Contents of file 'lect9/lect9.tex':
1 % File: nlopt/lect9/lect9.tex [pure TeX code]
2 % Last change: March 2, 2003
3 %
4 % Lecture No 9 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 %
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21 \def\Re{\mathop{\rm Re}\nolimits} % real part
22 \def\Im{\mathop{\rm Im}\nolimits} % imaginary part
23 \def\Tr{\mathop{\rm Tr}\nolimits} % quantum mechanical trace
24 \def\sinc{\mathop{\rm sinc}\nolimits} % the sinc(x)=sin(x)/x function
25 \def\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm
26 \topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt
27 \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
28 \else\hfill\fi}
29 \def\rightheadline{\tenrm{\it Lecture notes #1}
30 \hfil{\it Nonlinear Optics 5A5513 (2003)}}
31 \def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)}
32 \hfil{\it Lecture notes #1}}
33 \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps}
34 \vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}}
35 \hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1}
36 \vskip 24pt\noindent}
37 \def\section #1 {\medskip\goodbreak\noindent{\bf #1}
38 \par\nobreak\smallskip\noindent}
39 \def\subsection #1 {\smallskip\goodbreak\noindent{\it #1}
40 \par\nobreak\smallskip\noindent}
41
42 \lecture{9}
43 In this lecture, we will focus on examples of electromagnetic wave
44 propagation in nonlinear optical media, by applying the forms of Maxwell's
45 equations as obtained in the eighth lecture to a set of particular
46 nonlinear interactions as described by the previously formulated nonlinear
47 susceptibility formalism.
48 \medskip
49
50 \noindent The outline for this lecture is:
51 \item{$\bullet$}{General process for solving problems in nonlinear optics}
52 \item{$\bullet$}{Second harmonic generation (SHG)}
53 \item{$\bullet$}{Optical Kerr-effect}
54 \medskip
55
56 \section{General process for solving problems in nonlinear optics}
57 The typical steps in the process of solving a theoretical problem in
58 nonlinear optics typically involve:
59 $$
60 \bigg\{\matrix{
61 {\rm define\ the\ optical\ interaction\ of\ interest}\cr
62 {\rm (identifying\ the\ susceptibility)}\cr
63 }\bigg\}
64 $$
65 $$\Downarrow$$
66 $$
67 \bigg\{\matrix{
68 {\rm define\ in\ which\ medium\ the\ interaction\ take\ place}\cr
69 {\rm (identify\ crystallographic\ point\ symmetry\ group)}\cr
70 }\bigg\}
71 $$
72 $$\Downarrow$$
73 $$
74 \bigg\{\matrix{
75 {\rm consider\ eventual\ additional\ symmetries\ and\ constraints}\cr
76 {\rm (e.~g.~intrinsic,\ overall,\ or\ Kleinman\ symmetries)}\cr
77 }\bigg\}
78 $$
79 $$\Downarrow$$
80 $$
81 \bigg\{\matrix{
82 {\rm construct\ the\ polarization\ density}\cr
83 {\rm (the\ Butcher\ and\ Cotter\ convention)}\cr
84 }\bigg\}
85 $$
86 $$\Downarrow$$
87 $$
88 \bigg\{\matrix{
89 {\rm formulate\ the\ proper\ wave\ equation}\cr
90 {\rm (e.~g.~taking\ dispersion\ or\ diffraction\ into\ account)}\cr
91 }\bigg\}
92 $$
93 $$\Downarrow$$
94 $$
95 \bigg\{\matrix{
96 {\rm formulate\ the\ proper\ boundary}\cr
97 {\rm conditions\ for\ the\ wave\ equation}\cr
98 }\bigg\}
99 $$
100 $$\Downarrow$$
101 $$
102 \bigg\{\matrix{
103 {\rm solve\ the\ wave\ equation\ under}\cr
104 {\rm the\ boundary\ conditions}\cr
105 }\bigg\}
106 $$
107 \vfill\eject
108
109 \def\exercise#1#2{\medskip\goodbreak
110 \noindent{\bf Exercise #1.}\hskip 4pt({\it #2})\hskip 4pt}
111 \def\subexercise#1{\goodbreak
112 {\bf #1.}\hskip 4pt}
113
114 \section{Formulation of the exercises in this lecture}
115 In order to illustrate the scheme as previously outlined, the following
116 exercises serve as to give the connection between the susceptibilities,
117 as extensively analysed from a quantum-mechanical basis in earlier
118 lectures of this course, and the wave equation, derived from Maxwell's
119 equations of motion for electromagnetic fields.
120
121 \exercise{1}{Second harmonic generation in negative uniaxial media}
122 Consider a continuous pump wave at angular frequency $\omega$, initially
123 polarized in the $y$-direction and propagating in the positive $x$-direction
124 of a negative uniaxial crystal of crystallographic point symmetry group $3m$.
125 (Examples of crystals belonging to this class:
126 beta-${\rm Ba}{\rm B}_2{\rm O}_4$/BBO, ${\rm Li}{\rm Nb}{\rm O}_3$.)
127
128 \subexercise{1a} Formulate the polarization density of the medium for
129 the pump and second harmonic wave.
130
131 \subexercise{1b} Formulate the system of equations of motion for the
132 electromagnetic fields.
133
134 \subexercise{1c} Assuming no second harmonic signal present at the input,
135 solve the equations of motion for the second harmonic field, using the
136 non-depleted pump approximation, and derive an expression for the conversion
137 efficiency of the second harmonic generation.
138
139 \exercise{2}{Optical Kerr-effect -- continuous wave case}
140 In this setup, a monochromatic optical wave is propagating in the positive
141 $z$-direction of an isotropic optical Kerr-medium.
142
143 \subexercise{2a} Formulate the polarization density of the medium
144 for a wave polarized in the $xy$-plane.
145
146 \subexercise{2b} Formulate the polarization density of the medium
147 for a wave polarized in the $x$-direction.
148
149 \subexercise{2c} Formulate the wave equation for continuous wave propagation
150 in optical Kerr-media. The continuous wave is $x$-polarized and propagates
151 in the positive $z$-direction.
152
153 \subexercise{2d} For lossless media, solve the wave equation and give
154 an expression for the nonlinear, intensity-dependent refractive index
155 $n=n_0+n_2|{\bf E}_{\omega}|^2$.
