Contents of file 'lect3/lect3.tex':
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
10 %
11 % Read amssym to get the AMS {\Bbb E} font (strikethrough E) and
12 % the Euler fraktur font.
13 %
14 \input amssym
15 \font\ninerm=cmr9
16 \font\twelvesc=cmcsc10
17 %
18 % Use AMS Euler fraktur style for short-hand notation of Fourier transform
19 %
20 \def\fourier{\mathop{\frak F}\nolimits}
21 \def\Re{\mathop{\rm Re}\nolimits} % real part
22 \def\Im{\mathop{\rm Im}\nolimits} % imaginary part
23 \def\Tr{\mathop{\rm Tr}\nolimits} % quantum mechanical trace
24 \def\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm
25 \topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt
26 \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
27 \else\hfill\fi}
28 \def\rightheadline{\tenrm{\it Lecture notes #1}
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
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