Contents of file 'lect9/lect9.tex':




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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   %
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   \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
28       \else\hfill\fi}
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
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
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
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