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1   % File: nlopt/lect8/lect8.tex [pure TeX code]
2   % Last change: February 24, 2003
3   %
4   % Lecture No 8 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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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   %
25   % Define a handy macro for the list of symmetry operations
26   % in Schoenflies notation for point-symmetry groups.
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28   \newdimen\citemindent \citemindent=40pt
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30   \def\citem[#1]{\smallbreak\noindent\hbox to 20pt{}%
31     \hbox to\citemindent{#1\hfill}%
32     \hangindent\citemleftskip\ignorespaces}
33   \def\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm
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35     \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
36       \else\hfill\fi}
37     \def\rightheadline{\tenrm{\it Lecture notes #1}
38       \hfil{\it Nonlinear Optics 5A5513 (2003)}}
39     \def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)}
40       \hfil{\it Lecture notes #1}}
41     \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps}
42     \vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}}
43     \hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1}
44     \vskip 24pt\noindent}
45   \def\section #1 {\medskip\goodbreak\noindent{\bf #1}
46     \par\nobreak\smallskip\noindent}
47   \def\subsection #1 {\smallskip\goodbreak\noindent{\it #1}
48     \par\nobreak\smallskip\noindent}
49
50   \lecture{8}
51   In this lecture, the electric polarisation density of the medium is
52   finally inserted into Maxwell's equations, and the wave propagation
53   properties of electromagnetic waves in nonlinear optical media is
54   for the first time in this course analysed. As an example of wave
55   propagation in nonlinear optical media, the optical Kerr effect is
56   analysed for infinite plane continuous waves.
57   \medskip
58
59   \noindent The outline for this lecture is:
60   \item{$\bullet$}{Maxwells equations (general electromagnetic wave
61     propagation)}
62   \item{$\bullet$}{Time dependent processes (envelopes slowly varying
63     in space and time)}
64   \item{$\bullet$}{Time independent processes (envelopes slowly varying
65     in space but constant in time)}
66   \item{$\bullet$}{Examples (optical Kerr-effect $\Leftrightarrow$
67     $\chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega)$)}
68   \medskip
69
70   \section{Wave propagation in nonlinear media}
71   \subsection{Maxwell's equations}
72   The propagation of electromagnetic waves are, from a first principles
73   approach, governed by the {\sl Maxwell's equations} (here listed in their
74   real-valued form in SI units),
75   76 \eqalign{ 77 \nabla\times {\bf E}({\bf r},t) 78 &= -{{\partial{\bf B}({\bf r},t)}\over{\partial{t}}},\hskip 50pt 79 ({\textstyle\rm Faraday's\ law})\cr 80 \nabla\times{\bf H}({\bf r},t) 81 &= {\bf J}({\bf r},t) 82 + {{\partial{\bf D}({\bf r},t)}\over{\partial{t}}},\hskip 18.7pt 83 ({\textstyle\rm Ampere's\ law})\cr 84 \nabla\cdot{\bf D}({\bf r},t)&=\rho({\bf r},t),\cr 85 \nabla\cdot{\bf B}({\bf r},t)&=0,\cr 86 } 87
88   where $\rho({\bf r},t)$ is the density of free charges,
89   and ${\bf J}({\bf r},t)$ the corresponding current density of free charges.
90
91   \subsection{Constitutive relations}
92   The constitutive relations are in SI units formulated as
93   94 \eqalign{ 95 {\bf D}({\bf r},t)&=\varepsilon_0{\bf E}({\bf r},t)+{\bf P}({\bf r},t),\cr 96 {\bf B}({\bf r},t)&=\mu_0[{\bf H}({\bf r},t) + {\bf M}({\bf r},t)],\cr 97 } 98
99   where ${\bf P}({\bf r},t)={\bf P}[{\bf E}({\bf r},t),{\bf B}({\bf r},t)]$
100   is the macroscopic polarization density
101   (electric dipole moment per unit volume), and
102   ${\bf M}({\bf r},t)={\bf M}[{\bf E}({\bf r},t),{\bf B}({\bf r},t)]$
103   the magnetization (magnetic dipole moment per unit volume) of the medium.
104
105   Here ${\bf E}({\bf r},t)$ and ${\bf B}({\bf r},t)$ are considered as
106   the fundamental macroscopic electric and magnetic field quantities;
107   ${\bf D}({\bf r},t)$ and ${\bf H}({\bf r},t)$ are the corresponding
108   derived fields associated with the state of matter,
109   connected to ${\bf E}({\bf r},t)$ and ${\bf B}({\bf r},t)$
110   through the electric polarization density ${\bf P}({\bf r},t)$ and
111   magnetization (magnetic polarization density) ${\bf M}({\bf r},t)$
112   through the basic constitutive relations.
113   In fact, the constitutive equations above form the very
114   definitions\footnote{${}^1$}{J.~D.~Jackson, {\sl Classical Electrodynamics},
115   2nd ed.~(Wiley, New York, 1975); J.~A.~Stratton, {\sl Electromagnetic Theory}
116   (Mc\-Graw-Hill, New York, 1941).} of the electric polarization density and
117   magnetization.
