Contents of file 'lect5/lect5.tex':
1 % File: nlopt/lect5/lect5.tex [pure TeX code]
2 % Last change: February 2, 2003
3 %
4 % Lecture No 5 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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11 % Read amssym to get the AMS {\Bbb E} font (strikethrough E) and
12 % the Euler fraktur font.
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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{5}
42 In the previous lecture, the quantum mechanical origin of the linear and
43 nonlinear susceptibilities was discussed. In particular, a perturbation
44 analysis of the density operator was performed, and the resulting system
45 of equations was solved recursively for the $n$th order density operator
46 $\hat{\rho}_n(t)$ in terms of $\hat{\rho}_{n-1}(t)$, where the zeroth order
47 term (independent of the applied electric field of the light) is given by
48 the Boltzmann distribution at thermal equilibrium.
49
50 So far we have obtained the linear optical properties of the medium, in
51 terms of the first order susceptibility tensor (of rank-two), and we will
52 now proceed with the next order of interaction, giving the second order
53 electric susceptibility tensor (of rank three).
54
55 \section{The second order polarization density}
56 For the second order interaction, the corresponding term in the perturbation
57 series of the density operator {\sl in the interaction picture}
58 becomes\footnote{${}^1$}{It should be noticed that the form given in Eq.~(1)
59 not only applies to an ensemble of molecules, of arbitrary composition, but
60 also to {\sl any} kind of level of approximation for the interaction, such
61 as the inclusion of magnetic dipolar interactions or electric quadrupolar
62 interactions as well. These interactions should (of course) be incorporated
63 in the expression for the interaction Hamiltonian $\hat{H}'_{\rm I}(\tau)$,
64 here described in the interaction picture.}
65 $$
66 \eqalign{
67 \hat{\rho}'_2(t)&={{1}\over{i\hbar}}\int^t_{-\infty}
68 [\hat{H}'_{\rm I}(\tau_1),\hat{\rho}'_{1}(\tau_1)]\,d\tau_1\cr
69 &={{1}\over{i\hbar}}\int^t_{-\infty}
70 [\hat{H}'_{\rm I}(\tau_1),{{1}\over{i\hbar}}\int^{\tau_1}_{-\infty}
71 [\hat{H}'_{\rm I}(\tau_2),\hat{\rho}_0]\,d\tau_2\,]\,d\tau_1\cr
72 &={{1}\over{(i\hbar)^2}}\int^t_{-\infty}\int^{\tau_1}_{-\infty}
73 [\hat{H}'_{\rm I}(\tau_1),
74 [\hat{H}'_{\rm I}(\tau_2),\hat{\rho}_0]]\,d\tau_2\,d\tau_1\cr
75 }\eqno{(1)}
76 $$
77 In order to simplify the expression for the second order susceptibility,
78 we will in the following analysis make use of a generalization of the
79 cyclic perturbation of the terms in the commutator inside the trace, as
80 $$
81 \Tr\{[\hat{Q}_{\alpha}(\tau_1),[\hat{Q}_{\beta}(\tau_2),\hat{\rho}_0]]
82 \hat{Q}_{\mu}\}
83 =\Tr\{\hat{\rho}_0[[\hat{Q}_{\mu},\hat{Q}_{\alpha}(\tau_1)],
84 \hat{Q}_{\beta}(\tau_2)]\}.\eqno{(2)}
85 $$
86 By inserting the expression for the second order term of the
87 perturbation series for the density operator into the quantum mechanical
88 trace of the second order electric polarization density of the medium,
