Contents of file 'lect2/lect2.tex':
1 % File: nlopt/lect2/lect2.tex [pure TeX code]
2 % Last change: January 10, 2003
3 %
4 % Lecture No 2 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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14 \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
15 \else\hfill\fi}
16 \def\rightheadline{\tenrm{\it Lecture notes #1}
17 \hfil{\it Nonlinear Optics 5A5513 (2003)}}
18 \def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)}
19 \hfil{\it Lecture notes #1}}
20 \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps}
21 \vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}}
22 \hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1}
23 \vskip 24pt\noindent}
24 \def\section #1 {\medskip\goodbreak\noindent{\bf #1}
25 \par\nobreak\smallskip\noindent}
26 \def\subsection #1 {\smallskip\goodbreak\noindent{\it #1}
27 \par\nobreak\smallskip\noindent}
28
29 \lecture{2}
30 \section{Nonlinear polarization density}
31 From the introductory perturbation analysis of the all-classical anharmonic
32 oscillator in the previous lecture, we now {\sl a priori} know that
33 it is possible to express the electric polarization density as a power
34 series in the electric field of the optical wave.
35
36 Loosely formulated, the electric polarization density in complex notation
37 can be taken as the series
38 $$
39 \eqalign{
40 P_{\mu}(\omega_{\sigma})=\varepsilon_0[
41 \underbrace{\chi^{(1)}_{\mu\alpha}(-\omega_{\sigma};\omega_{\sigma})
42 E_{\alpha}(\omega_{\sigma})}_{
43 \sim P^{(1)}_{\mu}(\omega_{\sigma})}
44 &+\underbrace{\chi^{(2)}_{\mu\alpha\beta}
45 (-\omega_{\sigma};\omega_1,\omega_2)
46 E_{\alpha}(\omega_1)E_{\beta}(\omega_2)}_{
47 \sim P^{(2)}_{\mu}(\omega_{\sigma}),
48 \ \omega_{\sigma}=\omega_1+\omega_2}\cr
49 &+\underbrace{\chi^{(3)}_{\mu\alpha\beta\gamma}
50 (-\omega_{\sigma};\omega_1,\omega_2,\omega_3)
51 E_{\alpha}(\omega_1)E_{\beta}(\omega_2)E_{\gamma}(\omega_3)}_{
52 \sim P^{(3)}_{\mu}(\omega_{\sigma}),
53 \ \omega_{\sigma}=\omega_1+\omega_2+\omega_3}+\ldots\,]\cr
54 }
55 $$
56 where we adopted Einstein's convention of summation for terms with repeated
57 subscripts. (A more formal formulation of the polarization density will be
58 described later.)
59
60 \section{Symmetries in nonlinear optics}
61 There are essentially four classes of symmetries that we will
62 encounter in this course:
63 \smallskip
64
65 \item{$[1]$}{Intrinsic permutation symmetry:
66 $$
67 \eqalign{
68 \chi^{(3)}_{\mu\alpha\beta\gamma}
69 (-\omega_{\sigma};\omega_1,\omega_2,\omega_3)
70 &=\chi^{(3)}_{\mu\beta\alpha\gamma}
71 (-\omega_{\sigma};\omega_2,\omega_1,\omega_3)\cr
72 &=\chi^{(3)}_{\mu\beta\gamma\alpha}
73 (-\omega_{\sigma};\omega_2,\omega_3,\omega_1)\cr
74 &=\chi^{(3)}_{\mu\gamma\beta\alpha}
75 (-\omega_{\sigma};\omega_3,\omega_2,\omega_1),\cr
76 }
77 $$
78 that is to say, invariance under the $n!$ possible permutations of
79 $(\alpha_k,\omega_k)$, $k=1,2,\ldots,n$. This principle applies
80 generally to resonant as well as nonresonant media. ({\sl Described
81 in this lecture.})}
82 \item{$[2]$}{Overall permutation symmetry:
83 $$
84 \eqalign{
85 \chi^{(3)}_{\mu\alpha\beta\gamma}
86 (-\omega_{\sigma};\omega_1,\omega_2,\omega_3)
87 &=\chi^{(3)}_{\alpha\mu\beta\gamma}
88 (\omega_1;-\omega_{\sigma},\omega_2,\omega_3)\cr
89 &=\chi^{(3)}_{\alpha\beta\mu\gamma}
90 (\omega_1;\omega_2,-\omega_{\sigma},\omega_3)\cr
91 &=\chi^{(3)}_{\alpha\beta\gamma\mu}
92 (\omega_1;\omega_2,\omega_3,-\omega_{\sigma}),\cr
93 }
94 $$
95 that is to say, invariance under the $(n+1)!$ possible permutations of
96 $(\mu,-\omega_{\sigma})$, $(\alpha_k,\omega_k)$, $k=1,2,\ldots,n$.
