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    1   % File: nlopt/lect10/lect10.tex [pure TeX code]
    2   % Last change: March 10, 2003
    3   %
    4   % Lecture No 10 in the course ``Nonlinear optics'', held January-March,
    5   % 2003, at the Royal Institute of Technology, Stockholm, Sweden.
    6   %
    7   % Copyright (C) 2002-2003, Fredrik Jonsson
    8   %
    9   \input epsf
   10   %
   11   % Read amssym to get the AMS {\Bbb E} font (strikethrough E) and
   12   % the Euler fraktur font.
   13   %
   14   \input amssym
   15   \font\ninerm=cmr9
   16   \font\twelvesc=cmcsc10
   17   %
   18   % Use AMS Euler fraktur style for short-hand notation of Fourier transform
   19   %
   20   \def\fourier{\mathop{\frak F}\nolimits}
   21   \def\Re{\mathop{\rm Re}\nolimits}     % real part
   22   \def\Im{\mathop{\rm Im}\nolimits}     % imaginary part
   23   \def\Tr{\mathop{\rm Tr}\nolimits}     % quantum mechanical trace
   24   \def\sinc{\mathop{\rm sinc}\nolimits} % the sinc(x)=sin(x)/x function
   25   \def\sech{\mathop{\rm sech}\nolimits} % the sech(x)=... function
   26   \def\sgn{\mathop{\rm sgn}\nolimits}   % sgn(x)=0, if x<0, sgn(x)=1, otherwise
   27   \def\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm
   28     \topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt
   29     \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
   30       \else\hfill\fi}
   31     \def\rightheadline{\tenrm{\it Lecture notes #1}
   32       \hfil{\it Nonlinear Optics 5A5513 (2003)}}
   33     \def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)}
   34       \hfil{\it Lecture notes #1}}
   35     \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps}
   36     \vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}}
   37     \hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1}
   38     \vskip 24pt\noindent}
   39   \def\section #1 {\medskip\goodbreak\noindent{\bf #1}
   40     \par\nobreak\smallskip\noindent}
   41   \def\subsection #1 {\smallskip\goodbreak\noindent{\it #1}
   42     \par\nobreak\smallskip\noindent}
   43   
   44   \lecture{10}
   45   In this lecture, we will focus on examples of electromagnetic wave
   46   propagation in nonlinear optical media, by applying the forms of Maxwell's
   47   equations as obtained in the eighth lecture to a set of particular
   48   nonlinear interactions as described by the previously formulated nonlinear
   49   susceptibility formalism.
   50   \medskip
   51   
   52   \noindent The outline for this lecture is:
   53   \item{$\bullet$}{What are solitons?}
   54   \item{$\bullet$}{Basics of soliton theory}
   55   \item{$\bullet$}{Spatial and temporal solitons}
   56   \item{$\bullet$}{The mathematical equivalence between spatial and
   57     temporal solitons}
   58   \item{$\bullet$}{The creation of temporal and spatial solitons}
   59   \medskip
   60   
   61   \section{What are solitons?}
   62   The first reported observation of solitons was made in 1834 by
   63   John Scott Russell, a Scottish scientist and later famous Victorian
   64   engineer and shipbuilder, while studying water waves in the
   65   Glasgow-Edinburgh channel.
   66   As part of this investigation, he was observing a boat being pulled along,
   67   rapidly, by a pair of horses. For some reason, the horses stopped the boat
   68   rather suddenly, and the stopping of the boat caused a verystrong wave to
   69   be generated. This wave, in fact, a significant hump of water stretching
   70   across the rather narrow canal, rose up at the front of the boat and
   71   proceeded to travel, quite rapidly down the canal. Russell, immediately,
   72   realised that the wave was something very special. It was ``alone'', in
   73   the sense that it sat on the canal with no disturbance to the front or
   74   the rear, nor did it die away until he had followed it for quite a long
   75   way. The word ``alone'' is synonymous with ``solitary'', and Russell soon
   76   referred to his observation as the Great Solitary Wave.
   77   
   78   The word ``solitary'' is now routinely used, indeed even the word
   79   ``solitary'' tends to be replaced by the more generic word ``soliton''.
   80   Once the physics behind Russell's wave is understood, however, solitons,
   81   of one kind or another, appear to be everywhere but it is interesting that
   82   the underlying causes of soliton generation were not understood by Russell,
   83   and only partially by his contemporaries.
   84   
   85   \section{Classes of solitons}
   86   \subsection{Bright temporal envelope solitons}
   87   Pulses of light with a certain shape and energy that can propagate
   88   unchanged over large distances.
