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    1   % File: nlopt/bcconven/bcconven.tex [pure TeX code]
    2   % Last change: January 26, 2003
    3   %
    4   % Notes on the ``Butcher and Cotter convention'' in nonlinear optics,
    5   % and an example of its application to the formulation of the polarization
    6   % density in the case of optical Kerr-effect.  This memo is part of the
    7   % material handed out in the course ``Nonlinear optics'', held January-March,
    8   % 2003, at the Royal Institute of Technology, Stockholm, Sweden.
    9   %
   10   % This memo was in the 2003 course handed out during the third lecture.
   11   %
   12   % Copyright (C) 2003, Fredrik Jonsson
   13   %
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   25   \font\ninerm=cmr9
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   27   \def\Re{\mathop{\rm Re}\nolimits} % real part
   28   \def\Im{\mathop{\rm Im}\nolimits} % imaginary part
   29   \def\Tr{\mathop{\rm Tr}\nolimits} % quantum mechanical trace
   30   %
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   55   \def\section #1 {\medskip\goodbreak\noindent{\bf #1}
   56     \par\nobreak\smallskip\noindent}
   57   \def\subsection #1 {\medskip\goodbreak\noindent{\sl #1}
   58     \par\nobreak\smallskip\noindent}
   59   \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
   60     \else\hfill\fi}
   61   \def\rightheadline{\hfil{\it Nonlinear Optics 5A5513 (2003)}}
   62   \def\leftheadline{{\it Nonlinear Optics 5A5513 (2003)}\hfil}
   63   \voffset=2\baselineskip
   64   \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps}
   65   \vskip-18pt\hfill{\it Nonlinear Optics 5A5513 (2003)}
   66   \vskip 36pt
   67   \centerline{\twelvebf Notes on the ``Butcher and Cotter convention'' in
   68     nonlinear optics}
   69   \bigskip
   70   
   71   \section{Convention for description of nonlinear optical polarization}
   72   As a ``recipe'' in theoretical nonlinear optics, Butcher and Cotter
   73   provide a very useful convention which is well worth to hold on to.
   74   For a superposition of monochromatic waves, and by invoking the general
   75   property of the intrinsic permutation symmetry, the monochromatic
   76   form of the $n$th order polarization density can be written as
   77   $$
   78     (P^{(n)}_{\omega_{\sigma}})_{\mu}
   79       =\varepsilon_0\sum_{\alpha_1}\cdots\sum_{\alpha_n}\sum_{\omega}
   80        K(-\omega_{\sigma};\omega_1,\ldots,\omega_n)
   81        \chi^{(n)}_{\mu\alpha_1\cdots\alpha_n}
   82        (-\omega_{\sigma};\omega_1,\ldots,\omega_n)
   83        (E_{\omega_1})_{\alpha_1}\cdots(E_{\omega_n})_{\alpha_n}.
   84     \eqno{(1)}
   85   $$
   86   The first summations in Eq.~(1), over $\alpha_1,\ldots,\alpha_n$,
   87   is simply an explicit way of stating that the Einstein convention
   88   of summation over repeated indices holds.
   89   The summation sign $\sum_{\omega}$, however, serves as a reminder
   90   that the expression that follows is to be summed over
   91   {\sl all distinct sets of $\omega_1,\ldots,\omega_n$}.
   92   Because of the intrinsic permutation symmetry, the frequency arguments
   93   appearing in Eq.~(1) may be written in arbitrary order.
   94   
   95   By ``all distinct sets of $\omega_1,\ldots,\omega_n$'', we here mean
   96   that the summation is to be performed, as for example in the case of
   97   optical Kerr-effect, over the single set of nonlinear susceptibilities
   98   that contribute to a certain angular frequency as
   99   $(-\omega;\omega,\omega,-\omega)$ {\sl or} $(-\omega;\omega,-\omega,\omega)$
  100   {\sl or} $(-\omega;-\omega,\omega,\omega)$.
  101   In this example, each of the combinations are considered as {\sl distinct},
  102   and it is left as an arbitary choice which one of these sets that are
  103   most convenient to use (this is simply a matter of choosing notation,
  104   and does not by any means change the description of the interaction).