97 This principle applies to nonresonant media, where all optical frequencies
98 appearing in the formula for the susceptibility are removed far from the
99 transition frequencies of the medium. ({\sl Described in lecture~4.})}
100 \item{$[3]$}{Kleinman symmetry:
101 $$
102 \eqalign{
103 \chi^{(3)}_{\mu\alpha\beta\gamma}
104 (-\omega_{\sigma};\omega_1,\omega_2,\omega_3)
105 &=\chi^{(3)}_{\alpha\mu\beta\gamma}
106 (-\omega_{\sigma};\omega_1,\omega_2,\omega_3)\cr
107 &=\chi^{(3)}_{\alpha\beta\mu\gamma}
108 (-\omega_{\sigma};\omega_1,\omega_2,\omega_3)\cr
109 &=\chi^{(3)}_{\alpha\beta\gamma\mu}
110 (-\omega_{\sigma};\omega_1,\omega_2,\omega_3),\cr
111 }
112 $$
113 that is to say, invariance under the $(n+1)!$ possible permutations of
114 the subscripts $\mu,\alpha_1,\ldots,\alpha_n$.
115 This principle is a consequence of the overall permutation symmetry, and
116 applies in the low-frequency limit of nonresonant media.}
117 \item{$[4]$}{Spatial symmetries, given by the point symmetry class
118 of the medium. ({\sl Described in lecture~6.})}
119 \smallskip
120
121 \section{Conditions for observing nonlinear optical interactions}
122 Loosely formulated, a nonlinear response between light and matter
123 depends on one of two key indgredients: either there is a resonance
124 between the light wave and some natural oscillation mode of the
125 medium, or the light is sufficiently intense.
126 {\sl Direct resonance} can occur in isolated intervals of the electromagnetic
127 spectrum at
128 \item{$\bullet$}{ultraviolet and visible frequencies ($10^{15}$ ${\rm s}^{-1}$)
129 where the oscillator corresponds to an electronic transition of the medium,}
130 \item{$\bullet$}{infrared ($10^{13}$ ${\rm s}^{-1}$), where the medium has
131 vibrational modes, and}
132 \item{$\bullet$}{the far infrared-microwave range ($10^{11}$ ${\rm s}^{-1}$),
133 where there are rotational modes.}
134 These interactions are also called {\sl one-photon processes}, and
135 are schematically illustrated in Fig.~3.
136 \medskip
137 \centerline{\epsfxsize=70mm\epsfbox{../images/onephot/onephot.1}}
138 \medskip
139 \centerline{Figure 3. Transition scheme of the one-photon process.}
140 \medskip
141 \noindent
142 The lower frequency modes can be excited at optical frequencies
143 ($10^{15}$ ${\rm s}^{-1}$) through indirect resonant processes in which
144 the difference in frequencies and wave vectors of two light waves,
145 called the pump and Stokes wave, respectively, matches the frequency
146 and wave vector of one of these lower frequency modes.
147 These three-frequency interactions are sometime called {\sl two-photon
148 processes}. In the case where there ``lower frequency'' mode is an
149 electronic transition or in the vibrational range (in which case the
150 Stokes frequency can be of the same order of magnitude as that of
151 the light wave), this process is called {\sl Raman scattering}.
152 \medskip
153 \centerline{\epsfxsize=70mm\epsfbox{../images/raman/sraman.1}
154 \epsfxsize=70mm\epsfbox{../images/raman/asraman.1}}
155 \medskip
156 \centerline{Figure 4. Transition schemes of stimulated Raman Scattering.}
157 \medskip
158 \noindent
159 The stimulated Raman scattering is essentially a two-photon process in which
160 one photon at $\omega_1$ is absorbed and one photon at $\omega_2$ is emitted,
161 while the material makes a transition from the initial state $|a\rangle$
162 to the final state $|c\rangle$, as shown in Fig.~4.