156 \vfill\eject
157
158 \section{Second harmonic generation}
159 \subsection{The optical interaction}
160 In the case of second harmonic generation (SHG), two photons at angular
161 frequency~$\omega$ combine to a photon at twice the angular frequency,
162 $$
163 \hbar\omega+\hbar\omega\to\hbar(2\omega).
164 $$
165 This interaction is for the second harmonic wave (at angular
166 frequency~$\omega$)described by the second order susceptibility
167 $$
168 \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega,\omega),
169 $$
170 where $\omega_{\sigma}=2\omega$ is the generated second harmonic frequency
171 of the light.
172
173 \subsection{Symmetries of the medium}
174 In this example we consider second harmonic generation in trigonal media
175 of crystallographic point symmetry group $3m$.
176 (Example: ${\rm Li}{\rm Nb}{\rm O}_3$)
177 \bigskip
178 \centerline{\epsfxsize=150mm\epsfbox{../images/shgsetup/shgsetup.1}}
179 \medskip
180 \centerline{Figure 1. The setup for optical second harmonic generation in
181 ${\rm Li}{\rm Nb}{\rm O}_3$.}
182 \medskip
183 For this point symmetry group, the nonzero tensor elements of the first
184 order susceptibility are (for example according to Table A3.1 in
185 {\sl The Elements of Nonlinear Optics})
186 $$
187 \chi^{(1)}_{xx}=\chi^{(1)}_{yy},\qquad\chi^{(1)}_{zz},
188 $$
189 which gives the ordinary refractive indices
190 $$
191 n_x(\omega)=n_y(\omega)=[1+\chi^{(1)}_{xx}(-\omega;\omega)]^{1/2}
192 \equiv n_{\rm O}(\omega)
193 $$
194 for waves components polarized in the $x$- or $y$-directions, and
195 the extraordinary refractive index
196 $$
197 n_z(\omega)=[1+\chi^{(1)}_{zz}(-\omega;\omega)]^{1/2}
198 \equiv n_{\rm E}(\omega)
199 $$
200 for the wave component polarized in the $z$-direction. Since we here are
201 considering a negatively uniaxial crystal (see Butcher and Cotter, p.~214),
202 these refractive indices satisfy the inequality
203 $$n_{\rm E}(\omega)\le n_{\rm O}(\omega).$$
204
205 The nonzero tensor elements of the second order susceptibility are
206 (for example according to Table A3.2 in {\sl The Elements of Nonlinear Optics})
207 $$
208 \eqalign{
209 \chi^{(2)}_{yxx}&=\chi^{(2)}_{xyx}
210 =\chi^{(2)}_{xxy}=-\chi^{(2)}_{yyy},\qquad
211 \chi^{(2)}_{zzz},\cr
212 \chi^{(2)}_{zxx}&=\chi^{(2)}_{zyy},\qquad
213 \chi^{(2)}_{yyz}=\chi^{(2)}_{xxz},\qquad
214 \chi^{(2)}_{yzy}=\chi^{(2)}_{xzx},\cr
215 }\eqno{(1)}
216 $$
217
218 \subsection{Additional symmetries}
219 Intrinsic permutation symmetry for the case of second harmonic generation
220 gives
221 $$
222 \chi^{(2)}_{xxz}(-2\omega;\omega,\omega)
223 =\chi^{(2)}_{xzx}(-2\omega;\omega,\omega)
224 =\chi^{(2)}_{yzy}(-2\omega;\omega,\omega)
225 =\chi^{(2)}_{yyz}(-2\omega;\omega,\omega),
226 $$
227 which reduces the second order susceptibility in Eq.~(1) to a set
228 of 11 tensor elements, of which only 4 are independent.
229 (We recall that the intrinsic permutation symmetry is always applicable,
230 as being a consequence of the symmetrization described in lectures two
231 and five.)
232 Whenever Kleinman symmetry holds, the susceptibility is in addition symmetric
233 under any permutation of the indices, which hence gives the additional relation
234 $$
235 \chi^{(2)}_{zxx}(-2\omega;\omega,\omega)
236 =\chi^{(2)}_{xzx}(-2\omega;\omega,\omega)
237 =\chi^{(2)}_{xxz}(-2\omega;\omega,\omega),
238 $$
239 i.~e.~reducing the second order susceptibility to a set of 11 tensor
240 elements, of which only 3 are independent.
241
242 To summarize, the set of nonzero tensor elements describing second
243 harmonic generation under Kleinman symmetry is
244 $$
245 \eqalign{
246 \chi^{(2)}_{yxx}&=\chi^{(2)}_{xyx}
247 =\chi^{(2)}_{xxy}=-\chi^{(2)}_{yyy},\qquad
248 \chi^{(2)}_{zzz},\cr
249 \chi^{(2)}_{zxx}&=\chi^{(2)}_{zyy}=
250 \chi^{(2)}_{yyz}=\chi^{(2)}_{xxz}=
251 \chi^{(2)}_{yzy}=\chi^{(2)}_{xzx}.\cr
252 }\eqno{(2)}
253 $$
254
255 For the pump field at angular frequency $\omega$, the relevant susceptibility
256 describing the interaction with the second harmonic wave
257 is\footnote{${}^1$}{Keep in mind that in the convention of Butcher and Cotter,
258 the frequency arguments to the right of the semicolon may be writen in
259 arbitrary order, hence we may in an equal description instead use
260 $$\chi^{(2)}_{xxz}(-\omega;-\omega,2\omega)$$ for the description of
261 the second order interaction between light and matter.}
262 $$
263 \chi^{(2)}_{\mu\alpha\beta}(-\omega;2\omega,-\omega).
264 $$
265 For an arbitrary frequency argument, this is the proper form of the
266 susceptibility to use for the fundamental field, and this form generally
267 differ from that of the susceptibilities for the second harmonic field.
268 However, whenever Kleinman symmetry holds, the susceptibility for the
269 fundamental field can be cast into the same parameters as for the second
270 harmonic field, since
271 $$
272 \eqalign{
273 \chi^{(2)}_{\mu\alpha\beta}(-\omega;2\omega,-\omega)
274 &=\big\{{\rm Apply\ overall\ permutation\ symmetry}\big\}\cr
275 &=\chi^{(2)}_{\alpha\mu\beta}(2\omega;-\omega,-\omega)\cr
276 &=\big\{{\rm Apply\ Kleinman\ symmetry}\big\}\cr
277 &=\chi^{(2)}_{\mu\alpha\beta}(2\omega;-\omega,-\omega)\cr
278 &=\big\{{\rm Apply\ reality\ condition\ [B.\,\&C.\,Eq.\,(2.43)]}\big\}\cr
279 &=[\chi^{(2)}_{\mu\alpha\beta}(-2\omega;\omega,\omega)]^*\cr
280 &=\chi^{(2)}_{\mu\alpha\beta}(-2\omega;\omega,\omega).\cr
281 }
282 $$
283 Hence the second order interaction is described by the same set of tensor
284 elements for the fundamental as well as the second harmonic optical wave
285 whenever Kleinman symmetry applies.
