Contents of file 'lect3/lect3.tex':




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1   % File: nlopt/lect3/lect3.tex [pure TeX code]
2   % Last change: January 19, 2003
3   %
4   % Lecture No 3 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\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm
25     \topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt
27       \else\hfill\fi}
29       \hfil{\it Nonlinear Optics 5A5513 (2003)}}
30     \def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)}
31       \hfil{\it Lecture notes #1}}
32     \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps}
33     \vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}}
34     \hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1}
35     \vskip 24pt\noindent}
36   \def\section #1 {\medskip\goodbreak\noindent{\bf #1}
37     \par\nobreak\smallskip\noindent}
38   \def\subsection #1 {\smallskip\goodbreak\noindent{\it #1}
39     \par\nobreak\smallskip\noindent}
40
41   \lecture{3}
42   \section{Susceptibility tensors in the frequency domain}
43   The susceptibility tensors in the frequency domain arise when
44   the electric field $E_{\alpha}(t)$ of the light is expressed in terms
45   of its Fourier transform $E_{\alpha}(\omega)$, by means of the
46   Fourier integral identity
47   $$48 E_{\alpha}(t)=\int^{\infty}_{-\infty}E_{\alpha}(\omega) 49 \exp(-i\omega t)\,d\omega=\fourier^{-1}[E_{\alpha}](t),\eqno{(1')} 50$$
51   with inverse relation
52   $$53 E_{\alpha}(\omega)={{1}\over{2\pi}}\int^{\infty}_{-\infty}E_{\alpha}(\tau) 54 \exp(i\omega\tau)\,d\tau=\fourier[E_{\alpha}](\omega).\eqno{(1'')} 55$$
56   This convention of inclusion of the factor of $2\pi$, as well as the
57   sign convention, is commonly used in quantum mechanics; however, it should
58   be emphasized that this convention is not a commonly adopted standard
59   in optics, neither in linear nor in nonlinear optical regimes.
60
61   The sign convention here used leads to wave solutions of the form
62   $f(kz-\omega t)$ for monochromatic waves propagating in the positive
63   $z$-direction, which might be somewhat more intuitive than the alternative
64   form $f(\omega t-kz)$, which is obtained if one instead apply the
65   alternative sign convention.
66
67   The convention for the inclusion of $2\pi$ in the Fourier transform
68   in Eq.~($1''$) is here convenient for description of electromagnetic
69   wave propagation in the frequency domain (going from the time domain
70   description, in terms of the polarization response functions, to the
71   frequency domain, in terms of the linear and nonlinear susceptibilities),
72   since it enables us to omit any multiple of $2\pi$ of the Fourier
73   transformed fields.
74
75   \section{First order susceptibility tensor}
76   By inserting Eq.~($1'$) is inserted into the previously
77   obtained\footnote{${}^1$}{Expressions for the first order, second order,
78   and $n$th order polarization densities were obtained in lecture two.}
79   relation for the first order, linear polarization density, one obtains
80   81 \eqalign{ 82 P^{(1)}_{\mu}({\bf r},t) 83 &=\varepsilon_0\int^{\infty}_{-\infty} 84 R^{(1)}_{\mu\alpha}(\tau) E_{\alpha}({\bf r},t-\tau)\,d\tau,\cr 85 &=\{{\rm express}\ E_{\alpha}({\bf r},t-\tau) 86 \ {\rm in\ frequency\ domain}\}\cr 87 &=\varepsilon_0\int^{\infty}_{-\infty} 88 R^{(1)}_{\mu\alpha}(\tau)\int^{\infty}_{-\infty} 89 E_{\alpha}({\bf r},\omega) 90 \exp[-i\omega(t-\tau)]\,d\omega\,d\tau,\cr 91 &=\{{\rm change\ order\ of\ integration}\}\cr 92 &=\varepsilon_0\int^{\infty}_{-\infty}\int^{\infty}_{-\infty} 93 R^{(1)}_{\mu\alpha}(\tau) E_{\alpha}({\bf r},\omega) 94 \exp(i\omega\tau)\,d\tau\,\exp(-i\omega t)\,d\omega,\cr 95 &=\varepsilon_0\int^{\infty}_{-\infty} 96 \chi^{(1)}_{\mu\alpha}(-\omega;\omega) 97 E_{\alpha}({\bf r},\omega) 98 \exp(-i\omega t)\,d\omega,\cr 99 }\eqno{(2)} 100
101   where the {\sl linear electric dipolar susceptibility},
102   $$103 \chi^{(1)}_{\mu\alpha}(-\omega;\omega) 104 =\int^{\infty}_{-\infty}R^{(1)}_{\mu\alpha}(\tau) 105 \exp(i\omega\tau)\,d\tau 106 =\fourier[R^{(1)}_{\mu\alpha}](\omega),\eqno{(3)} 107$$
108   was introduced. In this expression for the susceptibility,
109   $\omega_{\sigma}=\omega$, and the reasons for the somewhat
110   peculiar notation of arguments of the susceptibility will
111   be explained later on in the context of nonlinear susceptibilities.
