitude of the second soliton with respect to the
other, $\vartheta$ the phase difference, and $2 q_0$ the initial,
normalized separation between the solitons.
A set of computer generated solutions to this pair of initial soliton
shapes are shown in the handed-out Fig.~10.6 of Govind P.~Agrawal
{\sl Fiber-Optic Communication systems} (Wiley, New York, 1997).
In this figure, the upper left graph shows that a pair of solitons may,
as a matter of fact, attract each other, forming a soliton pair which
oscillate around the center of the moving reference frame.
Another interesting point is that soliton pairs may be formed by spatial
solitons as well. In Fig.~9 of the handed-out material, the self-trapping
of two spatial solitons, launched with initial trajectories that do
not lie in the same plane, are shown.
In this experiment, carried out by Mitchell et.~al.~at
Princeton\footnote{${}^3$}{M.~Mitchell, Z.~Chen, M.~Shih, and M.~Sageev,
Phys.~Rev.~Lett. {\bf 77}, 490 (1996).}, the two solitons start spiraling
around each other in a helix, experiencing attractive forces that
together with the orbital momentum carried by the pulses form a stable
configuration.
\section{Dependence on initial conditions}
For a real situation, one might ask oneself how sensitive the forming
of solitons is, depending on perturbations on the preferred $\sech(s)$
initial shape. In a real situation, for example, we will rarely be able
to construct the exact pulse form required for launching a pulse that
will possess the soliton properties already from the beginning.
As a matter of fact, the soliton formation process accepts quite a broad
range of initial pulse shapes, and as long as the initial intensity is
sufficiently well matched to the energy content of the propagating soliton,
the generated soliton is remarkable stable against perurbations.
In a functional theoretical analogy, we may call this the soliton
``acceptance angle'' of initial functions that will be accepted
for soliton formation in a medium.
In order to illustrate the soliton formation, one may study Figs.~10.2
and~10.3 of Govind P.~Agrawal {\sl Fiber-Optic Communication systems}
(Wiley, New York, 1997)\footnote{${}^4$}{The same pictures can be found in
Govind P.~Agrawal {\sl Nonlinear Fiber Optics}
(Academic Press, New York, 1989).}
In Fig.~10.2, the input pulse shape is a Gaussian, rather than the
natural $\sech(s)$ initial shape. As can be seen in the figure, the
pulse shape gradually change towards the fundamental soliton, even
though the Gaussian shape is a quite bad approximation to the final
$\sech(s)$ form.
The forming of the soliton does not only depend on the initial shape of the
pulse, but also on the peak intensity of the pulse. In Fig.~10.3, an ideal
$\sech(s)$ pulse shape, though with a 20 percent higher pulse amplitude
than the ideal one of unity, is used as input. In this case the pulse
slightly oscillate in amplitude during the propagation, but finally
approaching the fundamental soliton solution.
Finally, as being an example of an even worse approximation to the~$\sech(s)$
shape, a square input pulse can also generate solitons, as shown in the
handed-out Fig.~16 of {\sl Beam Shaping and Control with Nonlinear Optics},
Eds.~F.~Kajzar and R.~Reinisch (Plenum Press, New York, 1998).
\bye