89 one obtains
90 $$
91 \eqalign{
92 P^{(2)}_{\mu}({\bf r},t)&={{1}\over{V}}\Tr[\hat{\rho}_2(t)\hat{Q}_{\mu}]\cr
93 &={{1}\over{V}}\Tr\Big[
94 \underbrace{\Big(\hat{U}_0(t)
95 \underbrace{
96 {{1}\over{(i\hbar)^2}}\int^t_{-\infty}\int^{\tau_1}_{-\infty}
97 [\hat{H}'_{\rm I}(\tau_1),
98 [\hat{H}'_{\rm I}(\tau_2),\hat{\rho}_0]]\,d\tau_2\,d\tau_1
99 }_{=\hat{\rho}'_2(t){\quad\hbox{\ninerm(interaction picture)}}}
100 \hat{U}_0(-t)\Big)
101 }_{=\hat{\rho}_2(t){\quad\hbox{\ninerm(Schr\"{o}dinger picture)}}}
102 \hat{Q}_{\mu}\Big]\cr
103 &=\{E_{\alpha}(\tau_1){\rm\ and\ }E_{\alpha}(\tau_1)
104 {\rm\ are\ classical\ fields\ (omit\ space\ dependence\ {\bf r})}\}\cr
105 &={{1}\over{V(i\hbar)^2}}\Tr\Big\{
106 \hat{U}_0(t)\int^t_{-\infty}\int^{\tau_1}_{-\infty}
107 [\hat{Q}_{\alpha}(\tau_1),
108 [\hat{Q}_{\beta}(\tau_2),\hat{\rho}_0]]
109 E_{\alpha}(\tau_1) E_{\beta}(\tau_2)\,d\tau_2\,d\tau_1
110 \hat{U}_0(-t)\,\hat{Q}_{\mu}\Big\}\cr
111 &=\{{\rm Pull\ out\ }E_{\alpha_1}(\tau_1) E_{\alpha_2}(\tau_2)
112 {\rm\ and\ the\ integrals\ outside\ the\ trace}\}\cr
113 }
114 $$
115 $$
116 \eqalign{
117 &={{1}\over{V(i\hbar)^2}}\int^t_{-\infty}\int^{\tau_1}_{-\infty}
118 \Tr\Big\{\hat{U}_0(t)
119 [\hat{Q}_{\alpha}(\tau_1),[\hat{Q}_{\beta}(\tau_2),\hat{\rho}_0]]
120 \hat{U}_0(-t)\,\hat{Q}_{\mu}\Big\}
121 E_{\alpha}(\tau_1) E_{\beta}(\tau_2)\,d\tau_2\,d\tau_1.\cr
122 }
123 $$
124 In analogy with the results as obtained for the first order (linear) optical
125 properties, now express the term $E_{\alpha_1}(\tau_1)E_{\alpha_2}(\tau_2)$
126 in the frequency domain, by using the Fourier identity
127 $$
128 E_{\alpha_k}(\tau_k)=\int^{\infty}_{-\infty}
129 E_{\alpha_k}(\omega_k)\exp(-i\omega\tau_k)\,d\omega,
130 $$
131 which hence gives the second order polarization density expressed in terms
132 of the electric field in the frequency domain as
133 $$
134 \eqalign{
135 P^{(2)}_{\mu}({\bf r},t)
136 &={{1}\over{V(i\hbar)^2}}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
137 \int^t_{-\infty}\int^{\tau_1}_{-\infty}
138 \Tr\Big\{\hat{U}_0(t)
139 [\hat{Q}_{\alpha}(\tau_1),[\hat{Q}_{\beta}(\tau_2),\hat{\rho}_0]]
140 \hat{U}_0(-t)\,\hat{Q}_{\mu}\Big\}
141 E_{\alpha}(\omega_1) E_{\beta}(\omega_2)
142 \cr&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times
143 \exp(-i\omega_1\tau_1)\exp(-i\omega_2\tau_2)
144 \,d\tau_2\,d\tau_1\,d\omega_2\,d\omega_1\cr
145 &=\{{\rm Use\ }\exp(-i\omega\tau)=\exp(-i\omega t)
146 \exp[-i\omega(\tau-t)]\}\cr
147 &={{1}\over{V(i\hbar)^2}}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
148 \int^t_{-\infty}\int^{\tau_1}_{-\infty}
149 \Tr\Big\{\hat{U}_0(t)
150 [\hat{Q}_{\alpha}(\tau_1),[\hat{Q}_{\beta}(\tau_2),\hat{\rho}_0]]
151 \hat{U}_0(-t)\,\hat{Q}_{\mu}\Big\}
152 E_{\alpha}(\omega_1) E_{\beta}(\omega_2)
153 \cr&\qquad\qquad\qquad\qquad\times
154 \exp[-i\omega_1(\tau_1-t)-i\omega_2(\tau_2-t)]
155 \,d\tau_2\,d\tau_1\,\exp[-i\underbrace{(\omega_1+\omega_2)}_{
156 =\omega_{\sigma}}t]
157 \,d\omega_2\,d\omega_1\cr
158 &=\varepsilon_0\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
159 \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega_1,\omega_2)