112
113   \section{Second order susceptibility tensor}
114   In similar to the linear susceptibility tensor, by inserting Eq.~($1'$)
115   into the previously obtained relation for the second order, quadratic
116   polarization density, one obtains
117   118 \eqalign{ 119 P^{(2)}_{\mu}({\bf r},t) 120 &=\varepsilon_0 121 \int^{\infty}_{-\infty}\int^{\infty}_{-\infty} 122 \int^{\infty}_{-\infty}\int^{\infty}_{-\infty} 123 R^{(2)}_{\mu\alpha\beta}(\tau_1,\tau_2) 124 E_{\alpha}({\bf r},\omega_1) E_{\beta}({\bf r},\omega_2) 125 \cr&\qquad\qquad\qquad\times 126 \exp[-i(\omega_1(t-\tau_1)+\omega_2(t-\tau_2))] 127 \,d\tau_1\,d\tau_2 128 \,d\omega_1\,d\omega_2\cr 129 &=\varepsilon_0 130 \int^{\infty}_{-\infty}\int^{\infty}_{-\infty} 131 \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega_1,\omega_2) 132 E_{\alpha}({\bf r},\omega_1) E_{\beta}({\bf r},\omega_2) 133 \exp[-i\underbrace{(\omega_1+\omega_2)}_{\equiv\omega_{\sigma}}t] 134 \,d\omega_1\,d\omega_2\cr 135 }\eqno{(4)} 136
137   where the {\sl quadratic electric dipolar susceptibility},
138   $$139 \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega_1,\omega_2) 140 =\int^{\infty}_{-\infty}\int^{\infty}_{-\infty} 141 R^{(2)}_{\mu\alpha\beta}(\tau_1,\tau_2) 142 \exp[i(\omega_1\tau_1+\omega_2\tau_2)]\,d\tau_1\,d\tau_2,\eqno{(5)} 143$$
144   was introduced. In this expression for the susceptibility,
145   $\omega_{\sigma}=\omega_1+\omega_2$, and the reason for the notation
146   of arguments should now be somewhat more clear: the first angular
147   frequency argument of the susceptibility tensor is simply the sum
148   of all driving angular frequencies of the optical field.
149
150   %Let us now consider the effects of the previously discussed requirements
151   %of reality and causality.
152
153   The intrinsic permutation symmetry of
154   $R^{(2)}_{\mu\alpha\beta}(\tau_1,\tau_2)$,
155   the second order polarization response function, carries over to the
156   second order susceptibility tensor as well, in the sense that
157   $$158 \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega_1,\omega_2) 159 =\chi^{(2)}_{\mu\beta\alpha}(-\omega_{\sigma};\omega_2,\omega_1), 160$$
161   i.~e.~the second order susceptibility is invariant under any
162   of the $2!=2$ pairwise permutations of $(\alpha,\omega_1)$
163   and $(\beta,\omega_2)$.