163 Energy conservation requires the Raman resonance frequency
164 (electronic or vibrational) to satisfy
165 $\hbar\Omega_{ca}=\hbar(\omega_1-\omega_2)$,
166 and hence we may classify the Stokes and Anti-Stokes transitions as
167 $$
168 \eqalign{
169 \hbar\Omega_{ca}>0\qquad&\Leftrightarrow
170 \qquad{\textstyle{\rm Stokes\ Raman}}\cr
171 \hbar\Omega_{ca}<0\qquad&\Leftrightarrow
172 \qquad{\textstyle{\rm Anti-Stokes\ Raman}}\cr
173 }
174 $$
175 When the lower frequency mode instead is an {\sl acoustic} mode of
176 the material, the process is instead called {\sl Brillouin scattering}.
177
178 As will be shown explicitly later on in the course, optical resonance
179 with transitions of the material is an important tool for ``boosting up''
180 the nonlinearities, with enhanced possibilities of applications.
181 However, at single-photon resonance we have a strong absorption
182 at the frequency of the optical field, and in most cases this
183 is a non-desirable effect, since it decreases the optical intensity,
184 even though it meanwhile also enhances the nonlinearity of the material.
185
186 However, by instead exploiting the two-photon resonance, the desired
187 nonlinearity can be significantly enhanced whilst at the same time
188 the competing absorption process can be minimized by avoiding
189 coincidences between optical frequencies and {\sl single-photon} resonances.
190
191 One general drawback with the resonant enhancement is that the response
192 time of the polarization density of the material is slowed down, affecting
193 applications such as optical switching or modulation, where speed
194 is of importance.
195
196 \section{Phenomenological description of the susceptibility tensors}
197 Before entering the full quantum-mechanical formalism, we will
198 assume the medium to possess a temporal response described by
199 {\sl time response functions} $R(t)$.
200
201 The very first step in the analysis is to express the electric
202 polarization density, which in classical electrodynamical terms
203 is expressed as the sum over all $M$ electric charges $q_k$ in a small
204 volume~$V$ centered at ${\bf r}$,
205 $$
206 {\bf P}({\bf r},t)={{1}\over{V}}\sum^{M}_{k=1} q_k {\bf r}_k(t),
207 $$
208 as a series expansion
209 $$
210 {\bf P}({\bf r},t)={\bf P}^{(0)}({\bf r},t)+{\bf P}^{(1)}({\bf r},t)
211 +{\bf P}^{(2)}({\bf r},t)+\ldots
212 +{\bf P}^{(n)}({\bf r},t)+\ldots,
213 $$
214 where ${\bf P}^{(1)}({\bf r},t)$ is linear in the electric field,
215 ${\bf P}^{(2)}({\bf r},t)$ is quadratic in the electric field, etc.
216 The field-independent ${\bf P}^{(0)}({\bf r},t)$ corresponds to
217 eventually appearing static polarization of the medium.
218 It should be emphasized that any of these terms may be linear as
219 well as nonlinear functions of the electric field of the optical
220 wave; i.~e.~an externally applied static electric field may
221 together with the electric field of the light interact through,
222 for example, ${\bf P}^{(3)}({\bf r},t)$ to induce an electric
223 polarization that is linear in the optical field.
224
225 \medskip
226 \centerline{\epsfxsize=85mm\epsfbox{../images/respfunc/respfunc.1}}
227 \centerline{Figure 9. Schematic form of a possible response function
228 in time domain.}
229 \medskip
230
231 In the analysis that now is to follow, we will a priori assume
232 that it is possible to express the polarization density as a
233 perturbation series. Later on, we will show how each of the
234 terms can be derived from a more stringent basis, where it
235 also will be stated which conditions that must hold in order
236 to apply perturbation theory.
237
238 \section{Linear polarization response function}
239 Since ${\bf P}^{(1)}({\bf r},t)$ is taken as linear in the electric field,
240 we may express the linear polarization density of the medium as
241 being related to the optical field as
242 $$
243 P^{(1)}_{\mu}({\bf r},t)=\varepsilon_0
244 \int^{\infty}_{-\infty} T^{(1)}_{\mu\alpha}(t;\tau)
245 E_{\alpha}({\bf r},\tau)\,d\tau,
246 \eqno{(7)}
247 $$
248 where $T^{(1)}_{\mu\alpha}(t;\tau)$ is a rank-two tensor that
249 weights all contributions in time from the electric field of
250 the light.