160 E_{\alpha}(\omega_1) E_{\beta}(\omega_2)
161 \exp(-i\omega_{\sigma} t)\,d\omega_2\,d\omega_1,\cr
162 }
163 $$
164 where the second order (quadratic) electric susceptibility is defined as
165 $$
166 \eqalign{
167 \chi^{(2)}_{\mu\alpha\beta}&(-\omega_{\sigma};\omega_1,\omega_2)\cr
168 &={{1}\over{\varepsilon_0 V(i\hbar)^2}}
169 \int^t_{-\infty}\int^{\tau_1}_{-\infty}
170 \Tr\Big\{\hat{U}_0(t)
171 [\hat{Q}_{\alpha}(\tau_1),[\hat{Q}_{\beta}(\tau_2),\hat{\rho}_0]]
172 \hat{U}_0(-t)\,\hat{Q}_{\mu}\Big\}
173 \cr&\qquad\qquad\qquad\qquad\times
174 \exp[-i\omega_1(\tau_1-t)-i\omega_2(\tau_2-t)]
175 \,d\tau_2\,d\tau_1\cr
176 &=\{{\rm Make\ use\ of\ Eq.\ (2)\ and\ take\ }\tau'_1=\tau_1-t\}\cr
177 &=\ldots\cr
178 &={{1}\over{\varepsilon_0 V(i\hbar)^2}}
179 \int^0_{-\infty}\int^{\tau'_1}_{-\infty}
180 \Tr\{\hat{\rho}_0[[\hat{Q}_{\mu},\hat{Q}_{\alpha}(\tau'_1)],
181 \hat{Q}_{\beta}(\tau'_2)]\}
182 \exp[-i(\omega_1\tau'_1+\omega_2\tau'_2)]
183 \,d\tau'_2\,d\tau'_1.\cr
184 }
185 $$
186 This obtained expression for the second order electric susceptibility does
187 not possess the property of intrinsic permutation symmetry. However, by
188 using the same arguments as discussed in the analysis of the polarization
189 response functions in lecture two, we can easily verify that this tensor
190 can be cast into a symmetric and antisymmetric part as
191 $$
192 \eqalign{
193 \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega_1,\omega_2)
194 &={{1}\over{2}}\underbrace{[
195 \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega_1,\omega_2)
196 +\chi^{(2)}_{\mu\beta\alpha}(-\omega_{\sigma};\omega_2,\omega_1)]}_{
197 {\rm symmetric\ part}}
198 \cr&\qquad\qquad
199 +{{1}\over{2}}\underbrace{[
200 \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega_1,\omega_2)
201 -\chi^{(2)}_{\mu\beta\alpha}(-\omega_{\sigma};\omega_2,\omega_1)]}_{
202 {\rm antisymmetric\ part}},\cr
203 }
204 $$
205 and since the antisymmetric part, again following the arguments for the
206 second order polarization response function, does not contribute to the
207 polarization density, it is customary (in the Butcher and Cotter convention
208 as well as all other conventions in nonlinear optics) to cast the
209 second order susceptibility into the form
210 $$
211 \eqalign{
212 \chi^{(2)}_{\mu\alpha\beta}&(-\omega_{\sigma};\omega_1,\omega_2)\cr
213 &={{1}\over{\varepsilon_0 V(i\hbar)^2}}{{1}\over{2!}}{\bf S}
214 \int^0_{-\infty}\int^{\tau'_1}_{-\infty}
215 \Tr\{\hat{\rho}_0[[\hat{Q}_{\mu},\hat{Q}_{\alpha}(\tau'_1)],
216 \hat{Q}_{\beta}(\tau'_2)]\}
217 \exp[-i(\omega_1\tau'_1+\omega_2\tau'_2)]
218 \,d\tau'_2\,d\tau'_1,\cr
219 }
220 $$
221 where ${\bf S}$, commonly called the {\sl symmetrizing operator}, denotes
222 that the expression that follows is to be summed over the $2!=2$ possible
223 pairwise permutations of $(\alpha,\omega_1)$ and $(\beta,\omega_2)$,
224 hence ensuring that the second order susceptibility possesses the
225 intrinsic permutation symmetry,
226 $$
227 \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega_1,\omega_2)
228 =\chi^{(2)}_{\mu\beta\alpha}(-\omega_{\sigma};\omega_2,\omega_1).