118
119   \section{Two frequent assumptions in nonlinear optics}
120   \item{$\bullet$}{No free charges present,
121     $$\rho({\bf r},t)=0,\qquad{\bf J}({\bf r},t)={\bf 0}.$$
122     (Any relaxation processes etc.~are included in
123     imaginary parts of the terms of the electric susceptibility.)}
124   \item{$\bullet$}{No magnetization of the medium,
125     $${\bf M}({\bf r},t)={\bf 0}.$$}
126
127   \section{The wave equation}
128   By taking the cross product with the nabla operator and Faraday's law,
129   one obtains
130   131 \eqalign{ 132 \nabla\times\nabla\times{\bf E}({\bf r},t) 133 &=-{{\partial}\over{\partial{t}}}\nabla\times{\bf B}({\bf r},t)\cr 134 &=-\mu_0{{\partial}\over{\partial{t}}}\nabla\times{\bf H}({\bf r},t)\cr 135 &=-\mu_0{{\partial}\over{\partial{t}}} 136 {{\partial{\bf D}({\bf r},t)}\over{\partial t}}\cr 137 &=-\mu_0\Big( 138 \varepsilon_0{{\partial^2{\bf E}({\bf r},t)}\over{\partial t^2}} 139 +{{\partial^2{\bf P}({\bf r},t)}\over{\partial t^2}}\Big).\cr 140 } 141
142   Since now $\mu_0\varepsilon_0=1/c^2$ in SI units, with $c$ being the speed
143   of light in vacuum, one hence obtains the basic wave equation, taken in
144   time domain, as
145   $$146 \nabla\times\nabla\times{\bf E}({\bf r},t) 147 +{{1}\over{c^2}}{{\partial^2{\bf E}({\bf r},t)}\over{\partial t^2}} 148 =-\mu_0{{\partial^2{\bf P}({\bf r},t)}\over{\partial t^2}},\eqno{(1)} 149$$
150   where, as in the previous lectures of this course, the polarization density
151   can be written in terms of the perturbation series as
152   153 \eqalign{ 154 {\bf P}({\bf r},t)&=\sum^{\infty}_{k=1}{\bf P}^{(k)}({\bf r},t) 155 =\underbrace{ 156 \varepsilon_0\int^{\infty}_{-\infty} 157 \chi^{(1)}_{\mu\alpha}(-\omega;\omega) 158 E_{\alpha}({\bf r},\omega)\exp(-i\omega t)\,d\omega 159 }_{={\bf P}^{(1)}({\bf r},t)} 160 +\underbrace{ 161 \sum^{\infty}_{k=2}{\bf P}^{(k)}({\bf r},t) 162 }_{={\bf P}^{({\rm NL})}({\bf r},t)}\cr 163 } 164
165   In the left hand side of Eq.~(1), we find the part of the homogeneous wave
166   equation for propagation of electromagnetic waves in vacuum, while the right
167   hand side described the modifications to the vacuum propagation due to the
168   interaction between light and matter. In this respect, it is now clear that
169   the electric polarisation effectively acts as a source term in the
170   mathematical description of electromagnetic wave propagation, making the
171   otherwise homogeneous vacuum problem an inhomogeneous problem (though with
172   known source terms).
173
174   It should be noticed that whenever the polarization density is calculated
175   from the Bloch equations (formulated later on, in lecture 10 of this course),
176   instead of by means of a perturbation series as above, the Maxwell
177   equations and the wave equation~(1) above are denoted {\sl Maxwell-Bloch
178   equations}.
179   In some sense, we can therefore see the choice of method for the calculation
180   of the polarization density as a switch point not only for using the
181   susceptibility formalism or not for the description of interaction between
182   light and matter, but also for the form of the wave propagation problem in
183   nonlinear media, which mathematically significantly differ between the
184   pure'' Maxwell's equations with susceptibilities and the Maxwell-Bloch
185   equations.
186
187   \section{The wave equation in frequency domain (optional)}
188   Frequently in this course, we have rather been studying the electric
189   fields and polarisation densities in frequency domain, since many
190   phenomena in optics are properly and conveniently described as static
191   (in which case the frequency dependence is simply reduced to the interaction
192   between discrete frequencies in the spectrum).
