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% File: nlopt/lect10/lect10.tex [pure TeX code]2% Last change: March 10, 20033%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 Jonsson8%9 \input epsf 10%11% Read amssym to get the AMS {\Bbb E} font (strikethrough E) and12% 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 transform19%20 \def\fourier{\mathop{\frak F}\nolimits} 21 \def\Re{\mathop{\rm Re}\nolimits}% real part22 \def\Im{\mathop{\rm Im}\nolimits}% imaginary part23 \def\Tr{\mathop{\rm Tr}\nolimits}% quantum mechanical trace24 \def\sinc{\mathop{\rm sinc}\nolimits}% the sinc(x)=sin(x)/x function25 \def\sech{\mathop{\rm sech}\nolimits}% the sech(x)=... function26 \def\sgn{\mathop{\rm sgn}\nolimits}% sgn(x)=0, if x<0, sgn(x)=1, otherwise27 \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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