164
165   \section{Higher order susceptibility tensors}
166   In similar to the linear and quadratic susceptibility tensors, by
167   inserting Eq.~($1'$) into the previously obtained relation for
168   the $n$th order polarization density, one obtains
169   170 \eqalign{ 171 P^{(n)}_{\mu}({\bf r},t) 172 &=\varepsilon_0 173 \int^{\infty}_{-\infty}\cdots\int^{\infty}_{-\infty} 174 \chi^{(n)}_{\mu\alpha_1\cdots\alpha_n} 175 (-\omega_{\sigma};\omega_1,\ldots,\omega_n) 176 E_{\alpha_1}({\bf r},\omega_1)\cdots 177 E_{\alpha_n}({\bf r},\omega_n) 178 \cr&\qquad\qquad\qquad\times 179 \exp[-i\underbrace{(\omega_1+\ldots+\omega_n)}_{ 180 \equiv\omega_{\sigma}}t] 181 \,d\omega_1\,\cdots\,d\omega_n,\cr 182 }\eqno{(6)} 183
184   where the {\sl $n$th order electric dipolar susceptibility},
185   $$186 \chi^{(n)}_{\mu\alpha_1\cdots\alpha_n} 187 (-\omega_{\sigma};\omega_1,\ldots,\omega_n) 188 =\int^{\infty}_{-\infty}\cdots\int^{\infty}_{-\infty} 189 R^{(n)}_{\mu\alpha_1\cdots\alpha_n}(\tau_1,\ldots,\tau_n) 190 \exp[i(\omega_1\tau_1+\ldots+\omega_n\tau_n)]\,d\tau_1\,\cdots\,d\tau_n, 191 \eqno{(7)} 192$$
193   was introduced, and where, as previously,
194   $$\omega_{\sigma}=\omega_1+\omega_2+\ldots+\omega_n.$$
195
196   The intrinsic permutation symmetry of
197   $R^{(n)}_{\mu\alpha_1\cdots\alpha_n}(\tau_1,\ldots,\tau_n)$,
198   the $n$th order polarization response function, also in this general
199   case carries over to the $n$th order susceptibility tensor as well,
200   in the sense that
201   $$202 \chi^{(n)}_{\mu\alpha_1\alpha_2\cdots\alpha_n} 203 (-\omega_{\sigma};\omega_1,\omega_2,\ldots,\omega_n) 204$$
205   is invariant under any of the $n!$ pairwise permutations of
206   $(\alpha_1,\omega_1)$, $(\alpha_2,\omega_2)$, $\ldots$, $(\alpha_n,\omega_n)$.
207
208   \section{Monochromatic fields}
209   It should at this stage be emphasized that even though the
210   electric field via the Fourier integral identity can be seen
211   as a superposition of infinitely many infinitesimally narrow
212   band monochromatic components, the superposition principle
213   of linear optics, which states that the wave equation may be
214   independently solved for each frequency component of the light,
215   generally does {\sl not} hold in nonlinear optics.