   89   This is the class of solitons which we will focus on in this lecture.
   90   \smallskip
   91   
   92   \subsection{Dark temporal envelope solitons}
   93   Pulses of ``darkness'' within a continuous wave, where the pulses are
   94   of a certain shape, and possess propagation properties similar to
   95   the bright solitons.
   96   \smallskip
   97   
   98   \subsection{Spatial solitons}
   99   Continuous wave beams or pulses, with a transverse extent of the beam that
  100   via the refractive index change due to optical Kerr-effect can compensate
  101   for the diffraction of the beam. The optically induced change of refractive
  102   index works as an effective waveguide for the light.
  103   \smallskip
  104   
  105   \section{The normalized nonlinear Schr\"{o}dinger equation for
  106     temporal solitons}
  107   The starting point for the analysis of temporal solitons is the time-dependent
  108   wave equation for the spatial envelopes of the electromagnetic fields in
  109   optical Kerr-media, here for simplicity taken for linearly polarized light
  110   in isotropic media,
  111   $$
  112     \Big(i{{\partial}\over{\partial z}}+i{{1}\over{v_{\rm g}}}
  113       {{\partial}\over{\partial t}}
  114       -{{\beta}\over{2}}{{\partial^2}\over{\partial t^2}}\Big)
  115        {\bf A}_{\omega}(z,t)
  116       =-{{\omega n_2}\over{c}}|{\bf A}_{\omega}(z,t)|^2
  117         {\bf A}_{\omega}(z,t),\eqno{(1)}
  118   $$
  119   where, as previously, $v_{\rm g}=(dk/d\omega)^{-1}$ is the linear group
  120   velocity, and where we introduced the notation
  121   $$
  122     \beta={{d^2 k}\over{d\omega^2}}\Big|_{\omega_{\sigma}}
  123   $$
  124   for the second order linear dispersion of the medium, and (in analogy
  125   with Butcher and Cotter Eq.~(6.63)),
  126   $$
  127     n_2=({{3}/{8n_0}})\chi^{(3)}_{xxxx}
  128   $$
  129   for the intensity-dependent refractive index $n=n_0+n_2|{\bf E}_{\omega}|^2$.
  130   Since we here are considering wave propagation in isotropic media,
  131   with linearly polarized light (for which no polarization state cross-talk
  132   occur), the wave equation~(1) is conveniently taken in a scalar form as
  133   $$
  134     \Big(i{{\partial}\over{\partial z}}+i{{1}\over{v_{\rm g}}}
  135       {{\partial}\over{\partial t}}
  136       -{{\beta}\over{2}}{{\partial^2}\over{\partial t^2}}\Big) A_{\omega}(z,t)
  137       =-{{\omega n_2}\over{c}}|A_{\omega}(z,t)|^2 A_{\omega}(z,t).\eqno{(2)}
  138   $$
  139   Equation (2) consists of three terms that interact. The first two terms
  140   contain first order derivatives of the envelope, and these terms can
  141   be seen as the homogeneous part of a wave equation for the envelope,
  142   giving travelling wave solutions that depend on the other two terms,
  143   which rather act like source terms.
  144   
  145   The third term contains a second order derivative of the envelope,
  146   and this terms is also linearly dependent on the dispersion $\beta$
  147   of the medium, that is to say, the change of the group velocity
  148   of the medium with respect to the angular frequency $\omega$ of the
  149   light. This term is generally responsible for smearing out a short pulse as
  150   it traverses a dispersive medium.
  151   
  152   Finally, the fourth term is a nonlinear source term, which depending on
  153   the sign of $n_2$ will concentrate higher frequency components either at
  154   the leading or trailing edge of the pulse, as soon will be shown.
  155   
  156   \subsection{The effect of dispersion}
  157   The {\sl group velocity dispersion} $d v_{\rm g}/d\omega$ is related to the
  158   introduced dispersion parameter $\beta\equiv d^2k/d\omega^2$ as
  159   $$
  160     {{dv_{\rm g}}\over{d\omega}}
  161        ={{d}\over{d\omega}}\left[\left(
  162         {{d k(\omega)}\over{d\omega}}\right)^{-1}\right]
  163        =-\underbrace{\left({{d k(\omega)}\over{d\omega}}\right)^{-2}}_{
  164            \equiv v^2_{\rm g}}
  165          \underbrace{{{d^2 k(\omega)}\over{d\omega^2}}}_{\equiv\beta}
  166        =-v^2_{\rm g}\beta,
  167   $$
  168   and hence the sign of the group velocity dispersion is the opposite
  169   of the sign of the dispersion parameter $\beta$.