193   By using the Fourier integral identity\footnote{${}^2$}{From the inverse
194   Fourier integral identity, it follows that the Fourier transform of a
195   derivative of a function $f(t)$ is
196   $$\fourier[f'(t)](\omega)=-i\omega\fourier[f(t)](\omega)\qquad 197 \Rightarrow\qquad\fourier[f''(t)](\omega)=-\omega^2\fourier[f(t)](\omega).$$}
198   $$199 E_{\alpha}(t)=\int^{\infty}_{-\infty}E_{\alpha}(\omega) 200 \exp(-i\omega t)\,d\omega=\fourier^{-1}[E_{\alpha}](t), 201$$
202   with inverse relation
203   $$204 E_{\alpha}(\omega)={{1}\over{2\pi}}\int^{\infty}_{-\infty}E_{\alpha}(\tau) 205 \exp(i\omega\tau)\,d\tau=\fourier[E_{\alpha}](\omega), 206$$
207   we obtain the wave equation~(1) as
208   $$209 \nabla\times\nabla\times{\bf E}({\bf r},\omega) 210 -{{\omega^2}\over{c^2}}{\bf E}({\bf r},\omega) 211 =\mu_0\omega^2{\bf P}({\bf r},\omega). 212$$
213
214   \section{Quasimonochromatic light - Time dependent problems}
215   By inserting the perturbation series for the electric polarisation density
216   into the general wave equation~(1), which apply to arbitrary electric field
217   distributions and field intensities of the light, one obtains the
218   equation
219   $$220 \nabla\times\nabla\times{\bf E}({\bf r},t) 221 +\underbrace{{{1}\over{c^2}}{{\partial^2}\over{\partial t^2}} 222 \int^{\infty}_{-\infty}{\bf e}_{\mu}\varepsilon_{\mu\alpha}(\omega) 223 E_{\alpha}({\bf r},\omega)\exp(-i\omega t)\,d\omega 224 }_{({\rm denote\ this\ integral\ as\ }I{\rm\ for\ later\ use})} 225 =-\mu_0{{\partial^2{\bf P}^{({\rm NL})}({\bf r},t)}\over{\partial t^2}}, 226 \eqno{(2)} 227$$
228   where
229   $$230 \varepsilon_{\mu\alpha}(\omega)=\delta_{\mu\alpha} 231 +\chi^{(1)}_{\mu\alpha}(-\omega;\omega) 232$$
233   is a parameter commonly denoted as the
234   {\sl relative electrical permittivity}.\footnote{${}^3$}{Notice that
235   for isotropic media, $\chi^{(1)}_{\mu\alpha}(-\omega;\omega) 236 =\chi^{(1)}_{xx}(-\omega;\omega)\delta_{\mu\alpha}$, which leads to
237   the simplified form
238   $$239 {\bf e}_{\mu}\varepsilon_{\mu\alpha}(\omega)E_{\alpha}({\bf r},\omega) 240 =\varepsilon(\omega){\bf E}({\bf r},\omega). 241$$
242   We will here, however, continue with the general form, in order not
243   to loose generality in discussion that is to follow.}
244   This wave equation is identical to Eq.~(7.14) in Butcher and Cotter's
245   book. (Notice though the printing error in Butcher and Cotter's Eq.~(7.14),
246   where the first $\mu_0$ should be replaced by $1/c^2$.)
247
248   The second term of the left hand side of Eq.~(2) gives all first order
249   optical contributions to the wave propagation, as well as all linear
250   optical dispersion effects.
251   This terms deserves some extra attention, and we will now proceed with
252   deriving the effect of the frequency dependence of the relative permittivity
253   upon the wave equation.
254   First of all, we notice that since
255   $E_{\alpha}({\bf r},-\omega)=E^*_{\alpha}({\bf r},\omega)$, which simply
256   is a consequence of the choice of complex Fourier transform of a real
257   valued field, the reality condition of Eq.~(2) requires that
258   $$259 \varepsilon_{\mu\alpha}(-\omega)=\varepsilon^*_{\mu\alpha}(\omega). 260$$
261   It should be emphasized that this property of the relative electrical
262   permittivity merely is a convenient mathematical construction, since we
263   in regular physical terms only consider positive angular frequencies as
264   argument for the refractive index, etc.
265
266   For quasimonochromatic light, the electric field and polarisation density
267   are taken as
268   269 \eqalign{ 270 {\bf E}({\bf r},t)&=\sum_{\omega_{\sigma}\ge 0} 271 \Re[{\bf E}_{\omega_{\sigma}}({\bf r},t)\exp(-i\omega_{\sigma} t)],\cr 272 {\bf P}({\bf r},t)&=\sum_{\omega_{\sigma}\ge 0} 273 \Re[{\bf P}_{\omega_{\sigma}}({\bf r},t)\exp(-i\omega_{\sigma} t)],\cr 274 } 275
276   where ${\bf E}_{\omega_{\sigma}}({\bf r},t)$ and
277   ${\bf P}_{\omega_{\sigma}}({\bf r},t)$ are slowly varying envelopes of
278   the fields. In the frequency domain, the quasimonochromatic fields are
279   expressed as
280   281 \eqalign{ 282 {\bf E}({\bf r},\omega)&={{1}\over{2}}\sum_{\omega_{\sigma}\ge 0} 283 [{\bf E}_{\omega_{\sigma}}({\bf r},\omega-\omega_{\sigma}) 284 +{\bf E}^*_{\omega_{\sigma}}({\bf r},-\omega-\omega_{\sigma})],\cr 285 {\bf P}({\bf r},\omega)&={{1}\over{2}}\sum_{\omega_{\sigma}\ge 0} 286 [{\bf P}_{\omega_{\sigma}}({\bf r},\omega-\omega_{\sigma}) 287 +{\bf P}^*_{\omega_{\sigma}}({\bf r},-\omega-\omega_{\sigma})],\cr 288 } 289
290   where the envelopes have some limited extent around the carrier
291   frequencies at $\pm\omega_{\sigma}$. Notice that the fields taken in the
292   frequency domain are expressed entirely in terms of their respective
293   temporal envelope, that is to say, without the exponential functions
294   that appear in their counterparts in time domain.