  105   
  106   In Eq.~(1), the degeneracy factor $K$ is formally described as
  107   $$K(-\omega_{\sigma};\omega_1,\ldots,\omega_n)=2^{l+m-n}p$$
  108   where
  109   $$
  110     \eqalign{
  111       p&={\rm the\ number\ of\ {\sl distinct}\ permutations\ of}
  112          \ \omega_1,\omega_2,\ldots,\omega_1,\cr
  113       n&={\rm the\ order\ of\ the\ nonlinearity},\cr
  114       m&={\rm the\ number\ of\ angular\ frequencies}\ \omega_k
  115          \ {\rm that\ are\ zero,\ and}\cr
  116       l&=\bigg\lbrace\matrix{1,\qquad{\rm if}\ \omega_{\sigma}\ne 0,\cr
  117                              0,\qquad{\rm otherwise}.}\cr
  118     }
  119   $$
  120   In other words, $m$ is the number of DC electric fields present,
  121   and $l=0$ if the nonlinearity we are analyzing gives a static, DC,
  122   polarization density, such as in the previously (in the spring model)
  123   described case of optical rectification in the presence of second
  124   harmonic fields (SHG).
  125   
  126   A list of frequently encountered nonlinear phenomena in nonlinear
  127   optics, including the degeneracy factors as conforming to the above
  128   convention, is given in Butcher and Cotters book, Table 2.1, on page 26.
  129   
  130   \section{Note on the complex representation of the optical field}
  131   Since the observable electric field of the light, in Butcher and
  132   Cotters notation taken as
  133   $$
  134     {\bf E}({\bf r},t)={{1}\over{2}}\sum_{\omega_k\ge 0}
  135     [{\bf E}_{\omega_k}\exp(-i\omega_k t)+{\bf E}^*_{\omega_k}\exp(i\omega_k t)],
  136   $$
  137   is a real-valued quantity, it follows that negative frequencies in the
  138   complex notation should be interpreted as the complex conjugate of the
  139   respective field component, or
  140   $$
  141     {\bf E}_{-\omega_k}={\bf E}^*_{\omega_k}.
  142   $$
  143   
  144   \section{Example: Optical Kerr-effect}
  145   Assume a monochromatic optical wave (containing forward and/or backward
  146   propagating components) polarized in the $xy$-plane,
  147   $$
  148     {\bf E}(z,t)=\Re[{\bf E}_{\omega}(z)\exp(-i\omega t)]\in{\Bbb R}^3,
  149   $$
  150   with all spatial variation of the field contained in
  151   $$
  152     {\bf E}_{\omega}(z)={\bf e}_x E^x_{\omega}(z)
  153       +{\bf e}_y E^y_{\omega}(z)\in{\Bbb C}^3.
  154   $$
  155   Optical Kerr-effect is in isotropic media described by the third order
  156   susceptibility
  157   $$
  158     \chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega),
  159   $$
  160   with nonzero components of interest for the $xy$-polarized beam given in
  161   Appendix 3.3 of Butcher and Cotters book as
  162   $$
  163     \chi^{(3)}_{xxxx}=\chi^{(3)}_{yyyy},\quad
  164     \chi^{(3)}_{xxyy}=\chi^{(3)}_{yyxx}
  165     =\bigg\{\matrix{{\rm intr.\ perm.\ symm.}\cr
  166                (\alpha,\omega)\rightleftharpoons(\beta,\omega)\cr}\bigg\}=
  167     \chi^{(3)}_{xyxy}=\chi^{(3)}_{yxyx},\quad
  168     \chi^{(3)}_{xyyx}=\chi^{(3)}_{yxxy},
  169   $$
  170   with
  171   $$
  172     \chi^{(3)}_{xxxx}=\chi^{(3)}_{xxyy}+\chi^{(3)}_{xyxy}+\chi^{(3)}_{xyyx}.