  170   In order to get a qualitative picture of the effect of linear dispersion,
  171   let us consider the effect of the sign of $\beta$:
  172   \medskip
  173   \item{$\bullet$}{$\beta>0$:
  174   
  175   For this case, the group velocity dispersion is negative, since
  176   $${{dv_{\rm g}}\over{d\omega}}=-v^2_{\rm g}\beta < 0.$$}
  177   \item{$~$}{
  178   This implies that the {\sl group velocity decreases} with an increasing
  179   angular frequency $\omega$. In other words, the ``blue'' frequency components
  180   of the pulse travel slower than the ``red'' components. Considering the
  181   effects on the pulse as it propagates, the leading edge of the pulse will
  182   after some distance contain a higher concentration of low (``red'')
  183   frequencies, while the trailing edge rather will contain a higher
  184   concentration of high (``blue'') frequencies. This effect is illustrated
  185   in Fig.~1.}
  186   \bigskip
  187   \centerline{\epsfxsize=50mm\epsfbox{nonchirp.eps}
  188   \qquad
  189   $\buildrel{\rm propagation}\over\longmapsto$
  190   \qquad
  191   \epsfxsize=50mm\epsfbox{poschirp.eps}}
  192   \medskip
  193   \centerline{Figure 1. Pulse propagation in a linearly dispersive medium
  194     with $\beta>0$.}
  195   \medskip
  196   \item{$~$}{
  197   Whenever ``red'' frequency components travel faster than ``blue'' components,
  198   we usually associate this with so-called {\sl normal dispersion}.}
  199   \medskip
  200   \item{$\bullet$}{$\beta<0$:
  201   
  202   For this case, the group velocity dispersion is instead positive, since now
  203   $${{dv_{\rm g}}\over{d\omega}}=-v^2_{\rm g}\beta > 0.$$}
  204   \item{$~$}{
  205   This implies that the {\sl group velocity increases} with an increasing
  206   angular frequency $\omega$. In other words, the ``blue'' frequency components
  207   of the pulse now travel {\sl faster} than the ``red'' components. Considering
  208   the effects on the pulse as it propagates, the leading edge of the pulse will
  209   after some distance hence contain a higher concentration of high (``blue'')
  210   frequencies, while the trailing edge rather will contain a higher
  211   concentration of low (``red'') frequencies. This effect, being the inverse
  212   of the one described for a negative group velocity dispersion, is illustrated
  213   in Fig.~2.}
  214   \bigskip
  215   \centerline{\epsfxsize=50mm\epsfbox{nonchirp.eps}
  216   \qquad
  217   $\buildrel{\rm propagation}\over\longmapsto$
  218   \qquad
  219   \epsfxsize=50mm\epsfbox{negchirp.eps}}
  220   \medskip
  221   \centerline{Figure 2. Pulse propagation in a linearly dispersive medium
  222     with $\beta<0$.}
  223   \medskip
  224   \item{$~$}{
  225   Whenever ``blue'' frequency components travel faster than ``red'' components,
  226   we usually associate this with so-called {\sl anomalous dispersion}.}
  227   \medskip
  228   \noindent
  229   Notice that depending on the distribution of the frequency components
  230   of the pulse as it enters a dispersive medium, the pulse may for some
  231   propagation distance actually undergo {\sl pulse compression}.
  232   For $\beta>0$, this occurs if the leading edge of the pulse contain
  233   a higher concentration of ``blue'' frequencies, while for $\beta<0$,
  234   this occurs if the leading edge of the pulse instead contain a higher
  235   concentration of ``red'' frequencies.
  236   \medskip
  237   
  238   \subsection{The effect of a nonlinear refractive index}
  239   Having sorted out the effects of the sign of $\beta$ on the pulse
  240   propagation, we will now focus on the effects of a nonlinear, optical
  241   field dependent refractive index of the medium.
  242   
  243   In order to extract the effect of the nonlinear refractive index, we
  244   will here go to the very definition of the instantaneous angular frequency
  245   of the light from its real-valued electric field,
  246   $$
  247     {\bf E}({\rm r},t)=\Re[{\rm E}_{\omega}({\bf r},t)\exp(-i\omega t)].
  248   $$
  249   For light propagating in a medium where the refractive index depend
  250   on the intensity as
  251   $$
  252     n(t)=n_0+n_2 I(t),
  253   $$
  254   the spatial envelope will typically be described by an effective
  255   propagation constant (see lecture notes as handed out
  256   during lecture nine)
  257   $$
  258     k_{\rm eff}(\omega,I(t))=(\omega/c)(n_0+n_2 I(t)),
  259   $$
  260   and the local, instantaneous angular frequency becomes
  261   $$
  262     \eqalign{
  263       \omega_{\rm loc}
  264         &=-{{d}\over{dt}}\bigg\{{\rm phase\ of\ the\ light}\bigg\}\cr
  265         &=-{{d}\over{dt}}\left[k_{\rm eff}(\omega,I(t))-\omega t\right]\cr
  266         &=-{{d}\over{dt}}\left[{{\omega}\over{c}}(n_0(\omega)+n_2(\omega)I(t))
  267              \right]+\omega\cr
  268         &=\omega-{{\omega n_2(\omega)}\over{c}}{{dI(t)}\over{dt}}.\cr
  269     }
  270   $$
  271   The typical behaviour of the instantaneous angular frequancy
  272   $\omega_{\rm loc}(t)$ on a typical pulse shape is shown in Fig.~3, for the
  273   case of $n_2>0$ and a Gaussian pulse.