295
296   For simplicity considering a medium that in the linear optical domain
297   is isotropic, with the relative electrical permittivity
298   $$299 \varepsilon_{\mu\alpha}(\omega) 300 =\varepsilon(\omega)\delta_{\mu\alpha} 301 =n^2_0(\omega)\delta_{\mu\alpha}, 302$$
303   where $n_0(\omega)$ is the first order contribution to the refractive
304   index of the medium, this leads to the middle term of the wave
305   equation~(1) in the form
306   307 \eqalign{ 308 I&\equiv{{1}\over{c^2}}{{\partial^2}\over{\partial t^2}} 309 \int^{\infty}_{-\infty}{\bf e}_{\mu}\varepsilon_{\mu\alpha}(\omega) 310 E_{\alpha}({\bf r},\omega)\exp(-i\omega t)\,d\omega\cr 311 &=-\int^{\infty}_{-\infty}{{\omega^2 n^2(\omega)}\over{c^2}} 312 \underbrace{ 313 {{1}\over{2}}\sum_{\omega_{\sigma}\ge 0} 314 [{\bf E}_{\omega_{\sigma}}({\bf r},\omega-\omega_{\sigma}) 315 +{\bf E}^*_{\omega_{\sigma}}({\bf r},-\omega-\omega_{\sigma})]}_{ 316 {\rm quasimonochromatic\ form\ of\ }{\bf E}({\bf r},\omega)} 317 \exp(-i\omega t)\,d\omega\cr 318 &=\{{\rm denote\ }\omega^2\varepsilon(\omega)/c^2 319 \equiv\omega^2 n^2_0(\omega)/c^2\equiv k^2(\omega)\}\cr 320 &=-{{1}\over{2}}\sum_{\omega_{\sigma}\ge 0} 321 \int^{\infty}_{-\infty}k^2(\omega) 322 [{\bf E}_{\omega_{\sigma}}({\bf r},\omega-\omega_{\sigma}) 323 +{\bf E}^*_{\omega_{\sigma}}({\bf r},-\omega-\omega_{\sigma})] 324 \exp(-i\omega t)\,d\omega.\cr 325 &=-{{1}\over{2}}\sum_{\omega_{\sigma}\ge 0} 326 \int^{\infty}_{-\infty}k^2(\omega) 327 {\bf E}_{\omega_{\sigma}}({\bf r},\omega-\omega_{\sigma}) 328 \exp(-i\omega t)\,d\omega + {\rm c.\,c.}\cr 329 } 330
331   If now the field envelopes decay to zero rapidly enough in the vicinity
332   of the carrier frequencies (as we would expect for quasimonochromatic light,
333   with a strong spectral confinement around the carrier frequency of the light),
334   then we may expect that a good approximation is to make a Taylor expansion
335   of $k^2(\omega)$, in the neighbourhood of respective carrier frequency of
336   the light, as
337   338 \eqalign{ 339 k^2(\omega)&\approx\Big(k(\omega_{\sigma}) 340 +{{dk}\over{d\omega}}\Big|_{\omega_{\sigma}}(\omega-\omega_{\sigma}) 341 +{{1}\over{2!}}{{d^2 k}\over{d\omega^2}} 342 \Big|_{\omega_{\sigma}}(\omega-\omega_{\sigma})^2 343 \Big)^2\cr 344 &\approx k^2_{\sigma} 345 +2k_{\sigma}{{dk}\over{d\omega}} 346 \Big|_{\omega_{\sigma}}(\omega-\omega_{\sigma}) 347 +k_{\sigma}{{d^2 k}\over{d\omega^2}} 348 \Big|_{\omega_{\sigma}}(\omega-\omega_{\sigma})^2, 349 \cr 350 } 351
352   where the notation $k_{\sigma}=k(\omega_{\sigma})$ was introduced, and
353   hence\footnote{${}^4$}{Notice that unless we apply the second approximation
354   in the Taylor expansion of $k^2(\omega)$, terms containing the {\sl squares}
355   of the derivatives will appear, which will lead to wave equations that
356   differ from the ones given by Butcher and Cotter.