286 \vfill\eject
287
288 \subsection{The polarization density}
289 Following the convention of Butcher and Cotter,\footnote{${}^2$}{See course
290 material on the Butcher and Cotter convention handed out during the third
291 lecture. Notice that for the first order polarization density, one at optical
292 frequencies {\sl always} has the trivial degeneracy factor
293 $$K(-2\omega;\omega)=2^{l+m-n}p=2^{1+0-1}\times 1=1.$$}
294 the degeneracy factor for the second harmonic signal at $2\omega$ is
295 $$
296 K(-2\omega;\omega,\omega)=2^{l+m-n}p,
297 $$
298 where
299 $$
300 \eqalign{
301 p&=\{{\rm the\ number\ of\ {\sl distinct}\ permutations\ of}
302 \ \omega,\omega\}=1,\cr
303 n&=\{{\rm the\ order\ of\ the\ nonlinearity}\}=2,\cr
304 m&=\{{\rm the\ number\ of\ angular\ frequencies}\ \omega_k
305 \ {\rm that\ are\ zero}\}=0,\cr
306 l&=\bigg\lbrace\matrix{1,\qquad{\rm if}\ 2\omega\ne 0,\cr
307 0,\qquad{\rm otherwise}.}\bigg\rbrace=1,\cr
308 }
309 $$
310 i.~e.
311 $$
312 K(-2\omega;\omega,\omega)=2^{1+0-2}\times 1=1/2.
313 $$
314 For the fundamental optical field at $\omega$, one might be mislead to assume
315 that since the second order interaction for this field is described by an
316 identical set of tensor elements as for the second harmonic wave, the
317 degeneracy factor must also be identical to the previously derived one.
318 This is, however, {\sl a very wrong assumption}, and one can easily verify
319 that the proper degeneracy factor for the fundamental field instead is
320 given as
321 $$
322 K(-\omega;2\omega,-\omega)=2^{l+m-n}p,
323 $$
324 where
325 $$
326 \eqalign{
327 p&=\{{\rm the\ number\ of\ {\sl distinct}\ permutations\ of}
328 \ 2\omega,-\omega\}=2,\cr
329 n&=\{{\rm the\ order\ of\ the\ nonlinearity}\}=2,\cr
330 m&=\{{\rm the\ number\ of\ angular\ frequencies}\ \omega_k
331 \ {\rm that\ are\ zero}\}=0,\cr
332 l&=\bigg\lbrace\matrix{1,\qquad{\rm if}\ \omega\ne 0,\cr
333 0,\qquad{\rm otherwise}.}\bigg\rbrace=1,\cr
334 }
335 $$
336 i.~e.
337 $$
338 K(-\omega;2\omega,-\omega)=2^{1+0-2}\times 2=1.
339 $$
340
341 The general second harmonic polarization density of the medium is hence
342 given as
343 $$
344 \eqalign{
345 [{\bf P}^{({\rm NL})}_{2\omega}]_z&=[{\bf P}^{(2)}_{2\omega}]_z
346 =\varepsilon_0 \underbrace{K(-2\omega;\omega,\omega)
347 \chi^{(2)}_{z\alpha\beta}(-2\omega;\omega,\omega)}_{
348 ={{1}\over{2}}\chi^{(2)}_{z\alpha\beta}(-2\omega;\omega,\omega)}
349 E^{\alpha}_{\omega}E^{\beta}_{\omega}\cr
350 &=(\varepsilon_0/2)[\chi^{(2)}_{zxx} E^x_{\omega} E^{x}_{\omega}
351 +\chi^{(2)}_{zyy} E^y_{\omega} E^{y}_{\omega}
352 +\chi^{(2)}_{zzz} E^z_{\omega} E^{z}_{\omega}]\cr
353 &=(\varepsilon_0/2)[\chi^{(2)}_{zxx}
354 (E^x_{\omega} E^{x}_{\omega}+E^y_{\omega} E^{y}_{\omega})
355 +\chi^{(2)}_{zzz} E^z_{\omega} E^{z}_{\omega}],\cr
356 [{\bf P}^{({\rm NL})}_{2\omega}]_y
357 &=(\varepsilon_0/2)[\chi^{(2)}_{yxx} E^x_{\omega} E^{x}_{\omega}
358 +\chi^{(2)}_{yyy} E^y_{\omega} E^{y}_{\omega}
359 +\chi^{(2)}_{yyz} E^y_{\omega} E^{z}_{\omega}
360 +\chi^{(2)}_{yzy} E^z_{\omega} E^{y}_{\omega}]\cr
361 &=(\varepsilon_0/2)[\chi^{(2)}_{yxx}
362 (E^x_{\omega} E^{x}_{\omega}-E^y_{\omega} E^{y}_{\omega})
363 +\chi^{(2)}_{zxx}
364 (E^y_{\omega} E^{z}_{\omega}+E^z_{\omega} E^{y}_{\omega})],\cr
365 [{\bf P}^{({\rm NL})}_{2\omega}]_x
366 &=(\varepsilon_0/2)[\chi^{(2)}_{xxy} E^x_{\omega} E^{y}_{\omega}
367 +\chi^{(2)}_{xyx} E^y_{\omega} E^{x}_{\omega}
368 +\chi^{(2)}_{xxz} E^x_{\omega} E^{z}_{\omega}
369 +\chi^{(2)}_{xzx} E^z_{\omega} E^{x}_{\omega}]\cr
370 &=(\varepsilon_0/2)[\chi^{(2)}_{yxx}
371 (E^x_{\omega} E^{y}_{\omega}+E^y_{\omega} E^{x}_{\omega})
372 +\chi^{(2)}_{zxx}
