Contents of file 'lect10/lect10.tex':




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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
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12   % the Euler fraktur font.
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14   \input amssym
15   \font\ninerm=cmr9
16   \font\twelvesc=cmcsc10
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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
30       \else\hfill\fi}
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}
189   $\buildrel{\rm propagation}\over\longmapsto$
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}
217   $\buildrel{\rm propagation}\over\longmapsto$
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}
277   \epsfxsize=50mm\epsfbox{nonlin2.eps}
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 $n_2$
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
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
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