357   In particular, this situation will arise even if one uses the suggested
358   expansion given by Eq.~(7.23) in Butcher and Cotter's book, which hence
359   should be taken with some care if one wish to build a strict foundation
360   for the time-dependent wave equation.}
361   362 \eqalign{ 363 I&\approx -{{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}\int^{\infty}_{-\infty} 364 \Big(k^2_{\sigma} 365 +2k_{\sigma}{{dk}\over{d\omega}} 366 \Big|_{\omega_{\sigma}}(\omega-\omega_{\sigma}) 367 +k_{\sigma}{{d^2 k}\over{d\omega^2}} 368 \Big|_{\omega_{\sigma}}(\omega-\omega_{\sigma})^2 369 \Big) 370 \cr&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times 371 {\bf E}_{\omega_{\sigma}}({\bf r},\omega-\omega_{\sigma}) 372 \exp(-i\omega t)\,d\omega + {\rm c.\,c.}\cr 373 &=\{{\rm change\ variable\ of\ integration\ } 374 \omega'=\omega-\omega_{\sigma}\}\cr 375 &=-{{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}\int^{\infty}_{-\infty} 376 \Big(k^2_{\sigma} 377 +2k_{\sigma}{{dk}\over{d\omega}} 378 \Big|_{\omega_{\sigma}}\omega' 379 +k_{\sigma}{{d^2 k}\over{d\omega^2}} 380 \Big|_{\omega_{\sigma}}\omega'^2 381 \Big) 382 \cr&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times 383 {\bf E}_{\omega_{\sigma}}({\bf r},\omega') 384 \exp(-i\omega't)\,d\omega'\,\exp(-i\omega_{\sigma} t) 385 + {\rm c.\,c.}\cr 386 &=\bigg\{{\rm use\ }\int^{\infty}_{-\infty}\omega^n f(\omega) 387 \exp(-i\omega t)\,d\omega=\fourier^{-1}[\omega^n f(\omega)](t) 388 =i^n{{d^n f(t)}\over{d t^n}}\bigg\}\cr 389 &=-{{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}\exp(-i\omega_{\sigma} t) 390 \Big(k^2_{\sigma} 391 +i2k_{\sigma}{{dk}\over{d\omega}} 392 \Big|_{\omega_{\sigma}}{{\partial}\over{\partial t}} 393 -k_{\sigma}{{d^2 k}\over{d\omega^2}} 394 \Big|_{\omega_{\sigma}}{{\partial^2}\over{\partial t^2}} 395 \Big){\bf E}_{\omega_{\sigma}}({\bf r},t) 396 + {\rm c.\,c.}\cr 397 } 398
399   As this result is inserted back into the wave equation~(2), one obtains
400   401 \eqalign{ 402 {{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}&\exp(-i\omega_{\sigma} t) 403 \Big[\nabla\times\nabla\times{\bf E}_{\omega_{\sigma}}({\bf r},t) 404 \cr&\qquad\qquad\qquad 405 -\Big(k^2_{\sigma} 406 +2ik_{\sigma}{{dk}\over{d\omega}} 407 \Big|_{\omega_{\sigma}}{{\partial}\over{\partial t}} 408 -k_{\sigma}{{d^2 k}\over{d\omega^2}} 409 \Big|_{\omega_{\sigma}}{{\partial^2}\over{\partial t^2}} 410 \Big){\bf E}_{\omega_{\sigma}}({\bf r},t)\Big] 411 + {\rm c.\,c.}\cr 412 &=-\mu_0{{\partial^2{\bf P}^{({\rm NL})}({\bf r},t)} 413 \over{\partial t^2}}\cr 414 &=-\mu_0{{\partial^2}\over{\partial t^2}} 415 {{1}\over{2}} 416 \sum_{\omega_{\sigma}\ge 0} 417 {\bf P}^{({\rm NL})}_{\omega_{\sigma}}({\bf r},t) 418 \exp(-i\omega_{\sigma} t)+{\rm c.\,c.}\cr 419 &\approx\mu_0{{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}\omega^2_{\sigma} 420 {\bf P}^{({\rm NL})}_{\omega_{\sigma}}({\bf r},t) 421 \exp(-i\omega_{\sigma} t)+{\rm c.\,c.}\cr 422 } 423
424   As we separate out the respective frequency components at
425   $\omega=\omega_{\sigma}$ of this equation, one obtains the time
426   dependent wave equation for the {\sl temporal envelope components}
427   of the electric field as
428   429 \eqalign{ 430 \nabla\times\nabla\times{\bf E}_{\omega_{\sigma}}({\bf r},t) 431 -\Big(k^2_{\sigma} 432 +i2k_{\sigma} 433 {{1}\over{v_{\rm g}}} 434 {{\partial}\over{\partial t}} 435 -k_{\sigma}{{d^2 k}\over{d\omega^2}}& 436 \Big|_{\omega_{\sigma}}{{\partial^2}\over{\partial t^2}} 437 \Big){\bf E}_{\omega_{\sigma}}({\bf r},t) 438 =\mu_0\omega^2_{\sigma} 439 {\bf P}^{({\rm NL})}_{\omega_{\sigma}}({\bf r},t),\cr 440 } 441 \eqno{(3)} 442
443   where
444   $$445 v_{\rm g}=\Big({{dk}\over{d\omega}}\Big|_{\omega_{\sigma}}\Big)^{-1}. 446$$
447   \vfill\eject
448
449   \section{Three practical approximations}
450   \item{$[1]$}{The infinite plane wave approximation,
451     $$452 {\bf E}_{\omega_{\sigma}}({\bf r},t) 453 ={\bf E}_{\omega_{\sigma}}(z,t)\bot{\bf e}_z 454 \qquad\Rightarrow\qquad 455 \nabla\times\nabla\times\to -{{\partial^2}\over{\partial z^2}}. 456$$}
457   \item{$[2]$}{Unidirectional propagation,
458     459 \eqalign{ 460 {\bf E}_{\omega_{\sigma}}(z,t)&={\bf A}_{\omega_{\sigma}}(z,t) 461 \exp(\pm ik_{\sigma}z)\cr 462 &\Downarrow\cr 463 \nabla\times\nabla\times{\bf E}_{\omega_{\sigma}}(z,t)&= 464 -[{{\partial^2{\bf A}_{\omega_{\sigma}}}\over{\partial z^2}} 465 \pm 2ik_{\sigma}{{\partial{\bf A}_{\omega_{\sigma}}}\over{\partial z}} 466 -k^2_{\sigma}{\bf A}_{\omega_{\sigma}} 467 ]\exp(\pm ik_{\sigma}z),\cr 468 } 469
470     for waves propagating in the positive/negative $z$-direction.