373 (E^x_{\omega} E^{z}_{\omega}+E^z_{\omega} E^{x}_{\omega})],\cr
374 }
375 $$
376 while the general polarization density at the angular frequency of the pump
377 field becomes\footnote{${}^3$}{Keep in mind that a negative frequency
378 argument to the right of the semicolon in the susceptibility is to be
379 associated with the complex conjugate of the respective electric field;
380 see Butcher and Cotter, section 2.3.2.}
381 $$
382 \eqalign{
383 [{\bf P}^{({\rm NL})}_{\omega}]_z&=[{\bf P}^{(2)}_{\omega}]_z
384 =\varepsilon_0 \underbrace{K(-\omega;2\omega,-\omega)
385 \chi^{(2)}_{z\alpha\beta}(-\omega;2\omega,-\omega)}_{
386 =\chi^{(2)}_{z\alpha\beta}(-2\omega;\omega,\omega)}
387 E^{\alpha}_{2\omega}E^{\beta}_{-\omega}\cr
388 &=\varepsilon_0[\chi^{(2)}_{zxx} E^x_{2\omega} E^{x*}_{\omega}
389 +\chi^{(2)}_{zyy} E^y_{2\omega} E^{y*}_{\omega}
390 +\chi^{(2)}_{zzz} E^z_{2\omega} E^{z*}_{\omega}]\cr
391 &=\varepsilon_0[\chi^{(2)}_{zxx}
392 (E^x_{2\omega} E^{x*}_{\omega}+E^y_{2\omega} E^{y*}_{\omega})
393 +\chi^{(2)}_{zzz} E^z_{2\omega} E^{z*}_{\omega}],\cr
394 [{\bf P}^{({\rm NL})}_{\omega}]_y
395 &=\varepsilon_0[\chi^{(2)}_{yxx} E^x_{2\omega} E^{x*}_{\omega}
396 +\chi^{(2)}_{yyy} E^y_{2\omega} E^{y*}_{\omega}
397 +\chi^{(2)}_{yyz} E^y_{2\omega} E^{z*}_{\omega}
398 +\chi^{(2)}_{yzy} E^z_{2\omega} E^{y*}_{\omega}]\cr
399 &=\varepsilon_0[\chi^{(2)}_{yxx}
400 (E^x_{2\omega} E^{x*}_{\omega}-E^y_{2\omega} E^{y*}_{\omega})
401 +\chi^{(2)}_{zxx}
402 (E^y_{2\omega} E^{z*}_{\omega}+E^z_{2\omega} E^{y*}_{\omega})],\cr
403 [{\bf P}^{({\rm NL})}_{\omega}]_x
404 &=\varepsilon_0[\chi^{(2)}_{xxy} E^x_{2\omega} E^{y*}_{\omega}
405 +\chi^{(2)}_{xyx} E^y_{2\omega} E^{x*}_{\omega}
406 +\chi^{(2)}_{xxz} E^x_{2\omega} E^{z*}_{\omega}
407 +\chi^{(2)}_{xzx} E^z_{2\omega} E^{x*}_{\omega}]\cr
408 &=\varepsilon_0[\chi^{(2)}_{yxx}
409 (E^x_{2\omega} E^{y*}_{\omega}+E^y_{2\omega} E^{x*}_{\omega})
410 +\chi^{(2)}_{zxx}
411 (E^x_{2\omega} E^{z*}_{\omega}+E^z_{2\omega} E^{x*}_{\omega})].\cr
412 }
413 $$
414 For a pump wave polarized in the $yz$-plane of the crystal frame, the
415 polarization density of the medium hence becomes
416 $$
417 \eqalign{
418 [{\bf P}^{({\rm NL})}_{2\omega}]_z
419 &=(\varepsilon_0/2)[\chi^{(2)}_{zxx}
420 E^y_{\omega} E^y_{\omega}
421 +\chi^{(2)}_{zzz} E^z_{\omega} E^{z}_{\omega}],\cr
422 [{\bf P}^{({\rm NL})}_{2\omega}]_y
423 &=(\varepsilon_0/2)[-\chi^{(2)}_{yxx} E^y_{\omega} E^{y}_{\omega}
424 +\chi^{(2)}_{zxx}
425 (E^y_{\omega} E^{z}_{\omega}+E^z_{\omega} E^{y}_{\omega})],\cr
426 [{\bf P}^{({\rm NL})}_{2\omega}]_x
427 &=0,\cr
428 }
429 $$
430 and
431 $$
432 \eqalign{
433 [{\bf P}^{({\rm NL})}_{\omega}]_z
434 &=\varepsilon_0[\chi^{(2)}_{zxx} E^y_{2\omega} E^{y*}_{\omega}
435 +\chi^{(2)}_{zzz} E^z_{2\omega} E^{z*}_{\omega}],\cr
436 [{\bf P}^{({\rm NL})}_{\omega}]_y
437 &=\varepsilon_0[-\chi^{(2)}_{yxx}E^y_{2\omega} E^{y*}_{\omega}
438 +\chi^{(2)}_{zxx}
439 (E^y_{2\omega} E^{z*}_{\omega}+E^z_{2\omega} E^{y*}_{\omega})],\cr
440 [{\bf P}^{({\rm NL})}_{\omega}]_x
441 &=0.\cr
442 }
443 $$
444
445 \subsection{The wave equation}
446 Strictly speaking, the previously formulated polarization density
447 gives a coupled system between the polarization states of both the
448 fundamental and second harmonic waves, since both the $y$- and $z$-components
449 of the polarization densities at $\omega$ and $2\omega$ contain components
450 of all other field components.
451 However, for simplicity we will here restrict the continued analysis to the
452 case of a $y$-polarized input pump wave, which through the
453 $\chi^{(2)}_{zyy}=\chi^{(2)}_{zxx}$ elements give rise to a $z$-polarized
454 second harmonic frequency component at $2\omega$.