251 A few required properties of $T^{(1)}_{\mu\alpha}(t;\tau)$ can
252 immediately be stated:
253 \smallskip
254
255 \item{$\bullet$}{{\sl Causality.} We require that no optically
256 induced contribution can occur {\sl before} the field is applied,
257 i.~e.~$T^{(1)}_{\mu\alpha}(t;\tau)=0$ for $t\le\tau$.}
258 \smallskip
259
260 \item{$\bullet$}{{\sl Time invariance.} Under most circumstances,
261 we may in addition assume that the material parameters are
262 constant in time, such that $P^{(1)}_{\mu}({\bf r},t')$
263 is identical to the polarisation as induced by the time-displaced
264 electric field $E_{\alpha}({\bf r},t')$.}
265 \smallskip
266 \noindent The second of these properties is essentially a manifestation
267 of an adiabatically following change of the carrier wave of the
268 optical field.
269
270 By using Eq.~(7), the time invariance of the constitutive relation
271 gives that
272 $$
273 \eqalign{
274 P^{(1)}_{\mu}({\bf r},t+t_0)
275 &=\varepsilon_0\int^{\infty}_{-\infty}
276 T^{(1)}_{\mu\alpha}(t+t_0;\tau) E_{\alpha}({\bf r},\tau)\,d\tau\cr
277 &=\big\{{\rm Should\ equal\ to\ polarization\ induced\ by}
278 \ E_{\alpha}({\bf r},\tau+t_0)\big\}\cr
279 &=\varepsilon_0\int^{\infty}_{-\infty}
280 T^{(1)}_{\mu\alpha}(t;\tau) E_{\alpha}({\bf r},\tau+t_0)\,d\tau\cr
281 &=\big\{\tau'=\tau+t_0\big\}\cr
282 &=\varepsilon_0\int^{\infty}_{-\infty}
283 T^{(1)}_{\mu\alpha}(t;\tau'-t_0) E_{\alpha}({\bf r},\tau')\,d\tau',\cr
284 }
285 $$
286 from which we, by changing the ``dummy'' variable of integration
287 back to $\tau$, obtain the relation
288 $$
289 T^{(1)}_{\mu\alpha}(t+t_0;\tau)=T^{(1)}_{\mu\alpha}(t;\tau-t_0).
290 $$
291 In particular, by setting $t=0$ and replacing the arbitrary time
292 displacement $t_0$ by $t$, one finds that the response of the
293 medium depends only of the time difference $\tau-t$,
294 $$
295 \eqalign{
296 T^{(1)}_{\mu\alpha}(t;\tau)
297 &=T^{(1)}_{\mu\alpha}(0;\tau-t)\cr
298 &=R^{(1)}_{\mu\alpha}(\tau-t)\cr
299 }
300 $$
301 where we defined the linear polarization response function
302 $R^{(1)}_{\mu\alpha}(\tau-t)$, being a rank-two tensor depending
303 only on the time difference $\tau-t$.
304
305 To summarize, the linear contribution to the electric polarization
306 density is given in terms of the linear polarization response
307 function as
308 $$
309 \eqalign{
310 P^{(1)}_{\mu}({\bf r},t)
311 &=\varepsilon_0\int^{\infty}_{-\infty}
312 R^{(1)}_{\mu\alpha}(t-\tau) E_{\alpha}({\bf r},\tau)\,d\tau,\cr
313 &=\varepsilon_0\int^{\infty}_{-\infty}
314 R^{(1)}_{\mu\alpha}(\tau') E_{\alpha}({\bf r},t-\tau')\,d\tau',\cr
315 }\eqno{(8)}
316 $$
317 and the causality condition for the linear polarization
318 response function requires that
319 $$
320 R^{(1)}_{\mu\alpha}(\tau-t)=0,\qquad t\le\tau,
321 $$
322 and in addition, since the relation~(8) is to hold for arbitrary
323 time evolution of the electrical field, and since the polarization
324 density $P^{(1)}_{\mu}({\bf r},t)$ and the electric field
325 $E^{(1)}_{\alpha}({\bf r},t)$ both are real-valued quantities,
326 we also require the polarization response function
327 $R^{(1)}_{\mu\alpha}(\tau-t)$ to be a real-valued function.