471     In this case, the real-valued electric field hence takes the form
472     473 \eqalign{ 474 {\bf E}({\bf r},t)&=\sum_{\omega_{\sigma}\ge 0} 475 \Re[{\bf E}_{\omega_{\sigma}}({\bf r},t)\exp(-i\omega_{\sigma} t)]\cr 476 &=\sum_{\omega_{\sigma}\ge 0} 477 \Re[{\bf A}_{\omega_{\sigma}}(z,t) 478 \exp(\pm ik_{\sigma}z-i\omega_{\sigma} t)]\cr 479 &=\sum_{\omega_{\sigma}\ge 0}|{\bf A}_{\omega_{\sigma}}(z,t)| 480 \Re\{\exp[ik_{\sigma}z\mp i\omega_{\sigma} t+i\phi(z)]\}\cr 481 &=\sum_{\omega_{\sigma}\ge 0}|{\bf A}_{\omega_{\sigma}}(z,t)| 482 \cos(k_{\sigma}z\mp \omega_{\sigma} t+\phi(z)),\cr 483 } 484
485     where $\phi(z,t)$ describes the spatially and temporally varying phase
486     of the complex-valued slowly varying envelope function
487     ${\bf A}_{\omega_{\sigma}}(z,t)$ of the electric field.
488   }
489   \medskip
490   \item{$[3]$}{The slowly varying envelope approximation,
491     $$\Big|{{\partial^2{\bf A}_{\omega_{\sigma}}}\over{\partial z^2}}\Big| 492 \ll \Big|k_{\sigma} 493 {{\partial{\bf A}_{\omega_{\sigma}}}\over{\partial z}}\Big|.$$}
494   \medskip
495   \noindent
496   These approximations, whenever applicable, further reduce the time dependent
497   wave equation to
498   499 \eqalign{ 500 \Big( 501 \pm i{{\partial}\over{\partial z}} 502 +i{{1}\over{v_{\rm g}}} 503 {{\partial}\over{\partial t}} 504 -{{1}\over{2}}{{d^2 k}\over{d\omega^2}}& 505 \Big|_{\omega_{\sigma}}{{\partial^2}\over{\partial t^2}} 506 \Big){\bf A}_{\omega_{\sigma}}(z,t) 507 =-{{\mu_0\omega^2_{\sigma}}\over{2k_{\sigma}}} 508 {\bf P}^{({\rm NL})}_{\omega_{\sigma}}({\bf r},t) 509 \exp(\mp ik_{\sigma}z).\cr 510 } 511 \eqno{(4)} 512
513   This form of the wave equation is identical to Butcher and Cotter's
514   Eq.~(7.24), with the exception that here waves propagating in positive
515   (upper signs) as well as negative (lower signs) $z$-direction are
516   considered.
517   \vfill\eject
518
519   \section{Monochromatic light}
520   \subsection{Monochromatic optical field}
521   522 \eqalign{ 523 {\bf E}({\bf r},t)&=\sum_{\sigma} 524 \Re[{\bf E}_{\omega_{\sigma}}({\bf r})\exp(-i\omega_{\sigma} t)], 525 \qquad\omega_{\sigma}\ge 0\cr 526 {\bf E}({\bf r},\omega) 527 &={{1}\over{2}}\sum_{\sigma} 528 [{\bf E}_{\omega_{\sigma}}({\bf r})\delta(\omega-\omega_{\sigma}) 529 +{\bf E}^*_{\omega_{\sigma}}({\bf r})\delta(\omega+\omega_{\sigma})]\cr 530 } 531
532
533   \subsection{Polarization density induced by monochromatic optical field}
534   $$535 {\bf P}^{(n)}({\bf r},t)=\sum_{\omega_{\sigma}\ge 0} 536 {\rm Re}[{\bf P}^{(n)}_{\omega_{\sigma}}\exp(-i\omega_{\sigma} t)], 537 \qquad\omega_{\sigma}=\omega_1+\omega_2+\ldots+\omega_n 538$$
