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    1   % File: nlopt/lect3/lect3.tex [pure TeX code]
    2   % Last change: January 19, 2003
    3   %
    4   % Lecture No 3 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{3}
   42   \section{Susceptibility tensors in the frequency domain}
   43   The susceptibility tensors in the frequency domain arise when
   44   the electric field $E_{\alpha}(t)$ of the light is expressed in terms
   45   of its Fourier transform $E_{\alpha}(\omega)$, by means of the
   46   Fourier integral identity
   47   $$
   48     E_{\alpha}(t)=\int^{\infty}_{-\infty}E_{\alpha}(\omega)
   49       \exp(-i\omega t)\,d\omega=\fourier^{-1}[E_{\alpha}](t),\eqno{(1')}
   50   $$
   51   with inverse relation
   52   $$
   53     E_{\alpha}(\omega)={{1}\over{2\pi}}\int^{\infty}_{-\infty}E_{\alpha}(\tau)
   54       \exp(i\omega\tau)\,d\tau=\fourier[E_{\alpha}](\omega).\eqno{(1'')}
   55   $$
   56   This convention of inclusion of the factor of $2\pi$, as well as the
   57   sign convention, is commonly used in quantum mechanics; however, it should
   58   be emphasized that this convention is not a commonly adopted standard
   59   in optics, neither in linear nor in nonlinear optical regimes.
   60   
   61   The sign convention here used leads to wave solutions of the form
   62   $f(kz-\omega t)$ for monochromatic waves propagating in the positive
   63   $z$-direction, which might be somewhat more intuitive than the alternative
   64   form $f(\omega t-kz)$, which is obtained if one instead apply the
   65   alternative sign convention.
   66   
   67   The convention for the inclusion of $2\pi$ in the Fourier transform
   68   in Eq.~($1''$) is here convenient for description of electromagnetic
   69   wave propagation in the frequency domain (going from the time domain
   70   description, in terms of the polarization response functions, to the
   71   frequency domain, in terms of the linear and nonlinear susceptibilities),
   72   since it enables us to omit any multiple of $2\pi$ of the Fourier
   73   transformed fields.
   74   
   75   \section{First order susceptibility tensor}
   76   By inserting Eq.~($1'$) is inserted into the previously
   77   obtained\footnote{${}^1$}{Expressions for the first order, second order,
   78   and $n$th order polarization densities were obtained in lecture two.}
   79   relation for the first order, linear polarization density, one obtains
   80   $$
   81     \eqalign{
   82       P^{(1)}_{\mu}({\bf r},t)
   83         &=\varepsilon_0\int^{\infty}_{-\infty}
   84           R^{(1)}_{\mu\alpha}(\tau) E_{\alpha}({\bf r},t-\tau)\,d\tau,\cr
   85         &=\{{\rm express}\ E_{\alpha}({\bf r},t-\tau)
   86           \ {\rm in\ frequency\ domain}\}\cr
   87         &=\varepsilon_0\int^{\infty}_{-\infty}
   88           R^{(1)}_{\mu\alpha}(\tau)\int^{\infty}_{-\infty}
   89           E_{\alpha}({\bf r},\omega)
   90           \exp[-i\omega(t-\tau)]\,d\omega\,d\tau,\cr
   91         &=\{{\rm change\ order\ of\ integration}\}\cr
   92         &=\varepsilon_0\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
   93           R^{(1)}_{\mu\alpha}(\tau) E_{\alpha}({\bf r},\omega)
   94           \exp(i\omega\tau)\,d\tau\,\exp(-i\omega t)\,d\omega,\cr
   95         &=\varepsilon_0\int^{\infty}_{-\infty}
   96             \chi^{(1)}_{\mu\alpha}(-\omega;\omega)
   97             E_{\alpha}({\bf r},\omega)
   98           \exp(-i\omega t)\,d\omega,\cr
   99       }\eqno{(2)}
  100   $$
  101   where the {\sl linear electric dipolar susceptibility},
  102   $$
  103     \chi^{(1)}_{\mu\alpha}(-\omega;\omega)
  104       =\int^{\infty}_{-\infty}R^{(1)}_{\mu\alpha}(\tau)
  105        \exp(i\omega\tau)\,d\tau
  106       =\fourier[R^{(1)}_{\mu\alpha}](\omega),\eqno{(3)}
  107   $$
  108   was introduced. In this expression for the susceptibility,
  109   $\omega_{\sigma}=\omega$, and the reasons for the somewhat
  110   peculiar notation of arguments of the susceptibility will
  111   be explained later on in the context of nonlinear susceptibilities.