216
217   For monochromatic light, the electric field can be written as a
218   superposition of a set of distinct terms in time domain as
219   $$220 {\bf E}({\bf r},t)=\sum_k \Re[{\bf E}_{\omega_k}\exp(-i\omega_k t)], 221$$
222   with the convention that the involved angular frequencies all are taken as
223   positive, $\omega_k\ge 0$, and the electric field in the frequency domain
224   simply becomes a superposition of delta peaks in the spectrum,
225   226 \eqalign{ 227 {\bf E}({\bf r},\omega) 228 &={{1}\over{2\pi}}\int^{\infty}_{-\infty}{\bf E}({\bf r},\tau) 229 \exp(i\omega\tau)\,d\tau\cr 230 &=\{{\rm express\ as\ monochromatic\ field}\}\cr 231 &={{1}\over{2\pi}}\sum_k \int^{\infty}_{-\infty} 232 \Re[{\bf E}_{\omega_k}\exp(-i\omega_k \tau)] 233 \exp(i\omega\tau)\,d\tau\cr 234 &=\{{\rm by\ definition}\}\cr 235 &={{1}\over{4\pi}}\sum_k \int^{\infty}_{-\infty} 236 [{\bf E}_{\omega_k}\exp(i(\omega-\omega_k)\tau) 237 +{\bf E}^*_{\omega_k}\exp(i(\omega+\omega_k)\tau)]\,d\tau\cr 238 &={{1}\over{2}}\sum_k \bigg[ 239 {\bf E}_{\omega_k}\underbrace{{{1}\over{2\pi}}\int^{\infty}_{-\infty} 240 \exp(i(\omega-\omega_k)\tau)\,d\tau}_{ 241 \equiv\delta(\omega-\omega_k)} 242 +{\bf E}^*_{\omega_k} 243 \underbrace{{{1}\over{2\pi}}\int^{\infty}_{-\infty} 244 \exp(i(\omega+\omega_k)\tau)\,d\tau}_{ 245 \equiv\delta(\omega+\omega_k)}\bigg]\cr 246 &=\{{\rm definition\ of\ the\ delta\ function}\}\cr 247 &={{1}\over{2}}\sum_k [{\bf E}_{\omega_k}\delta(\omega-\omega_k) 248 +{\bf E}^*_{\omega_k}\delta(\omega+\omega_k)].\cr 249 } 250
251   By inserting this form of the electric field (taken in the frequency domain)
252   into the polarization density, one obtains the polarization density
253   in the monochromatic form
254   $$255 {\bf P}^{(n)}({\bf r},t)=\sum_{\omega_{\sigma}\ge 0} 256 \Re[{\bf P}^{(n)}_{\omega_{\sigma}}\exp(-i\omega_{\sigma} t)], 257$$
258   with complex-valued Cartesian components at angular frequency
259   $\omega_{\sigma}$ given as
260   261 \eqalign{ 262 ({\bf P}^{(n)}_{\omega_{\sigma}})_{\mu} 263 =2\varepsilon_0\sum_{\alpha_1}\cdots\sum_{\alpha_n}& 264 \chi^{(n)}_{\mu\alpha_1\alpha_2\cdots\alpha_n} 265 (-\omega_{\sigma};\omega_1,\omega_2,\ldots,\omega_n) 266 (E_{\omega_1})_{\alpha_1} 267 (E_{\omega_2})_{\alpha_2} 268 \cdots(E_{\omega_n})_{\alpha_n}\cr 269 &+\chi^{(n)}_{\mu\alpha_1\alpha_2\cdots\alpha_n} 270 (-\omega_{\sigma};\omega_2,\omega_1,\ldots,\omega_n) 271 (E_{\omega_2})_{\alpha_1} 272 (E_{\omega_1})_{\alpha_2} 273 \cdots(E_{\omega_n})_{\alpha_n}\cr 274 &+{\rm all\ other\ distinguishable\ terms}\cr 275 } 276 \eqno{(8)} 277
278   where, as previously, $\omega_{\sigma}=\omega_1+\omega_2+\ldots+\omega_n$.
279   In the right hand side of Eq.~(8), the summation is performed over
280   all distinguishble terms, that is to say over all the possible
281   combinations of $\omega_1,\omega_2,\ldots,\omega_n$ that give
282   rise to the particular $\omega_{\sigma}$.
283   Within this respect, a certain frequency and its negative counterpart
284   are to be considered as distinct frequencies when appearing in the
285   set. In general, there are several possible combinations that
286   give rise to a certain $\omega_{\sigma}$; for example
287   $(\omega,\omega,-\omega)$, $(\omega,-\omega,\omega)$, and
288   $(-\omega,\omega,\omega)$ form the set of distinct combinations
289   of optical frequencies that give rise to optical Kerr-effect
290   (a field dependent contribution to the polarization density
291   at $\omega_{\sigma}=\omega+\omega-\omega=\omega$).