  274   \bigskip
  275   \centerline{\epsfxsize=50mm\epsfbox{nonlin1.eps}
  276   \qquad
  277   \epsfxsize=50mm\epsfbox{nonlin2.eps}
  278   \qquad
  279   \epsfxsize=50mm\epsfbox{nonlin3.eps}}
  280   \medskip
  281   \centerline{Figure 3. Effect of a intensity dependent refractive index
  282   $n=n_0+n_2I(t)$ on frequency content of the pulse.}
  283   \medskip
  284   As seen in the figure, the leading edge of the pulse has a slight decrease
  285   in angular frequency, while the trailing edge has a slight increase.
  286   This means that in the presence of an intensity dependent refractive index,
  287   for $n_2>0$, the pulse will have a concentration of ``red'' frequencies at
  288   the leading edge, while the trailing edge will have a concentration of
  289   ``blue'' frequencies. This is illustrated in Fig.~4.
  290   \vfill\eject
  291   
  292   \centerline{\epsfxsize=90mm\epsfbox{poschirp.eps}}
  293   \centerline{Figure 4. Typical frequency chirp of an optical pulse in
  294     a nonlinear medium with $n_2>0$.}
  295   \medskip
  296   
  297   If instead $n_2<0$, i.~e.~for an intensity dependent refractive index that
  298   decrease with an increasing intensity, the roles of the ``red'' and ``blue''
  299   edges of the pulse are reversed.
  300   
  301   \subsection{The basic idea behind temporal solitons}
  302   As seen from Figs.~2 and 4, the effect of anomalous dispersion
  303   (with $\beta<0$) and the effect of a nonlinear, intensity dependent
  304   refractive index (with $n_2>0$) are opposite of each other.
  305   When combined, that is to say, considering pulse propagation in a medium
  306   which simultaneously possesses anomalous dispersion and $n_2>0$,
  307   these effects can combine, {\sl giving a pulse that can propagate without
  308   altering its shape}. This is the basic pronciple of the {\sl temporal soliton}.
  309   
  310   \subsection{Normalization of the nonlinear Schr\"{o}dinger equation}
  311   Equation~(2) can now
  312   be cast into a normalized form, the so-called {\sl nonlinear Schr\"{o}dinger
  313   equation}, by applying the change of variables\footnote{${}^1$}{Please note
  314   that there is a printing error in Butcher and Cotter's book in the section
  315   that deals with the normalization of the nonlinear Schr\"odinger equation.
  316   In the first line of Eq.~(7.55), there is an ambiguity of the denominator,
  317   as well as an erroneous dispersion term, and the equation
  318   $$u=\tau\sqrt{n_2\omega/c|d^2 k/d\omega^2|^2}\widehat{E}$$
  319   should be replaced by
  320   $$u=\tau\sqrt{n_2\omega/(c|d^2 k/d\omega^2|)}\widehat{E}.$$
  321   (The other lines of Eq.~(7.55) in Butcher and Cotter are correct.)}
  322   $$
  323     u=\tau\sqrt{{{n_2\omega}\over{c|\beta|}}} A_{\omega},\qquad
  324     s=(t-z/v_{\rm g})/\tau,\qquad
  325     \zeta=|\beta|z/\tau^2,
  326   $$
  327   where $\tau$ is some characteristic time of the evolution of the pulse,
  328   usually taken as the pulse duration time, which gives the normalized form
  329   $$
  330     \Big(i{{\partial}\over{\partial\zeta}}
  331       -{{1}\over{2}}\sgn(\beta)
  332        {{\partial^2}\over{\partial s^2}}\Big)u(\zeta,s)
  333       +|u(\zeta,s)|^2 u(\zeta,s)=0.\eqno{(3)}
  334   $$
  335   This normalized equation has many interesting properties, and for some
  336   cases even analytical solutions exist, as we will see in the following
  337   sections. Before actually solving the equation, however, we will consider
  338   another mechanism for the generation of solitons.
  339   
  340   Before leaving the temporal pulse propagation, a few remarks on the signs
  341   of the dispersion term $\beta$ and the nonlinear refractive index $n_2$
  342   should be made.