539   (For construction of ${\bf P}^{(n)}_{\omega_{\sigma}}$, see notes on the
540   Butcher and Cotter convention handed out during the third lecture.)
541
542   \section{Monochromatic light - Time independent problems}
543   For strictly monochromatic light, as for example the output light of
544   continuous wave lasers, the temporal field envelopes are constants in time,
545   and the wave equation~(3) is reduced to
546   547 \eqalign{ 548 \nabla\times\nabla\times{\bf E}_{\omega_{\sigma}}({\bf r}) 549 -k^2_{\sigma}{\bf E}_{\omega_{\sigma}}({\bf r}) 550 =\mu_0\omega^2_{\sigma} 551 {\bf P}^{({\rm NL})}_{\omega_{\sigma}}({\bf r}).\cr 552 } 553 \eqno{(5)} 554
555   By applying the above listed approximations, one immediately finds
556   the monochromatic, time independent form of Eq.~(3) in the infinite
557   plane wave limit and slowly varying approximation as
558   559 \eqalign{ 560 {{\partial}\over{\partial z}}{\bf A}_{\omega_{\sigma}} 561 =\pm i{{\mu_0\omega^2_{\sigma}}\over{2k_{\sigma}}} 562 {\bf P}^{({\rm NL})}_{\omega_{\sigma}} 563 \exp(\mp ik_{\sigma}z),\cr 564 } 565 \eqno{(6)} 566
567   where the upper/lower sign correspond to a wave propagating in the
568   positive/negative $z$-direction. This equation corresponds to Butcher
569   and Cotter's Eq.~(7.17).
570   \vfill\eject
571
572   \section{Example I: Optical Kerr-effect - Time independent case}
573   In this example, we consider continuous wave
574   propagation\footnote{${}^5$}{That is to say, a time independent problem
575   with the temporal envelope of the electrical field being constant in time.}
576   in optical Kerr-media, using
577   light polarized in the $x$-direction and propagating along the positive
578   direction of the $z$-axis,
579   $$580 {\bf E}({\bf r},t)=\Re[{\bf E}_{\omega}(z)\exp(-i\omega t)], 581 \qquad{\bf E}_{\omega}(z)={\bf A}_{\omega}(z)\exp(ikz) 582 ={\bf e}_x A^x_{\omega}(z)\exp(ikz), 583$$
584   where, as previously, $k=\omega n_0/c$.
585   From material handed out during the third lecture (notes on the Butcher
586   and Cotter convention), the nonlinear polarization density for $x$-polarized
587   light is given as ${\bf P}^{({\rm NL})}_{\omega}={\bf P}^{(3)}_{\omega}$, with
588   589 \eqalign{ 590 {\bf P}^{(3)}_{\omega} 591 &=\varepsilon_0(3/4){\bf e}_x\chi^{(3)}_{xxxx} 592 (-\omega;\omega,\omega,-\omega) 593 |E^x_{\omega}|^2 E^x_{\omega}\cr 594 &=\varepsilon_0(3/4)\chi^{(3)}_{xxxx} 595 |{\bf E}_{\omega}|^2 {\bf E}_{\omega}\cr 596 &=\varepsilon_0(3/4)\chi^{(3)}_{xxxx} 597 |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\exp(ikz),\cr 598 } 599
600   and the time independent wave equation for the field envelope
601   ${\bf A}_{\omega}$, using Eq.~(6), becomes
602   603 \eqalign{ 604 {{\partial}\over{\partial z}}{\bf A}_{\omega} 605 &=i{{\mu_0\omega^2}\over{2k}} 606 \underbrace{\varepsilon_0(3/4)\chi^{(3)}_{xxxx} 607 |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\exp(ikz)}_{ 608 ={\bf P}^{({\rm NL})}_{\omega}(z)}\exp(-ikz)\cr 609 &=i{{3\omega^2}\over{8c^2k}}\chi^{(3)}_{xxxx} 610 |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\cr 611 &=\{{\rm since\ }k=\omega n_0(\omega)/c\}\cr 612 &=i{{3\omega}\over{8cn_0}} 613 \chi^{(3)}_{xxxx} 614 |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\cr 615 &=\{{\rm in\ analogy\ with\ Butcher\ and\ Cotter\ Eq.~(6.63),\ } 616 n_2=({{3}/{8n_0}})\chi^{(3)}_{xxxx}\}\cr 617 &=i {{\omega n_2}\over{c}} |{\bf A}_{\omega}|^2 {\bf A}_{\omega},\cr 618 } 619
620   or, equivalently, in its scalar form
621   $$622 {{\partial}\over{\partial z}}A^x_{\omega} 623 =i {{\omega n_2}\over{c}} |A^x_{\omega}|^2 A^x_{\omega}. 624$$
625   If the medium of interest now is analyzed at an angular frequency far
626   from any resonance, we may look for solutions to this equation with
627   $|{\bf A}_{\omega}|$ being constant (for a lossless medium).