455
456 The electric fields of the fundamental and second harmonic optical waves
457 are for the forward propagating configuration expressed in their
458 {\sl spatial envelopes} ${\bf A}_{\omega}$ and ${\bf A}_{2\omega}$ as
459 $$
460 \eqalign{
461 {\bf E}_{\omega}(x)&={\bf e}_y A^y_{\omega}(x)\exp(ik_{\omega_y} x),
462 \qquad k_{\omega_y}\equiv\omega n_{\omega_y}/c
463 \equiv\omega n_{\rm O}(\omega)/c\cr
464 {\bf E}_{2\omega}(x)&={\bf e}_z A^z_{2\omega}(x)\exp(ik_{2\omega_z} x),
465 \qquad k_{2\omega_z}\equiv 2\omega n_{2\omega_z}/c
466 \equiv 2\omega n_{\rm E}(2\omega)/c\cr
467 }
468 $$
469 Using the above separation of the natural, spatial oscillation of the
470 light, in the infinite plane wave approximation and by using the slowly
471 varying envelope approximation, the wave equation for the envelope of the
472 second harmonic optical field becomes (see Eq.~(6) in the notes from
473 lecture eight)
474 $$
475 \eqalign{
476 {{\partial A^z_{2\omega}}\over{\partial x}}
477 &=i{{\mu_0(2\omega)^2}\over{2k_{2\omega_z}}}
478 [{\bf P}^{({\rm NL})}_{2\omega}]_z
479 \exp(-ik_{2\omega_z}x)\cr
480 &=i{{\mu_0(2\omega)^2}\over{2(2\omega n_{2\omega_z}/c)}}
481 \underbrace{
482 {{\varepsilon_0}\over{2}}
483 \chi^{(2)}_{zxx} A^y_{\omega}{}^2\exp(2ik_{\omega_y}x)}_{
484 =[{\bf P}^{({\rm NL})}_{2\omega}]_z}
485 \exp(-ik_{2\omega_z}x)\cr
486 &=i{{\omega\chi^{(2)}_{zxx}}\over{2 n_{2\omega_z} c}}
487 A^y_{\omega}{}^2\exp[i(2k_{\omega_y}-k_{2\omega_z})x],\cr
488 }
489 $$
490 while for the fundamental wave,
491 $$
492 \eqalign{
493 {{\partial A^y_{\omega}}\over{\partial x}}
494 &=i{{\mu_0 \omega^2}\over{2k_{\omega_y}}}
495 [{\bf P}^{({\rm NL})}_{\omega}]_y
496 \exp(-ik_{\omega_y}x)\cr
497 &=i{{\mu_0 \omega^2}\over{2(\omega n_{\omega_y}/c)}}
498 \underbrace{\varepsilon_0\chi^{(2)}_{zxx}
499 A^z_{2\omega}\exp(ik_{2\omega_z}x)
500 A^{y*}_{\omega}\exp(-ik_{\omega_y}x)}_{
501 =[{\bf P}^{({\rm NL})}_{\omega}]_y}
502 \exp(-ik_{\omega_y}x)\cr
503 &=i{{\omega\chi^{(2)}_{zxx}}\over{2 n_{\omega_y} c}}
504 A^z_{2\omega}A^{y*}_{\omega}
505 \exp[-i(2k_{\omega_y}-k_{2\omega_z})x].\cr
506 }
507 $$
508 These equations can hence be summarized by the coupled system
509 $$
510 \eqalignno{
511 {{\partial A^z_{2\omega}}\over{\partial x}}
512 &=i{{\omega\chi^{(2)}_{zxx}}\over{2 n_{2\omega_z} c}}
513 A^y_{\omega}{}^2\exp(i\Delta k x),&({\rm 3a})\cr
514 {{\partial A^y_{\omega}}\over{\partial x}}
515 &=i{{\omega\chi^{(2)}_{zxx}}\over{2 n_{\omega_y} c}}
516 A^z_{2\omega}A^{y*}_{\omega}
517 \exp(-i\Delta k x).&({\rm 3b})\cr
518 }
519 $$
520 where
521 $$
522 \eqalign{
523 \Delta k
524 &=2k_{\omega_y}-k_{2\omega_z}\cr
525 &=2\omega n_{\omega_y}/c-2\omega n_{2\omega_z}/c\cr
526 &=(2\omega/c)(n_{\omega_y}-\omega n_{2\omega_z})\cr
527 }
528 $$
529 is the so-called phase mismatch between the pump and second harmonic wave.
530
531 \subsection{Boundary conditions}
532 Here the boundary conditions are simply that no second harmonic signal
533 is present at the input,
534 $$A^z_{2\omega}(0)=0,$$
535 together with a known input field at the fundamental frequency,
536 $$A^y_{\omega}(0)=\{{\rm known}\}.$$
537
538 \subsection{Solving the wave equation}
539 Considering a nonzero $\Delta k$, the conversion efficiency is regularly
540 quite small, and one may approximately take the spatial distribution of
541 the pump wave to be constant, $A^y_{\omega}(x)\approx A^y_{\omega}(0)$.
542 Using this approximation\footnote{${}^4$}{For an outline of the method of
543 solving the coupled system~(1) exactly in terms of Jacobian elliptic
544 functions (thus allowing for a depleted pump as well), see J.~A.~Armstrong,
545 N.~Bloembergen, J.~Ducuing, and P.~S.~Pershan, Phys.~Rev.~{\bf 127},
546 {1918--1939}, (1962).}, and by applying the initial condition
547 $A^z_{2\omega}(0)=0$ of the second harmonic signal, one finds
548 $$
549 \eqalign{
550 A^z_{2\omega}(L)
551 &=\int^L_0{{\partial A^z_{2\omega}(z)}\over{\partial x}}\,dx\cr
552 &=\int^L_0 i{{\omega\chi^{(2)}_{zxx}}\over{2 n_{{2\omega}_z}} c}
553 A^y_{\omega}{}^2(0)\exp(i\Delta k x)\,dx\cr
554 &={{\omega\chi^{(2)}_{zxx}}\over{2 n_{{2\omega}_z}} c}
555 A^y_{\omega}{}^2(0){{1}\over{\Delta k}}[\exp(i\Delta k L)-1]\cr
556 &=\big\{{\rm Use\ }[\exp(i\Delta k L)-1]/\Delta k
557 =iL\exp(i\Delta k L/2)\sinc(\Delta k L/2)\big\}\cr
558 &=i{{\omega\chi^{(2)}_{zxx} L}\over{2 n_{{2\omega}_z}} c}
559 A^y_{\omega}{}^2(0)\exp(i\Delta k L/2)\sinc(\Delta k L/2).\cr
560 }
561 $$
562 \vfill\eject
563
564 %%\bigskip
565 \centerline{\epsfxsize=110mm\epsfbox{type1.eps}}
566 \medskip
567 \noindent{Figure 2. Conversion efficiency $I_{2\omega}(L)/I_{\omega}(0)$
568 as function of normalized crystal length $\Delta k L/2$.