  343   Whenever $\beta>0$, the group velocity dispersion
  344   $$
  345     {{d v_{\rm g}}\over{d\omega}}
  346       \equiv{{d}\over{d\omega}}
  347       \left[\left({{dk}\over{d\omega}}\right)^{-1}\right]
  348      =-\left({{dk}\over{d\omega}}\right)^{-2}{{d^2 k}\over{d\omega^2}}
  349      =-v^2_{\rm g}{{d^2 k}\over{d\omega^2}}
  350   $$
  351   will be negative, and the pulse will experience what we call a
  352   {\sl normal} dispersion, for which the refractive index of the
  353   medium decrease with an increasing wavelength of the light.
  354   This is the ``common'' way dispersion enters in optical processes, where
  355   the pulse is broadened as it traverses the medium.
  356   
  357   \section{Spatial solitons}
  358   As a light beam with some limited spatial extent in the transverse direction
  359   enter an optical Kerr media, the intensity variation across the beam will
  360   via the intensity dependent refractive index $n=n_0+n_2 I$ form a lensing
  361   through the medium.
  362   Depending on the sign of the coefficient $n_2$ (the ``nonlinear refractive
  363   index''), the beam will either experience a defocusing lensing effect
  364   (if $n_2<0$) or a focusing lensing effect (if $n_2>0$); in the latter
  365   case the beam itself will create a self-induced waveguide in the medium
  366   (see Fig.~5).
  367   \bigskip
  368   \centerline{\epsfxsize=130mm\epsfbox{../images/selfocus/selfocus.1}}
  369   \medskip
  370   \centerline{Figure 5. An illustration of the effect of self-focusing.}
  371   \medskip
  372   As being the most important case for beams with maximum intensity in
  373   the middle of the beam (as we usually encounter them in most situations),
  374   we will focus on the case $n_2>0$. For this case, highly intense beams
  375   may cause such a strong focusing that the beam eventually break up again,
  376   due to strong diffraction effects for very narrow beams, or even due
  377   to material damage in the nonlinear crystal.
  378   
  379   For some situations, however, there exist stationary solutions to the
  380   spatial light distribution that exactly balance between the self-focusing
  381   and the diffraction of the beam. We can picture this as a balance between
  382   two lensing effects, with the first one due to self-focusing, with an
  383   effective focal length $f_{\rm foc}$ (see Fig.~6), and the second one
  384   due to diffraction, with an effective focal length of $f_{\rm defoc}$
  385   (see Fig.~7).
  386   \vfill\eject
  387   
  388   \centerline{\epsfxsize=130mm\epsfbox{../images/equilens/equilens.1}}
  389   \medskip
  390   \centerline{Figure 6. Self-focusing seen as an effective lensing of the
  391     optical beam.}
  392   \medskip
  393   \centerline{\epsfxsize=130mm\epsfbox{../images/defolens/defolens.1}}
  394   \medskip
  395   \centerline{Figure 7. Diffraction seen as an effective defocusing of the
  396     optical beam.}
  397   \medskip
  398   \noindent
  399   Whenever these effects balance each other, we in this picture have the
  400   effective focal length $f_{\rm foc}+f_{\rm defoc}=0$.
  401   
  402   In the electromagnetic wave picture, the propagation of an optical
  403   continuous wave in optical Kerr-media is governed by the wave equation
  404   $$
  405     \eqalign{
  406       \nabla\times\nabla\times{\bf E}_{\omega}({\bf r})
  407         -k^2{\bf E}_{\omega}({\bf r})
  408         &=\mu_0\omega^2{\bf P}^{({\rm NL})}_{\omega}({\bf r})\cr
  409         &={{3}\over{4}}{{\omega^2}\over{c^2}}\chi^{(3)}_{xxxx}
  410            |{\bf E}_{\omega}({\bf r})|^2{\bf E}_{\omega}({\bf r}),\cr
  411     }\eqno{(4)}
  412   $$
  413   with $k=\omega n_0/c$, using notations as previously introduced in this
  414   course. For simplicity we will from now on consider the spatial extent of
  415   the beam in only one transverse Cartesian coordinate~$x$.