229 $$
230
231 \section{Higher order polarization densities}
232 The previously described principle of deriving the susceptibilities
233 of first and second order are straightforward to extend to the $n$th
234 order interaction. In this case, we will make use of the following
235 generalization of Eq.~(2),
236 $$
237 \eqalign{
238 \Tr\{[&\hat{Q}_{\alpha_1}(\tau_1),[\hat{Q}_{\alpha_2}(\tau_2),\ldots,
239 [\hat{Q}_{\alpha_n}(\tau_n),\hat{\rho}_0]]\ldots]\hat{Q}_{\mu}\}\cr
240 &\qquad=\Tr\{\hat{\rho}_0[\ldots[[\hat{Q}_{\mu},\hat{Q}_{\alpha_1}(\tau_1)],
241 \hat{Q}_{\alpha_2}(\tau_2)],\ldots\hat{Q}_{\alpha_n}(\tau_n)]\},\cr
242 }
243 $$
244 which, when applied in the evaluation of the expectation value of the
245 electric dipole operator of the ensemble, gives the $n$th order electric
246 susceptibility as
247 $$
248 \eqalign{
249 \chi^{(n)}_{\mu\alpha_1\cdots\alpha_n}
250 &(-\omega_{\sigma};\omega_1,\ldots,\omega_n)\cr
251 &={{1}\over{\varepsilon_0 V (-i\hbar)^n}}{{1}\over{n!}}{\bf S}
252 \int^0_{-\infty}\int^{\tau_1}_{-\infty}\cdots\int^{\tau_{n-1}}_{-\infty}
253 \Tr\{\hat{\rho}_0[\ldots[[\hat{Q}_{\mu},\hat{Q}_{\alpha_1}(\tau_1)],
254 \hat{Q}_{\alpha_2}(\tau_2)],\ldots\hat{Q}_{\alpha_n}(\tau_n)]\}
255 \cr&\qquad\qquad\qquad\qquad\qquad\qquad\times
256 \exp[-i(\omega_1\tau_1+\omega_2\tau_2+\ldots+\omega_n\tau_n)]
257 \,d\tau_n\,\cdots\,d\tau_2\,d\tau_1,\cr
258 }
259 $$
260 where now the symmetrizing operator ${\bf S}$ indicates that the expression
261 following it should be summed over all the $n!$ pairwise permutations of
262 $(\alpha_1,\omega_1),\ldots,(\alpha_n,\omega_n)$.
263
264 It should be emphasized the symmetrizing operator ${\bf S}$ always
265 implies summation over {\sl all} the $n!$ pairwise permutations of
266 $(\alpha_1,\omega_1),\ldots,(\alpha_n,\omega_n)$, {\sl irregardless of
267 whether the permutations are distinct or not}. This is due to that eventually
268 occuring degenerate permutations are taken care of in the degeneracy
269 coefficient $K(-\omega_{\sigma};\omega_1,\ldots,\omega_n)$ in Butcher
270 and Cotters convention, as described in lecture three and in additional
271 notes that has been handed out in lecture four.
272 \bye
273
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