569 The conversion efficiency is in the phase mismatched case ($\Delta k\ne 0$)
570 a periodic function, with period $2L_{\rm c}$, with
571 $L_{\rm c}=\pi/\Delta k$ being the {\sl coherence length}.}
572 \medskip
573 \noindent
574 In terms if the light intensities of the waves, one after a propagation
575 distance $x=L$ hence has the second harmonic signal with intensity
576 $I_{2\omega}(L)$ expressed in terms of the input intensity $I_{\omega}$ as
577 $$
578 \eqalign{
579 I_{2\omega}(L)
580 &={{1}\over{2}}\varepsilon_0 c n_{2\omega_z}|A^z_{2\omega}(L)|^2\cr
581 &={{1}\over{2}}\varepsilon_0 c n_{2\omega_z}
582 \Big|i{{\omega\chi^{(2)}_{zxx} L}\over{2 n_{{2\omega}_z}} c}
583 A^y_{\omega}{}^2(0)\exp(i\Delta k L/2)\sinc(\Delta k L/2)\Big|^2\cr
584 &=\varepsilon_0
585 {{\omega^2 L^2}\over{8 n_{{2\omega}_z}} c}
586 |\chi^{(2)}_{zxx}|^2
587 |A^y_{\omega}(0)|^4\sinc^2(\Delta k L/2)\cr
588 &=\bigg\{{\rm Use\ }|A^y_{\omega}(0)|^2
589 ={{2 I_{\omega}(0)}\over{\varepsilon_0 c n_{\omega_y}}}\bigg\}\cr
590 &=\varepsilon_0
591 {{\omega^2 L^2}\over{8 n_{{2\omega}_z}} c}
592 |\chi^{(2)}_{zxx}|^2
593 {{4 I^2_{\omega}(0)}\over{\varepsilon^2_0 c^2 n^2_{\omega_y}}}
594 \sinc^2(\Delta k L/2)\cr
595 &={{\omega^2 L^2}\over{2 \varepsilon_0 c^3}}
596 {{|\chi^{(2)}_{zxx}(-2\omega;\omega,\omega)|^2}
597 \over{n_{{2\omega}_z} n^2_{\omega_y}}}
598 I^2_{\omega}(0)\sinc^2(\Delta k L/2),\cr
599 }
600 $$
601 i.~e.~with the conversion efficiency
602 $$
603 {{I_{2\omega}(L)}\over{I_{\omega}(0)}}
604 ={{\omega^2 L^2}\over{2 \varepsilon_0 c^3}}
605 {{|\chi^{(2)}_{zxx}(-2\omega;\omega,\omega)|^2}
606 \over{n_{{2\omega}_z} n^2_{\omega_y}}}
607 I_{\omega}(0)\sinc^2(\Delta k L/2).
608 $$
609 \vfill\eject
610
611 %\bigskip
612 \centerline{\epsfxsize=110mm\epsfbox{neguniax.eps}}
613 \medskip
614 \noindent{Figure 3. Ordinary and extraordinary refractive indices
615 of a negative uniaxial crystal as function of vacuum wavelength of
616 the light, in the case of normal dispersion. Phase matching between
617 the pump and second harmonic wave is obtained whenever
618 $n_{\omega_y}\equiv n_{\rm O}(\omega)=n_{\rm E}(2\omega)\equiv
619 n_{2\omega_z}$.}
620 \medskip
621
622 \bigskip
623 \centerline{\epsfxsize=110mm\epsfbox{qpm.eps}}
624 \medskip
625 \noindent{Figure 4. Conversion efficiency $I_{2\omega}(L)/I_{\omega}(0)$
626 as function of normalized crystal length $\Delta k L/2$ when the material
627 properties are periodically reversed, with a half-period of $L_{\rm c}$.}
628 \medskip
629 \vfill\eject
630
631 \section{Optical Kerr-effect - Field corrected refractive index}
632 As a start, we assume a monochromatic optical wave (containing forward
633 and/or backward propagating components) polarized in the $xy$-plane,
634 $$
635 {\bf E}(z,t)=\Re[{\bf E}_{\omega}(z)\exp(-i\omega t)]\in{\Bbb R}^3,
636 $$
637 with all spatial variation of the field contained in
638 $$
639 {\bf E}_{\omega}(z)={\bf e}_x E^x_{\omega}(z)
640 +{\bf e}_y E^y_{\omega}(z)\in{\Bbb C}^3.
641 $$
642
643 \subsection{The optical interaction}
644 Optical Kerr-effect is in isotropic media described by the third order
645 susceptibility\footnote{${}^5$}{Again, keep in mind that in the convention
646 of Butcher and Cotter, the frequency arguments to the right of the semicolon
647 may be writen in arbitrary order, hence we may in an equal description
648 instead use
649 $$\chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,-\omega,\omega)$$
650 or
651 $$\chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;-\omega,\omega,\omega)$$
652 for this description of the third order interaction between light and matter.}
653 $$
654 \chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega).
655 $$
656
657 \subsection{Symmetries of the medium}
658 The general set of nonzero components of $\chi^{(3)}_{\mu\alpha\beta\gamma}$
659 for isotropic media are from Appendix A3.3 of Butcher and Cotters book
660 given as
661 $$
662 \eqalign{
663 \chi^{(3)}_{xxxx}&=\chi^{(3)}_{yyyy}=\chi^{(3)}_{zzzz},\cr
664 \chi^{(3)}_{yyzz}&=\chi^{(3)}_{zzyy}
665 =\chi^{(3)}_{zzxx}=\chi^{(3)}_{xxzz}
666 =\chi^{(3)}_{xxyy}=\chi^{(3)}_{yyxx}\cr
667 \chi^{(3)}_{yzyz}&=\chi^{(3)}_{zyzy}
668 =\chi^{(3)}_{zxzx}=\chi^{(3)}_{xzxz}
669 =\chi^{(3)}_{xyxy}=\chi^{(3)}_{yxyx}\cr
670 \chi^{(3)}_{yzzy}&=\chi^{(3)}_{zyyz}
671 =\chi^{(3)}_{zxxz}=\chi^{(3)}_{xzzx}
672 =\chi^{(3)}_{xyyx}=\chi^{(3)}_{yxxy}\cr
673 }\eqno{(4)}
674 $$
675 with
676 $$
677 \chi^{(3)}_{xxxx}=\chi^{(3)}_{xxyy}+\chi^{(3)}_{xyxy}+\chi^{(3)}_{xyyx},
678 $$
679 i.~e.~a general set of 21 elements, of which only 3 are independent.
