Contents of file 'lect5/lect5.tex':




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1   % File: nlopt/lect5/lect5.tex [pure TeX code]
2   % Last change: February 2, 2003
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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
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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{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