Contents of file 'lect8/lect8.tex':
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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12 % the Euler fraktur font.
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18 % Use AMS Euler fraktur style for short-hand notation of Fourier transform
19 %
20 \def\fourier{\mathop{\frak F}\nolimits}
21 \def\Re{\mathop{\rm Re}\nolimits} % real part
22 \def\Im{\mathop{\rm Im}\nolimits} % imaginary part
23 \def\Tr{\mathop{\rm Tr}\nolimits} % quantum mechanical trace
24 %
25 % Define a handy macro for the list of symmetry operations
26 % in Schoenflies notation for point-symmetry groups.
27 %
28 \newdimen\citemindent \citemindent=40pt
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35 \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
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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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