  416   
  417   By introducing the spatial envelope ${\bf A}_{\omega}(x,z)$ according to
  418   $$
  419     {\bf E}_{\omega}({\bf r})={\bf A}_{\omega}(x,z)\exp(ikz),
  420   $$
  421   and using the slowly varying envelope approximation in the direction of
  422   propagation $z$, the wave equation~(4) takes the form
  423   $$
  424     i{{\partial{\bf A}_{\omega}(x,z)}\over{\partial z}}
  425      +{{1}\over{2k}}
  426       {{\partial^2{\bf A}_{\omega}(x,z)}\over{\partial x^2}}
  427       =-{{\omega n_2}\over{c}}
  428        |{\bf A}_{\omega}(x,z)|^2{\bf A}_{\omega}(x,z).\eqno{(5)}
  429   $$
  430   Notice the strong similarity between this equation for continuous wave
  431   propagation and the equation~(3) for the envelope of a infinite plane wave
  432   pulse. The only significant difference, apart from the physical dimensions
  433   of the involved parameters, is that here nu additional first order
  434   derivative with respect to $x$ is present.
  435   In all other respects, Eqs.~(3) and (5) are identical, if we interchange
  436   the roles of the time $t$ in Eq.~(3) with the transverse spatial coordinate
  437   $x$ in Eq.~(5).
  438   
  439   While the sign of the dispersion parameter $\beta$ occurring in Eq.~(3)
  440   has significance for the compression or broadening of the pulse, no
  441   such sign option appear in Eq.~(5) for the spatial envelope of the
  442   continuous wave beam.
  443   This follows naturally, since the spatial broadening mechanism
  444   (in contrary to the temporal compression or broadening of the pulse)
  445   is due to diffraction, a non-reversible process which in nature always
  446   tend to broaden a collimated light beam.
  447   
  448   As with Eq.~(3) for the temporal pulse propagation, we may now for
  449   the continuous wave case cast Eq.~(5) into a normalized form, by
  450   applying the change of variables
  451   $$
  452     u=L\sqrt{{{n_2\omega k}\over{c}}} A_{\omega},\qquad
  453     s=x/L,\qquad
  454     \zeta=z/(k L^2),
  455   $$
  456   where $L$ is some characteristic length of the evolution of the beam,
  457   usually taken as the transverse beam width, which gives the normalized form
  458   $$
  459     \Big(i{{\partial}\over{\partial\zeta}}
  460       +{{1}\over{2}}{{\partial^2}\over{\partial s^2}}\Big)u(\zeta,s)
  461       +|u(\zeta,s)|^2 u(\zeta,s)=0.\eqno{(6)}
  462   $$
  463   
  464   \section{Mathematical equivalence between temporal and spatial solitons}
  465   As seen in the above derivation of the normalized forms of the equations
  466   governing wave propagation of temporal and spatial solitons, they are
  467   described by exactly the same normalized nonlinear Schr\"odinger equation.
  468   The only difference between the two cases are the ways the normalization
  469   is being carried out.
  470   In the interpretation of the solutions to the nonlinear Schr\"odinger
  471   equation, the $s$ variable could for the temporal solitons be taken
  472   as a normalized time variable, while for the spatial solitons, the $s$
  473   variable could instead be taken as a normalized transverse coordinate.
  474   
  475   \section{Soliton solutions}
  476   The nonlinear Schr\"odinger equations given by Eqs.~(3) and (6) possess
  477   infinitely many solutions, of which only a few are possible to obtain
  478   analytically. In the regime where $dv_{\rm g}/d\omega>0$ (i.~e.~for
  479   which $\beta<0$), an exact temporal soliton solution to Eq.~(3) is
  480   though obtained when the pulse $u(\zeta,s)$ has the initial shape
  481   $$
  482     u(0,s)=N\sech(s),
  483   $$
  484   where $N\ge 1$ is an integer number. Depending on the value of $N$,
  485   solitons of different order can be formed, and the so-called ``fundamental
  486   soliton'' is given for $N=1$. For higher values of $N$, the solitons
  487   are hence called ``higher order solitons''.
  488   
  489   The first analytical solution to the nonlinear Schr\"odinger
  490   equation is given for $N=1$ as\footnote{${}^2$}{Please note that there is
  491   a printing error in Butcher and Cotter's {\sl The Elements of Nonlinear
  492   Optics} in their expression for this solution, on page 241, row 30,
  493   where their erroneous equation ``$u(\zeta,s)=\sech(s)\exp(-i\zeta/2)$''
  494   should be replaced by the proper one, {\sl without} the minus sign in
  495   the exponential.}
  496   $$
  497     u(\zeta,s)=\sech(s)\exp(i\zeta/2).
  498   $$
  499   The shape of this fundamental solution is shown in Fig.~8.
  500   \vfill\eject
  501   
  502   \centerline{\epsfxsize=70mm\epsfbox{fund3d.eps}\qquad
  503     \epsfxsize=70mm\epsfbox{fund2d.eps}}
  504   \medskip
  505   \centerline{Figure 8. The fundamental bright soliton solution to the NLSE.}
  506   \medskip
  507   \noindent
  508   For higher order solitons, the behaviour is usually not stable
  509   with respect to the normalized distance $\zeta$, but rather of an
  510   oscillatory nature, as shown in Fig.~10.1 of the handed out material.
