Contents of file 'lect10/lect10.tex':
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
Generated by ::viewsrc::