328
329 \section{Quadratic polarization response function}
330 The second-order, quadratic polarization density can in similar to the
331 linear one be phenomenologically be written as the sum of all
332 infinitesimal previous contributions in time.
333 In this case, we must though include the possibility that not all
334 contributions origin in the same time scale, and hence the proper
335 formulation of the second order polarization density is as a
336 two-dimensional integral,
337 $$
338 P^{(2)}_{\mu}({\bf r},t)=\varepsilon_0
339 \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
340 T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
341 E_{\alpha}({\bf r},\tau_1) E_{\beta}({\bf r},\tau_2)
342 \,d\tau_1\,d\tau_2.
343 \eqno{(9)}
344 $$
345 The tensor $T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)$ uniquely
346 determines the quadratic, second-order polarization of the medium.
347 However, the tensor $T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)$
348 is not itself unique. To see this, we may express the tensor
349 as a sum of a symmetric part and an antisymmetric part,
350 $$
351 \eqalign{
352 T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
353 &={{1}\over{2}}
354 \underbrace{
355 \bigg[T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
356 +T^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)\bigg]
357 }_{\rm symmetric}
358 +{{1}\over{2}}
359 \underbrace{
360 \bigg[T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
361 -T^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)\bigg]
362 }_{\rm antisymmetric}\cr
363 &=\underbrace{S^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)}_{
364 =S^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)}
365 +\underbrace{A^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)}_{
366 =-A^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)}\cr
367 }
368 $$
369 As this form of the response function is inserted into the
370 original expression (9) for the second order polarization density,
371 we immediately find that it is left invariant under the interchange
372 of the dummy variables $(\alpha,\tau_1)$ and $(\beta,\tau_2)$
373 (since the integration is performed from minus to plus infinity
374 in time). In particular, it from this follows that
375
376
377 $$
378 \eqalign{
379 P^{(2)}_{\mu}({\bf r},t)&=\varepsilon_0
380 \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
381 [S^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
382 +A^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)]
383 E_{\alpha}({\bf r},\tau_1) E_{\beta}({\bf r},\tau_2)
384 \,d\tau_1\,d\tau_2\cr
385 &=\varepsilon_0
386 \underbrace{\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
387 S^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
388 E_{\alpha}({\bf r},\tau_1) E_{\beta}({\bf r},\tau_2)
389 \,d\tau_1\,d\tau_2}_{\equiv I_1}\cr&\qquad
390 +\varepsilon_0
391 \underbrace{\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
392 A^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
393 E_{\alpha}({\bf r},\tau_1) E_{\beta}({\bf r},\tau_2)
394 \,d\tau_1\,d\tau_2}_{\equiv I_2}\cr
395 &=\varepsilon_0
396 \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
397 S^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)
398 E_{\alpha}({\bf r},\tau_1) E_{\beta}({\bf r},\tau_2)
399 \,d\tau_1\,d\tau_2\cr&\qquad
400 -\varepsilon_0
401 \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
402 A^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)
403 E_{\alpha}({\bf r},\tau_1) E_{\beta}({\bf r},\tau_2)
404 \,d\tau_1\,d\tau_2\cr
405 &=\varepsilon_0
406 \underbrace{\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
407 S^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)
408 E_{\beta}({\bf r},\tau_2) E_{\alpha}({\bf r},\tau_1)
409 \,d\tau_1\,d\tau_2}_{=I_1}\cr&\qquad
410 -\varepsilon_0
411 \underbrace{\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
412 A^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)
413 E_{\beta}({\bf r},\tau_2) E_{\alpha}({\bf r},\tau_1)
414 \,d\tau_1\,d\tau_2}_{=I_2}\cr
415 }
416 $$
417 i.~e.~the symmetric part satisfy the trivial identity $I_1=I_1$,
418 while the antisymmetric part satisfy $I_2=-I_2$, i.~e.~the antisymmetric
419 part does not contribute to the polarization density, and that we
420 may set the antiymmetric part to zero, without imposing any
421 constraint on the validity of the theory.
422
423 The time response $T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)$
424 is then unique and now chosen to be symmetric,
425 $$
426 T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
427 =T^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1).
428 $$
429 This procedure of symmetrization may seem like an all theoretical
430 contruction, somewhat out of focus of what lies ahead, but in fact
431 it turns out to be an extremely useful property that we later
432 on will exploit extensively.