680
681 \subsection{Additional symmetries}
682 By applying the intrinsic permutation symmetry in the middle indices
683 for optical Kerr-effect, one generally has
684 $$
685 \chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega)
686 =\chi^{(3)}_{\mu\beta\alpha\gamma}(-\omega;\omega,\omega,-\omega),
687 $$
688 which hence slightly reduce the set~(4) to still 21 nonzero elements,
689 but of which now only two are independent.
690 For a beam polarized in the $xy$-plane, the elements of interest are
691 only those which only contain $x$ or $y$ in the indices, i.~e.~the subset
692 $$
693 \chi^{(3)}_{xxxx}=\chi^{(3)}_{yyyy},\quad
694 \chi^{(3)}_{xxyy}=\chi^{(3)}_{yyxx}
695 =\bigg\{\matrix{{\rm intr.\ perm.\ symm.}\cr
696 (\alpha,\omega)\rightleftharpoons(\beta,\omega)\cr}\bigg\}=
697 \chi^{(3)}_{xyxy}=\chi^{(3)}_{yxyx},\quad
698 \chi^{(3)}_{xyyx}=\chi^{(3)}_{yxxy},
699 $$
700 with
701 $$
702 \chi^{(3)}_{xxxx}=\chi^{(3)}_{xxyy}+\chi^{(3)}_{xyxy}+\chi^{(3)}_{xyyx},
703 $$
704 i.~e.~a set of eight elements, of which only two are independent.
705
706 \subsection{The polarization density}
707 The degeneracy factor $K(-\omega;\omega,\omega,-\omega)$ is calculated as
708 $$
709 K(-\omega;\omega,\omega,-\omega)=2^{l+m-n}p=2^{1+0-3}3=3/4.
710 $$
711 From the reduced set of nonzero susceptibilities for the beam polarized in
712 the $xy$-plane, and by using the calculated
713 value of the degeneracy factor in the convention of Butcher and Cotter,
714 we hence have the third order electric polarization density at
715 $\omega_{\sigma}=\omega$ given as ${\bf P}^{(n)}({\bf r},t)=
716 \Re[{\bf P}^{(n)}_{\omega}\exp(-i\omega t)]$, with
717 $$
718 \eqalign{
719 {\bf P}^{(3)}_{\omega}
720 &=\sum_{\mu}{\bf e}_{\mu}(P^{(3)}_{\omega})_{\mu}\cr
721 &=\{{\rm Using\ the\ convention\ of\ Butcher\ and\ Cotter}\}\cr
722 &=\sum_{\mu}{\bf e}_{\mu}
723 \bigg[\varepsilon_0{{3}\over{4}}\sum_{\alpha}\sum_{\beta}\sum_{\gamma}
724 \chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega)
725 (E_{\omega})_{\alpha}(E_{\omega})_{\beta}(E_{-\omega})_{\gamma}\bigg]\cr
726 &=\{{\rm Evaluate\ the\ sums\ over\ } (x,y,z)
727 {\rm\ for\ field\ polarized\ in\ the\ }xy{\rm\ plane}\}\cr
728 &=\varepsilon_0{{3}\over{4}}\{
729 {\bf e}_x[
730 \chi^{(3)}_{xxxx} E^x_{\omega} E^x_{\omega} E^x_{-\omega}
731 +\chi^{(3)}_{xyyx} E^y_{\omega} E^y_{\omega} E^x_{-\omega}
732 +\chi^{(3)}_{xyxy} E^y_{\omega} E^x_{\omega} E^y_{-\omega}
733 +\chi^{(3)}_{xxyy} E^x_{\omega} E^y_{\omega} E^y_{-\omega}]\cr
734 &\qquad\quad
735 +{\bf e}_y[
736 \chi^{(3)}_{yyyy} E^y_{\omega} E^y_{\omega} E^y_{-\omega}
737 +\chi^{(3)}_{yxxy} E^x_{\omega} E^x_{\omega} E^y_{-\omega}
738 +\chi^{(3)}_{yxyx} E^x_{\omega} E^y_{\omega} E^x_{-\omega}
739 +\chi^{(3)}_{yyxx} E^y_{\omega} E^x_{\omega} E^x_{-\omega}]\}\cr
740 &=\{{\rm Make\ use\ of\ }{\bf E}_{-\omega}={\bf E}^*_{\omega}
741 {\rm\ and\ relations\ }\chi^{(3)}_{xxyy}=\chi^{(3)}_{yyxx},
742 {\rm\ etc.}\}\cr
743 &=\varepsilon_0{{3}\over{4}}\{
744 {\bf e}_x[
745 \chi^{(3)}_{xxxx} E^x_{\omega} |E^x_{\omega}|^2
746 +\chi^{(3)}_{xyyx} E^y_{\omega}{}^2 E^{x*}_{\omega}
747 +\chi^{(3)}_{xyxy} |E^y_{\omega}|^2 E^x_{\omega}
748 +\chi^{(3)}_{xxyy} E^x_{\omega} |E^y_{\omega}|^2]\cr
749 &\qquad\quad
750 +{\bf e}_y[
751 \chi^{(3)}_{xxxx} E^y_{\omega} |E^y_{\omega}|^2
752 +\chi^{(3)}_{xyyx} E^x_{\omega}{}^2 E^{y*}_{\omega}
753 +\chi^{(3)}_{xyxy} |E^x_{\omega}|^2 E^y_{\omega}
754 +\chi^{(3)}_{xxyy} E^y_{\omega} |E^x_{\omega}|^2]\}\cr
755 &=\{{\rm Make\ use\ of\ the\ intrinsic\ permutation\ symmetry}\}\cr
756 &=\varepsilon_0{{3}\over{4}}\{
757 {\bf e}_x[
758 (\chi^{(3)}_{xxxx} |E^x_{\omega}|^2
759 +2\chi^{(3)}_{xxyy} |E^y_{\omega}|^2) E^x_{\omega}
760 +(\chi^{(3)}_{xxxx}-2\chi^{(3)}_{xxyy})
761 E^y_{\omega}{}^2 E^{x*}_{\omega}\cr
762 &\qquad\quad
763 {\bf e}_y[
764 (\chi^{(3)}_{xxxx} |E^y_{\omega}|^2
765 +2\chi^{(3)}_{xxyy} |E^x_{\omega}|^2) E^y_{\omega}
766 +(\chi^{(3)}_{xxxx}-2\chi^{(3)}_{xxyy})
767 E^x_{\omega}{}^2 E^{y*}_{\omega}.\cr
768 }
769 $$
770 For the optical field being linearly polarized, say in the $x$-direction,
771 the expression for the polarization density is significantly simplified,
772 to yield
773 $$
774 {\bf P}^{(3)}_{\omega}=\varepsilon_0(3/4){\bf e}_x
775 \chi^{(3)}_{xxxx} |E^x_{\omega}|^2 E^x_{\omega},
776 $$
777 i.~e.~taking a form that can be interpreted as an intensity-dependent
778 ($\sim|E^x_{\omega}|^2$) contribution to the refractive index
779 (cf.~Butcher and Cotter \S 6.3.1).