  173   $$
  174   The degeneracy factor $K(-\omega;\omega,\omega,-\omega)$ is calculated as
  175   $$
  176     K(-\omega;\omega,\omega,-\omega)=2^{l+m-n}p=2^{1+0-3}3=3/4.
  177   $$
  178   From this set of nonzero susceptibilities, and using the calculated
  179   value of the degeneracy factor in the convention of Butcher and Cotter,
  180   we hence have the third order electric polarization density at
  181   $\omega_{\sigma}=\omega$ given as ${\bf P}^{(n)}({\bf r},t)=
  182   \Re[{\bf P}^{(n)}_{\omega}\exp(-i\omega t)]$, with
  183   $$
  184     \eqalign{
  185       {\bf P}^{(3)}_{\omega}
  186       &=\sum_{\mu}{\bf e}_{\mu}(P^{(3)}_{\omega})_{\mu}\cr
  187       &=\{{\rm Using\ the\ convention\ of\ Butcher\ and\ Cotter}\}\cr
  188       &=\sum_{\mu}{\bf e}_{\mu}
  189         \bigg[\varepsilon_0{{3}\over{4}}\sum_{\alpha}\sum_{\beta}\sum_{\gamma}
  190          \chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega)
  191          (E_{\omega})_{\alpha}(E_{\omega})_{\beta}(E_{-\omega})_{\gamma}\bigg]\cr
  192       &=\{{\rm Evaluate\ the\ sums\ over\ } (x,y,z)
  193           {\rm\ for\ field\ polarized\ in\ the\ }xy{\rm\ plane}\}\cr
  194       &=\varepsilon_0{{3}\over{4}}\{
  195         {\bf e}_x[
  196           \chi^{(3)}_{xxxx} E^x_{\omega} E^x_{\omega} E^x_{-\omega}
  197           +\chi^{(3)}_{xyyx} E^y_{\omega} E^y_{\omega} E^x_{-\omega}
  198           +\chi^{(3)}_{xyxy} E^y_{\omega} E^x_{\omega} E^y_{-\omega}
  199           +\chi^{(3)}_{xxyy} E^x_{\omega} E^y_{\omega} E^y_{-\omega}]\cr
  200        &\qquad\quad
  201        +{\bf e}_y[
  202           \chi^{(3)}_{yyyy} E^y_{\omega} E^y_{\omega} E^y_{-\omega}
  203           +\chi^{(3)}_{yxxy} E^x_{\omega} E^x_{\omega} E^y_{-\omega}
  204           +\chi^{(3)}_{yxyx} E^x_{\omega} E^y_{\omega} E^x_{-\omega}
  205           +\chi^{(3)}_{yyxx} E^y_{\omega} E^x_{\omega} E^x_{-\omega}]\}\cr
  206       &=\{{\rm Make\ use\ of\ }{\bf E}_{-\omega}={\bf E}^*_{\omega}
  207           {\rm\ and\ relations\ }\chi^{(3)}_{xxyy}=\chi^{(3)}_{yyxx},
  208           {\rm\ etc.}\}\cr
  209       &=\varepsilon_0{{3}\over{4}}\{
  210         {\bf e}_x[
  211           \chi^{(3)}_{xxxx} E^x_{\omega} |E^x_{\omega}|^2
  212           +\chi^{(3)}_{xyyx} E^y_{\omega}{}^2 E^{x*}_{\omega}
  213           +\chi^{(3)}_{xyxy} |E^y_{\omega}|^2 E^x_{\omega}
  214           +\chi^{(3)}_{xxyy} E^x_{\omega} |E^y_{\omega}|^2]\cr
  215        &\qquad\quad
  216        +{\bf e}_y[
  217           \chi^{(3)}_{xxxx} E^y_{\omega} |E^y_{\omega}|^2
  218           +\chi^{(3)}_{xyyx} E^x_{\omega}{}^2 E^{y*}_{\omega}
  219           +\chi^{(3)}_{xyxy} |E^x_{\omega}|^2 E^y_{\omega}
  220           +\chi^{(3)}_{xxyy} E^y_{\omega} |E^x_{\omega}|^2]\}\cr
  221       &=\{{\rm Make\ use\ of\ intrinsic\ permutation\ symmetry}\}\cr
  222       &=\varepsilon_0{{3}\over{4}}\{
  223         {\bf e}_x[
  224           (\chi^{(3)}_{xxxx} |E^x_{\omega}|^2
  225             +2\chi^{(3)}_{xxyy} |E^y_{\omega}|^2) E^x_{\omega}
  226           +(\chi^{(3)}_{xxxx}-2\chi^{(3)}_{xxyy})
  227            E^y_{\omega}{}^2 E^{x*}_{\omega}\cr
  228        &\qquad\quad
  229         {\bf e}_y[
  230           (\chi^{(3)}_{xxxx} |E^y_{\omega}|^2
  231             +2\chi^{(3)}_{xxyy} |E^x_{\omega}|^2) E^y_{\omega}
  232           +(\chi^{(3)}_{xxxx}-2\chi^{(3)}_{xxyy})
  233            E^x_{\omega}{}^2 E^{y*}_{\omega}.\cr
  234     }
  235   $$
  236   For the optical field being linearly polarized, say in the $x$-direction,
  237   the expression for the polarization density is significantly simplified,
  238   to yield
  239   $$
  240     {\bf P}^{(3)}_{\omega}=\varepsilon_0(3/4){\bf e}_x
  241       \chi^{(3)}_{xxxx} |E^x_{\omega}|^2 E^x_{\omega},
  242   $$
  243   i.~e.~taking a form that can be interpreted as an intensity-dependent
  244   ($\sim|E^x_{\omega}|^2$) contribution to the refractive index
  245   (cf.~Butcher and Cotter \S 6.3.1).
  246   \bye
  247   

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