433 The previously described symmetric property will later on, for
434 susceptibility tensors in the frequency domain, be denoted as
435 the {\sl intrinsic permutation symmetry}, a general property
436 that will hold irregardless of whether the nonlinear interaction
437 under analysis is highly resonant or far from resonance.
438
439 By again applying the arguments of time invariance, as previously
440 for the linear response function, we find that
441 $$
442 T^{(2)}_{\mu\alpha\beta}(t+t_0;\tau_1,\tau_2)
443 =T^{(2)}_{\mu\alpha\beta}(t;\tau_1-t_0,\tau_2-t_0)
444 $$
445 for all $t$, $\tau_1$, and $\tau_2$.
446 Hence, by setting $t=0$ and then replacing th arbitrary time $t_0$
447 by $t$, again as previously done for the linear case, one finds
448 that $T^{(2)}(t;\tau_1,\tau_2)$ depends only on the two time
449 differences $t-\tau_1$ and $t-\tau_2$.
450 To make this fact explicit, we may hence write the response function as
451 $$
452 T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
453 =R^{(2)}_{\mu\alpha\beta}(t-\tau_1,t-\tau_2),
454 $$
455 giving the canonical form of the quadratic polarization density as
456 $$
457 \eqalign{
458 P^{(2)}_{\mu}({\bf r},t)
459 &=\varepsilon_0
460 \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
461 R^{(2)}_{\mu\alpha\beta}(t-\tau_1,t-\tau_2)
462 E_{\alpha}({\bf r},\tau_1) E_{\beta}({\bf r},\tau_2)
463 \,d\tau_1\,d\tau_2\cr
464 &=\varepsilon_0
465 \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
466 R^{(2)}_{\mu\alpha\beta}(\tau'_1,\tau'_2)
467 E_{\alpha}({\bf r},t-\tau'_1) E_{\beta}({\bf r},t-\tau'_2)
468 \,d\tau'_1\,d\tau'_2.\cr
469 }\eqno{(10)}
470 $$
471 The tensor $R^{(2)}_{\mu\alpha\beta}(\tau_1,\tau_2)$ is called
472 the quadratic electric polarization response of the medium,
473 and in similar with the linear response function, arguments
474 of causality require the response function to be zero whenever
475 $\tau_1$ and/or $\tau_2$ is negative. Similarly, the reality
476 condition on $E_{\alpha}({\bf r},t)$ and $P_{\alpha}({\bf r},t)$
477 requires that $R^{(2)}_{\mu\alpha\beta}(\tau_1,\tau_2)$ is a
478 real-valued function.
479
480 \section{Higher order polarization response functions}
481 The $n$th order polarization density can in similar to the linear ($n=1$
482 and quadratic ($n=2$) ones be written as
483 $$
484 \eqalign{
485 P^{(n)}_{\mu}({\bf r},t)
486 &=\varepsilon_0
487 \int^{\infty}_{-\infty}\cdots\int^{\infty}_{-\infty}
488 R^{(n)}_{\mu\alpha_1\cdots\alpha_n}
489 (t-\tau_1,\ldots,t-\tau_n)
490 E_{\alpha_1}({\bf r},\tau_1)\cdots E_{\alpha_n}({\bf r},\tau_n)
491 \,d\tau_1\,\cdots\,d\tau_n\cr
492 &=\varepsilon_0
493 \int^{\infty}_{-\infty}\cdots\int^{\infty}_{-\infty}
494 R^{(n)}_{\mu\alpha_1\cdots\alpha_n}
495 (\tau'_1,\ldots,\tau'_n)
496 E_{\alpha_1}({\bf r},t-\tau'_1)\cdots E_{\alpha_n}({\bf r},t-\tau'_n)
497 \,d\tau'_1\,\cdots\,d\tau'_n,\cr
498 }\eqno{(11)}
499 $$
500 where the $n$th order response function is a tensor of rank $n+1$,
501 and a real-valued function of the $n$ parameters $\tau_1,\ldots,\tau_n$,
502 vanishing whenever any $\tau_k<0$, $k=1,2,\ldots,n$, and with the
503 {\sl intrinsic permutation symmetry} that it is left invariant under
504 any of the $n!$ pairwise permutations
505 of $(\alpha_1,\tau_1),\ldots,(\alpha_n,\tau_n)$.
506 \bye
507
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