292
293   A general conclusion of the form of Eq.~(8), keeping the intrinsic
294   permutation symmetry in mind, is that only one term needs to be
295   written, and the number of times this term appears in the expression
296   for the polarization density should consequently be equal to
297   the number of distinguishable combinations of
298   $\omega_1,\omega_2,\ldots,\omega_n$.
299
300   \section{Convention for description of nonlinear optical polarization}
301   As a recipe'' in theoretical nonlinear optics, Butcher and Cotter
302   provide a very useful convention which is well worth to hold on to.
303   For a superposition of monochromatic waves, and by invoking the general
304   property of the intrinsic permutation symmetry, the monochromatic
305   form of the $n$th order polarization density can be written as
306   $$307 (P^{(n)}_{\omega_{\sigma}})_{\mu} 308 =\varepsilon_0\sum_{\alpha_1}\cdots\sum_{\alpha_n}\sum_{\omega} 309 K(-\omega_{\sigma};\omega_1,\ldots,\omega_n) 310 \chi^{(n)}_{\mu\alpha_1\cdots\alpha_n} 311 (-\omega_{\sigma};\omega_1,\ldots,\omega_n) 312 (E_{\omega_1})_{\alpha_1}\cdots(E_{\omega_n})_{\alpha_n}. 313 \eqno{(9)} 314$$
315   The first summations in Eq.~(9), over $\alpha_1,\ldots,\alpha_n$,
316   is simply an explicit way of stating that the Einstein convention
317   of summation over repeated indices holds.
318   The summation sign $\sum_{\omega}$, however, serves as a reminder
319   that the expression that follows is to be summed over
320   {\sl all distinct sets of $\omega_1,\ldots,\omega_n$}.
321   Because of the intrinsic permutation symmetry, the frequency arguments
322   appearing in Eq.~(9) may be written in arbitrary order.
323
324   By all distinct sets of $\omega_1,\ldots,\omega_n$'', we here mean
325   that the summation is to be performed, as for example in the case of
326   optical Kerr-effect, over the single set of nonlinear susceptibilities
327   that contribute to a certain angular frequency as
328   $(-\omega;\omega,\omega,-\omega)$ {\sl or} $(-\omega;\omega,-\omega,\omega)$
329   {\sl or} $(-\omega;-\omega,\omega,\omega)$.
330   In this example, each of the combinations are considered as {\sl distinct},
331   and it is left as an arbitary choice which one of these sets that are
332   most convenient to use (this is simply a matter of choosing notation,
333   and does not by any means change the description of the interaction).
334
335   In Eq.~(9), the degeneracy factor $K$ is formally described as
336   $$K(-\omega_{\sigma};\omega_1,\ldots,\omega_n)=2^{l+m-n}p$$
337   where
338   339 \eqalign{ 340 p&={\rm the\ number\ of\ {\sl distinct}\ permutations\ of} 341 \ \omega_1,\omega_2,\ldots,\omega_1,\cr 342 n&={\rm the\ order\ of\ the\ nonlinearity},\cr 343 m&={\rm the\ number\ of\ angular\ frequencies}\ \omega_k 344 \ {\rm that\ are\ zero,\ and}\cr 345 l&=\bigg\lbrace\matrix{1,\qquad{\rm if}\ \omega_{\sigma}\ne 0,\cr 346 0,\qquad{\rm otherwise}.}\cr 347 } 348
349   In other words, $m$ is the number of DC electric fields present,
350   and $l=0$ if the nonlinearity we are analyzing gives a static, DC,
351   polarization density, such as in the previously (in the spring model)
352   described case of optical rectification in the presence of second
353   harmonic fields (SHG).
354
355   A list of frequently encountered nonlinear phenomena in nonlinear
356   optics, including the degeneracy factors as conforming to the above
357   convention, is given in Butcher and Cotters book, Table 2.1, on page 26.
