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    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
   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{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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