628   For such a case it is straightforward to integrate the final wave
629   equation to yield the general solution
630   $$631 {\bf A}_{\omega}(z)= 632 {\bf A}_{\omega}(z_0)\exp[i\omega n_2 |{\bf A}_{\omega}(z_0)|^2 z/c], 633$$
634   or, again equivalently, in the scalar form
635   $$636 A^x_{\omega}(z)= 637 A^x_{\omega}(z_0)\exp[i\omega n_2 |A^x_{\omega}(z_0)|^2 z/c], 638$$
639   which hence gives the solution for the real-valued electric field
640   ${\bf E}({\bf r},t)$ as
641   642 \eqalign{ 643 {\bf E}({\bf r},t)&=\Re[{\bf E}_{\omega}(z)\exp(-i\omega t)]\cr 644 &=\Re\{{\bf A}_{\omega}(z)\exp[i(kz-\omega t)]\}\cr 645 &=\Re\{{\bf A}_{\omega}(z_0) 646 \exp[i(kz+\omega n_2 |{\bf A}_{\omega}(z_0)|^2 z/c-\omega t)]\}.\cr 647 } 648
649   From this solution, one immediately finds that the wave propagates
650   with an effective propagation constant
651   $$652 k+\omega n_2 |{\bf A}_{\omega}(z_0)|^2/c 653 = (\omega/c) (n_0+ n_2 |{\bf A}_{\omega}(z_0)|^2), 654$$
655   that is to say, experiencing the intensity dependent refractive index
656   $$657 n_{\rm eff}=n_0+ n_2 |{\bf A}_{\omega}(z_0)|^2. 658$$
659   \vfill\eject
660
661   \section{Example II: Optical Kerr-effect - Time dependent case}
662   We now consider a {\sl time dependent} envelope ${\bf E}_{\omega}(z,t)$,
663   of an optical wave propagating in the same medium and geometry as in the
664   previous example, for which now
665   $$666 {\bf E}({\bf r},t)=\Re[{\bf E}_{\omega}(z,t)\exp(-i\omega t)], 667 \qquad{\bf E}_{\omega}(z,t)={\bf A}_{\omega}(z,t)\exp(ikz) 668 ={\bf e}_x A^x_{\omega}(z,t)\exp(ikz). 669$$
670   The proper wave equation to apply for this case is the time dependent
671   wave equation~(4), and since the nonlinear polarization density of
672   the medium still is given by the optical Kerr-effect, we obtain
673   674 \eqalign{ 675 \Big( 676 i{{\partial}\over{\partial z}} 677 &+i{{1}\over{v_{\rm g}}} 678 {{\partial}\over{\partial t}} 679 -{{1}\over{2}}{{d^2 k}\over{d\omega^2}} 680 \Big|_{\omega_{\sigma}}{{\partial^2}\over{\partial t^2}} 681 \Big){\bf A}_{\omega}(z,t)\cr 682 &=-{{\mu_0\omega^2}\over{2k}} 683 \underbrace{\varepsilon_0(3/4)\chi^{(3)}_{xxxx} 684 |{\bf A}_{\omega}(z,t)|^2 {\bf A}_{\omega}(z,t)\exp(ikz)}_{ 685 ={\bf P}^{({\rm NL})}_{\omega}(z,t)} 686 \exp(-ikz)\cr 687 &=-{{3\omega^2}\over{8c^2k}} 688 \chi^{(3)}_{xxxx} 689 |{\bf A}_{\omega}(z,t)|^2 {\bf A}_{\omega}(z,t)\cr 690 &=\{{\rm as\ in\ previous\ example}\}\cr 691 &=-{{\omega n_2}\over{c}} 692 |{\bf A}_{\omega}(z,t)|^2 {\bf A}_{\omega}(z,t).\cr 693 } 694
695   The resulting wave equation
696   697 \eqalign{ 698 \Big( 699 i{{\partial}\over{\partial z}} 700 &+i{{1}\over{v_{\rm g}}} 701 {{\partial}\over{\partial t}} 702 -{{1}\over{2}}{{d^2 k}\over{d\omega^2}} 703 \Big|_{\omega_{\sigma}}{{\partial^2}\over{\partial t^2}} 704 \Big){\bf A}_{\omega} 705 =-{{\omega n_2}\over{c}} 706 |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\cr 707 } 708
709   is the starting point for analysis of solitons and solitary waves
710   in optical Kerr-media.
711   The obtained equation is the non-normalized form of the
712   in nonlinear physics (not only nonlinear optics!) often
713   encountered {\sl nonlinear Schr\"odinger equation}
714   (or NLSE, as its common acronym yields).
715
716   For a discussion on the transformation that cast the nonlinear Schr\"odinger
717   equation into the {\sl normalized nonlinear Schr\"odinger equation},
718   see Butcher and Cotter's book, page~240.
719   \bye
720   



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