358
359   \section{Note on the complex representation of the optical field}
360   Since the observable electric field of the light, in Butcher and
361   Cotters notation taken as
362   $$363 {\bf E}({\bf r},t)={{1}\over{2}}\sum_{\omega_k\ge 0} 364 [{\bf E}_{\omega_k}\exp(-i\omega_k t)+{\bf E}^*_{\omega_k}\exp(i\omega_k t)], 365$$
366   is a real-valued quantity, it follows that negative frequencies in the
367   complex notation should be interpreted as the complex conjugate of the
368   respective field component, or
369   $$370 {\bf E}_{-\omega_k}={\bf E}^*_{\omega_k}. 371$$
372
373   \section{Example: Optical Kerr-effect}
374   Assume a monochromatic optical wave (containing forward and/or backward
375   propagating components) polarized in the $xy$-plane,
376   $$377 {\bf E}(z,t)=\Re[{\bf E}_{\omega}(z)\exp(-i\omega_t)]\in{\Bbb R}^3, 378$$
379   with all spatial variation of the field contained in
380   $$381 {\bf E}_{\omega}(z)={\bf e}_x E^x_{\omega}(z) 382 +{\bf e}_y E^y_{\omega}(z)\in{\Bbb C}^3. 383$$
384   Optical Kerr-effect is in isotropic media described by the third order
385   susceptibility
386   $$387 \chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega), 388$$
389   with nonzero components of interest for the $xy$-polarized beam given in
390   Appendix 3.3 of Butcher and Cotters book as
391   $$392 \chi^{(3)}_{xxxx}=\chi^{(3)}_{yyyy},\quad 393 \chi^{(3)}_{xxyy}=\chi^{(3)}_{yyxx} 394 =\bigg\{\matrix{{\rm intr.\ perm.\ symm.}\cr 395 (\alpha,\omega)\rightleftharpoons(\beta,\omega)\cr}\bigg\}= 396 \chi^{(3)}_{xyxy}=\chi^{(3)}_{yxyx},\quad 397 \chi^{(3)}_{xyyx}=\chi^{(3)}_{yxxy}, 398$$
399   with
400   $$401 \chi^{(3)}_{xxxx}=\chi^{(3)}_{xxyy}+\chi^{(3)}_{xyxy}+\chi^{(3)}_{xyyx}. 402$$
403   The degeneracy factor $K(-\omega;\omega,\omega,-\omega)$ is calculated as
404   $$405 K(-\omega;\omega,\omega,-\omega)=2^{l+m-n}p=2^{1+0-3}3=3/4. 406$$
407   From this set of nonzero susceptibilities, and using the calculated
408   value of the degeneracy factor in the convention of Butcher and Cotter,
409   we hence have the third order electric polarization density at
410   $\omega_{\sigma}=\omega$ given as ${\bf P}^{(n)}({\bf r},t)= 411 \Re[{\bf P}^{(n)}_{\omega}\exp(-i\omega t)]$, with