  511   (Figure 10.1 is copied from Govind P.~Agrawal {\sl Fiber-Optic
  512   Communication systems} (Wiley, New York, 1997).)
  513   This figure shows the fundamental soliton together with the third
  514   order ($N=3$) soliton, and one can see that there is a continuous,
  515   oscillatory energy transfer in the $s$-direction of the pulse.
  516   (See also Butcher and Cotter's Fig.~7.8 on page 242, where the $N=4$
  517   soliton is shown.)
  518   
  519   The solutions so far discussed belong to a class called ``bright solitons''.
  520   The reason for using the term ``bright soliton'' becomes more clear if
  521   we consider another type of solutions to the nonlinear Schrödinger
  522   equation, namely the ``dark'' solitons, given as the solutions
  523   $$
  524     u(\zeta,s)=[\eta\tanh(\eta(s-\kappa\zeta))-i\kappa]\exp(iu^2_0\zeta),
  525   $$
  526   with $u_0$ being the normalized amplitude of the continuous-wave background,
  527   $\phi$ is an internal phase angle in the range $0\le\phi\le\pi/2$, and
  528   $$
  529     \eta=u_0\cos\phi,\qquad\kappa=u_0\sin\phi.
  530   $$
  531   For the dark solitons, one makes a distinction between the ``black''
  532   soliton for $\phi=0$, which drops down to zero intensity in the middle
  533   of the pulse, and the ``grey'' solitons for $\phi\ne 0$, which do not
  534   drop down to zero.
  535   For the black solitons, the solution for $\phi=0$ takes ths simpler form
  536   $$
  537     u(\zeta,s)=u_0\tanh(u_0 s)\exp(iu^2_0\zeta).
  538   $$
  539   The shape of the black fundamental soliton is shown in Fig.~9.
  540   \bigskip
  541   \centerline{\epsfxsize=70mm\epsfbox{dark3d.eps}\qquad
  542     \epsfxsize=70mm\epsfbox{dark2d.eps}}
  543   \medskip
  544   \centerline{Figure 9. The fundamental dark (black) soliton solution
  545     to the NLSE.}
  546   \vfill\eject
  547   
  548   Another important difference between the bright and the dark soliton, apart
  549   from their obvious difference in appearances, is that the velocity of a
  550   dark soliton depends on its amplitude, through the internal phase angle
  551   $u^2_0\zeta$. This is not the case for the bright solitons, which propagate
  552   with the same velocity irregardless of the amplitude.
  553   
  554   The darks soliton is easily pictured as a dark travelling pulse
  555   in an otherwise continuous level background intensity. The described
  556   dark solitons, however, are equally well applied to spatial solitons
  557   as well, for the case $n_2>0$, where a dark center of the beam causes
  558   a slightly {\sl lower} refractive index than for the illuminated
  559   surroundings, hence generating an effective ``anti-waveguide'' that
  560   compensates for the diffraction experianced by the black center.
  561   
  562   \section{General travelling wave solutions}
  563   It should be emphasized that the nonlinear Schr\"odinger equation permits
  564   travelling wave solutions as well. On example of such an exact solution
  565   is given by
  566   $$
  567     u(\zeta,s)=a\sech[a(s-c\zeta/\sqrt{2})]
  568     \exp[ic(s\sqrt{2}-c\zeta)/2+in\zeta]
  569   $$
  570   where $n=(1/2)(a^2+c^2/2)$. That this in fact {\sl is} a solution to the
  571   nonlinear Schr\"odinger equation,
  572   $$
  573     \Big(i{{\partial}\over{\partial\zeta}}
  574       +{{1}\over{2}}
  575        {{\partial^2}\over{\partial s^2}}\Big)u(\zeta,s)
  576       +|u(\zeta,s)|^2 u(\zeta,s)=0,
  577   $$
  578   (here for simplicity taken for the special case $\sgn(\beta)=-1$)
  579   is straightforward to verify by, for example, using the following MapleV
  580   blocks:
  581   \medskip
  582   {\obeyspaces\obeylines\tt
  583   ~  restart:
  584   ~  assume(s,real);
  585   ~  assume(zeta,real);
  586   ~  assume(a,real);
  587   ~  assume(c,real);
  588   ~  n:=(1/2)*(a\^ 2+c\^ 2/2);
  589   ~  u(zeta,s):=a*sech(a*(s*sqrt(2)-c*zeta)/sqrt(2))
  590   ~                *exp(I*((c/2)*(s*sqrt(2)-c*zeta)+n*zeta));
  591   ~  nlse:=I*diff(u(zeta,s),zeta)+(1/2)*diff(u(zeta,s),s\$2)
  592   ~                +conjugate(u(zeta,s))*u(zeta,s)\^ 2;
  593   ~  simplify(nlse);
  594   }
  595   \medskip
  596   \noindent
  597   For further information regarding travelling wave solutions and higher
  598   order soliton solutions to the nonlinear Schr\"odinger equation, see
  599   P.~G.~Johnson and R.~S.~Drazin, {\sl Solitons: an introduction}
  600   (Cambridge Univrsity Press, Cambridge, 1989).