780
781 \subsection{The wave equation -- Time independent case}
782 In this example, we consider continuous wave
783 propagation\footnote{${}^6$}{That is to say, a time independent problem
784 with the temporal envelope of the electrical field being constant in time.}
785 in optical Kerr-media, using
786 light polarized in the $x$-direction and propagating along the positive
787 direction of the $z$-axis,
788 $$
789 {\bf E}({\bf r},t)=\Re[{\bf E}_{\omega}(z)\exp(-i\omega t)],
790 \qquad{\bf E}_{\omega}(z)={\bf A}_{\omega}(z)\exp(ikz)
791 ={\bf e}_x A^x_{\omega}(z)\exp(ikz),
792 $$
793 where, as previously, $k=\omega n_0/c$.
794 From material handed out during the third lecture (notes on the Butcher
795 and Cotter convention), the nonlinear polarization density for $x$-polarized
796 light is given as ${\bf P}^{({\rm NL})}_{\omega}={\bf P}^{(3)}_{\omega}$, with
797 $$
798 \eqalign{
799 {\bf P}^{(3)}_{\omega}
800 &=\varepsilon_0(3/4){\bf e}_x\chi^{(3)}_{xxxx}
801 (-\omega;\omega,\omega,-\omega)
802 |E^x_{\omega}|^2 E^x_{\omega}\cr
803 &=\varepsilon_0(3/4)\chi^{(3)}_{xxxx}
804 |{\bf E}_{\omega}|^2 {\bf E}_{\omega}\cr
805 &=\varepsilon_0(3/4)\chi^{(3)}_{xxxx}
806 |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\exp(ikz),\cr
807 }
808 $$
809 and the time independent wave equation for the field envelope
810 ${\bf A}_{\omega}$, using Eq.~(6), becomes
811 $$
812 \eqalign{
813 {{\partial}\over{\partial z}}{\bf A}_{\omega}
814 &=i{{\mu_0\omega^2}\over{2k}}
815 \underbrace{\varepsilon_0(3/4)\chi^{(3)}_{xxxx}
816 |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\exp(ikz)}_{
817 ={\bf P}^{({\rm NL})}_{\omega}(z)}\exp(-ikz)\cr
818 &=i{{3\omega^2}\over{8c^2k}}\chi^{(3)}_{xxxx}
819 |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\cr
820 &=\{{\rm since\ }k=\omega n_0(\omega)/c\}\cr
821 &=i{{3\omega}\over{8cn_0}}
822 \chi^{(3)}_{xxxx}
823 |{\bf A}_{\omega}|^2 {\bf A}_{\omega},\cr
824 }
825 $$
826 or, equivalently, in its scalar form
827 $$
828 {{\partial}\over{\partial z}}A^x_{\omega}
829 =i{{3\omega}\over{8cn_0}}\chi^{(3)}_{xxxx}
830 |A^x_{\omega}|^2 A^x_{\omega}.
831 $$
832
833 \subsection{Boundary conditions -- Time independent case}
834 For this special case of unidirectional wave propagation, the
835 boundary condition is simply a known optical field at the input,
836 $$A^x_{\omega}(0)=\{{\rm known}\}.$$
837
838 \subsection{Solving the wave equation -- Time independent case}
839 If the medium of interest now is analyzed at an angular frequency far
840 from any resonance, we may look for solutions to this equation with
841 $|{\bf A}_{\omega}(z)|$ being constant (for a lossless medium).
842 For such a case it is straightforward to integrate the final wave
843 equation to yield the general solution
844 $$
845 {\bf A}_{\omega}(z)={\bf A}_{\omega}(0)
846 \exp[i{{3\omega}\over{8cn_0}}\chi^{(3)}_{xxxx}|{\bf A}_{\omega}(0)|^2 z],
847 $$
848 or, again equivalently, in the scalar form
849 $$
850 A^x_{\omega}(z)=
851 A^x_{\omega}(0)
852 \exp[i{{3\omega}\over{8cn_0}}\chi^{(3)}_{xxxx}|A^x_{\omega}(0)|^2 z],
853 $$
854 which hence gives the solution for the real-valued electric field
855 ${\bf E}({\bf r},t)$ as
856 $$
857 \eqalign{
858 {\bf E}({\bf r},t)&=\Re[{\bf E}_{\omega}(z)\exp(-i\omega t)]\cr
859 &=\Re\{{\bf A}_{\omega}(z)\exp[i(kz-\omega t)]\}\cr
860 &=\Re\{{\bf A}_{\omega}(0)
861 \exp[i(\underbrace{{{\omega n_0}\over{c}}z
862 +{{3\omega}\over{8cn_0}}\chi^{(3)}_{xxxx}|A^x_{\omega}(0)|^2 z}_{
863 \equiv k_{\rm eff}z}
864 -\omega t)]\}.\cr
865 }
866 $$
867 From this solution, one immediately finds that the wave propagates
868 with an effective propagation constant
869 $$
870 k_{\rm eff}={{\omega}\over{c}}
871 [n_0+{{3}\over{8n_0}}\chi^{(3)}_{xxxx}|A^x_{\omega}(0)|^2],
872 $$
873 that is to say, experiencing the intensity dependent refractive index
874 $$
875 \eqalign{
876 n_{\rm eff}
877 &=n_0+{{3}\over{8n_0}}\chi^{(3)}_{xxxx}|A^x_{\omega}(0)|^2\cr
878 &=n_0+ n_2 |A^x_{\omega}(0)|^2,\cr
879 }
880 $$
881 with
882 $$n_2={{3}\over{8n_0}}\chi^{(3)}_{xxxx}.$$
883
884 \bye
885
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