412   413 \eqalign{ 414 {\bf P}^{(3)}_{\omega} 415 &=\sum_{\mu}{\bf e}_{\mu}(P^{(3)}_{\omega})_{\mu}\cr 416 &=\{{\rm Using\ the\ convention\ of\ Butcher\ and\ Cotter}\}\cr 417 &=\sum_{\mu}{\bf e}_{\mu} 418 \bigg[\varepsilon_0{{3}\over{4}}\sum_{\alpha}\sum_{\beta}\sum_{\gamma} 419 \chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega) 420 (E_{\omega})_{\alpha}(E_{\omega})_{\beta}(E_{-\omega})_{\gamma}\bigg]\cr 421 &=\{{\rm Evaluate\ the\ sums\ over\ } (x,y,z) 422 {\rm\ for\ field\ polarized\ in\ the\ }xy{\rm\ plane}\}\cr 423 &=\varepsilon_0{{3}\over{4}}\{ 424 {\bf e}_x[ 425 \chi^{(3)}_{xxxx} E^x_{\omega} E^x_{\omega} E^x_{-\omega} 426 +\chi^{(3)}_{xyyx} E^y_{\omega} E^y_{\omega} E^x_{-\omega} 427 +\chi^{(3)}_{xyxy} E^y_{\omega} E^x_{\omega} E^y_{-\omega} 428 +\chi^{(3)}_{xxyy} E^x_{\omega} E^y_{\omega} E^y_{-\omega}]\cr 429 &\qquad\quad 430 +{\bf e}_y[ 431 \chi^{(3)}_{yyyy} E^y_{\omega} E^y_{\omega} E^y_{-\omega} 432 +\chi^{(3)}_{yxxy} E^x_{\omega} E^x_{\omega} E^y_{-\omega} 433 +\chi^{(3)}_{yxyx} E^x_{\omega} E^y_{\omega} E^x_{-\omega} 434 +\chi^{(3)}_{yyxx} E^y_{\omega} E^x_{\omega} E^x_{-\omega}]\}\cr 435 &=\{{\rm Make\ use\ of\ }{\bf E}_{-\omega}={\bf E}^*_{\omega} 436 {\rm\ and\ relations\ }\chi^{(3)}_{xxyy}=\chi^{(3)}_{yyxx}, 437 {\rm\ etc.}\}\cr 438 &=\varepsilon_0{{3}\over{4}}\{ 439 {\bf e}_x[ 440 \chi^{(3)}_{xxxx} E^x_{\omega} |E^x_{\omega}|^2 441 +\chi^{(3)}_{xyyx} E^y_{\omega}{}^2 E^{x*}_{\omega} 442 +\chi^{(3)}_{xyxy} |E^y_{\omega}|^2 E^x_{\omega} 443 +\chi^{(3)}_{xxyy} E^x_{\omega} |E^y_{\omega}|^2]\cr 444 &\qquad\quad 445 +{\bf e}_y[ 446 \chi^{(3)}_{xxxx} E^y_{\omega} |E^y_{\omega}|^2 447 +\chi^{(3)}_{xyyx} E^x_{\omega}{}^2 E^{y*}_{\omega} 448 +\chi^{(3)}_{xyxy} |E^x_{\omega}|^2 E^y_{\omega} 449 +\chi^{(3)}_{xxyy} E^y_{\omega} |E^x_{\omega}|^2]\}\cr 450 &=\{{\rm Make\ use\ of\ intrinsic\ permutation\ symmetry}\}\cr 451 &=\varepsilon_0{{3}\over{4}}\{ 452 {\bf e}_x[ 453 (\chi^{(3)}_{xxxx} |E^x_{\omega}|^2 454 +2\chi^{(3)}_{xxyy} |E^y_{\omega}|^2) E^x_{\omega} 455 +(\chi^{(3)}_{xxxx}-2\chi^{(3)}_{xxyy}) 456 E^y_{\omega}{}^2 E^{x*}_{\omega}\cr 457 &\qquad\quad 458 {\bf e}_y[ 459 (\chi^{(3)}_{xxxx} |E^y_{\omega}|^2 460 +2\chi^{(3)}_{xxyy} |E^x_{\omega}|^2) E^y_{\omega} 461 +(\chi^{(3)}_{xxxx}-2\chi^{(3)}_{xxyy}) 462 E^x_{\omega}{}^2 E^{y*}_{\omega}.\cr 463 } 464
465   For the optical field being linearly polarized, say in the $x$-direction,
466   the expression for the polarization density is significantly simplified,
467   to yield
468   $$469 {\bf P}^{(3)}_{\omega}=\varepsilon_0(3/4){\bf e}_x 470 \chi^{(3)}_{xxxx} |E^x_{\omega}|^2 E^x_{\omega}, 471$$
472   i.~e.~taking a form that can be interpreted as an intensity-dependent
473   ($\sim|E^x_{\omega}|^2$) contribution to the refractive index
474   (cf.~Butcher and Cotter \S 6.3.1).
475   \bye
476