  601   
  602   \section{Soliton interactions}
  603   One can understand the implications of soliton interaction by solving the
  604   NLSE numerically with the input amplitude consisting of a soliton pair
  605   $$
  606     u(0,\tau)=\sech(\tau-q_0)+r\sech[r(\tau+q_0)]\exp(i\vartheta)
  607   $$
  608   with, as previously, $\sech(x)\equiv 1/\cosh(x)$, and
  609   $r$ is the relative amplitude of the second soliton with respect to the
  610   other, $\vartheta$ the phase difference, and $2 q_0$ the initial,
  611   normalized separation between the solitons.
  612   
  613   A set of computer generated solutions to this pair of initial soliton
  614   shapes are shown in the handed-out Fig.~10.6 of Govind P.~Agrawal
  615   {\sl Fiber-Optic Communication systems} (Wiley, New York, 1997).
  616   In this figure, the upper left graph shows that a pair of solitons may,
  617   as a matter of fact, attract each other, forming a soliton pair which
  618   oscillate around the center of the moving reference frame.
  619   
  620   Another interesting point is that soliton pairs may be formed by spatial
  621   solitons as well. In Fig.~9 of the handed-out material, the self-trapping
  622   of two spatial solitons, launched with initial trajectories that do
  623   not lie in the same plane, are shown.
  624   In this experiment, carried out by Mitchell et.~al.~at
  625   Princeton\footnote{${}^3$}{M.~Mitchell, Z.~Chen, M.~Shih, and M.~Sageev,
  626   Phys.~Rev.~Lett. {\bf 77}, 490 (1996).}, the two solitons start spiraling
  627   around each other in a helix, experiencing attractive forces that
  628   together with the orbital momentum carried by the pulses form a stable
  629   configuration.
  630   
  631   \section{Dependence on initial conditions}
  632   For a real situation, one might ask oneself how sensitive the forming
  633   of solitons is, depending on perturbations on the preferred $\sech(s)$
  634   initial shape. In a real situation, for example, we will rarely be able
  635   to construct the exact pulse form required for launching a pulse that
  636   will possess the soliton properties already from the beginning.
  637   
  638   As a matter of fact, the soliton formation process accepts quite a broad
  639   range of initial pulse shapes, and as long as the initial intensity is
  640   sufficiently well matched to the energy content of the propagating soliton,
  641   the generated soliton is remarkable stable against perurbations.
  642   In a functional theoretical analogy, we may call this the soliton
  643   ``acceptance angle'' of initial functions that will be accepted
  644   for soliton formation in a medium.
  645   
  646   In order to illustrate the soliton formation, one may study Figs.~10.2
  647   and~10.3 of Govind P.~Agrawal {\sl Fiber-Optic Communication systems}
  648   (Wiley, New York, 1997)\footnote{${}^4$}{The same pictures can be found in
  649   Govind P.~Agrawal {\sl Nonlinear Fiber Optics}
  650   (Academic Press, New York, 1989).}
  651   In Fig.~10.2, the input pulse shape is a Gaussian, rather than the
  652   natural $\sech(s)$ initial shape. As can be seen in the figure, the
  653   pulse shape gradually change towards the fundamental soliton, even
  654   though the Gaussian shape is a quite bad approximation to the final
  655   $\sech(s)$ form.
  656   
  657   The forming of the soliton does not only depend on the initial shape of the
  658   pulse, but also on the peak intensity of the pulse. In Fig.~10.3, an ideal
  659   $\sech(s)$ pulse shape, though with a 20 percent higher pulse amplitude
  660   than the ideal one of unity, is used as input. In this case the pulse
  661   slightly oscillate in amplitude during the propagation, but finally
  662   approaching the fundamental soliton solution.
  663   
  664   Finally, as being an example of an even worse approximation to the~$\sech(s)$
  665   shape, a square input pulse can also generate solitons, as shown in the
  666   handed-out Fig.~16 of {\sl Beam Shaping and Control with Nonlinear Optics},
  667   Eds.~F.~Kajzar and R.~Reinisch (Plenum Press, New York, 1998).
  668   \bye
  669   

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