  112   
  113   \section{Second order susceptibility tensor}
  114   In similar to the linear susceptibility tensor, by inserting Eq.~($1'$)
  115   into the previously obtained relation for the second order, quadratic
  116   polarization density, one obtains
  117   $$
  118     \eqalign{
  119       P^{(2)}_{\mu}({\bf r},t)
  120       &=\varepsilon_0
  121         \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  122         \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  123           R^{(2)}_{\mu\alpha\beta}(\tau_1,\tau_2)
  124           E_{\alpha}({\bf r},\omega_1) E_{\beta}({\bf r},\omega_2)
  125   \cr&\qquad\qquad\qquad\times
  126           \exp[-i(\omega_1(t-\tau_1)+\omega_2(t-\tau_2))]
  127         \,d\tau_1\,d\tau_2
  128         \,d\omega_1\,d\omega_2\cr
  129       &=\varepsilon_0
  130         \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  131           \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega_1,\omega_2)
  132           E_{\alpha}({\bf r},\omega_1) E_{\beta}({\bf r},\omega_2)
  133           \exp[-i\underbrace{(\omega_1+\omega_2)}_{\equiv\omega_{\sigma}}t]
  134           \,d\omega_1\,d\omega_2\cr
  135       }\eqno{(4)}
  136   $$
  137   where the {\sl quadratic electric dipolar susceptibility},
  138   $$
  139     \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega_1,\omega_2)
  140       =\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  141        R^{(2)}_{\mu\alpha\beta}(\tau_1,\tau_2)
  142        \exp[i(\omega_1\tau_1+\omega_2\tau_2)]\,d\tau_1\,d\tau_2,\eqno{(5)}
  143   $$
  144   was introduced. In this expression for the susceptibility,
  145   $\omega_{\sigma}=\omega_1+\omega_2$, and the reason for the notation
  146   of arguments should now be somewhat more clear: the first angular
  147   frequency argument of the susceptibility tensor is simply the sum
  148   of all driving angular frequencies of the optical field.
  149   
  150   %Let us now consider the effects of the previously discussed requirements
  151   %of reality and causality.
  152   
  153   The intrinsic permutation symmetry of
  154   $R^{(2)}_{\mu\alpha\beta}(\tau_1,\tau_2)$,
  155   the second order polarization response function, carries over to the
  156   second order susceptibility tensor as well, in the sense that
  157   $$
  158     \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega_1,\omega_2)
  159       =\chi^{(2)}_{\mu\beta\alpha}(-\omega_{\sigma};\omega_2,\omega_1),
  160   $$
  161   i.~e.~the second order susceptibility is invariant under any
  162   of the $2!=2$ pairwise permutations of $(\alpha,\omega_1)$
  163   and $(\beta,\omega_2)$.
  164   
  165   \section{Higher order susceptibility tensors}
  166   In similar to the linear and quadratic susceptibility tensors, by
  167   inserting Eq.~($1'$) into the previously obtained relation for
  168   the $n$th order polarization density, one obtains
  169   $$
  170     \eqalign{
  171       P^{(n)}_{\mu}({\bf r},t)
  172       &=\varepsilon_0
  173         \int^{\infty}_{-\infty}\cdots\int^{\infty}_{-\infty}
  174           \chi^{(n)}_{\mu\alpha_1\cdots\alpha_n}
  175           (-\omega_{\sigma};\omega_1,\ldots,\omega_n)
  176           E_{\alpha_1}({\bf r},\omega_1)\cdots
  177           E_{\alpha_n}({\bf r},\omega_n)
  178   \cr&\qquad\qquad\qquad\times
  179           \exp[-i\underbrace{(\omega_1+\ldots+\omega_n)}_{
  180             \equiv\omega_{\sigma}}t]
  181         \,d\omega_1\,\cdots\,d\omega_n,\cr
  182       }\eqno{(6)}
  183   $$
  184   where the {\sl $n$th order electric dipolar susceptibility},
  185   $$
  186     \chi^{(n)}_{\mu\alpha_1\cdots\alpha_n}
  187       (-\omega_{\sigma};\omega_1,\ldots,\omega_n)
  188       =\int^{\infty}_{-\infty}\cdots\int^{\infty}_{-\infty}
  189        R^{(n)}_{\mu\alpha_1\cdots\alpha_n}(\tau_1,\ldots,\tau_n)
  190        \exp[i(\omega_1\tau_1+\ldots+\omega_n\tau_n)]\,d\tau_1\,\cdots\,d\tau_n,
  191     \eqno{(7)}
  192   $$
  193   was introduced, and where, as previously,
  194   $$\omega_{\sigma}=\omega_1+\omega_2+\ldots+\omega_n.$$
  195   
  196   The intrinsic permutation symmetry of
  197   $R^{(n)}_{\mu\alpha_1\cdots\alpha_n}(\tau_1,\ldots,\tau_n)$,
  198   the $n$th order polarization response function, also in this general
  199   case carries over to the $n$th order susceptibility tensor as well,
  200   in the sense that
  201   $$
  202     \chi^{(n)}_{\mu\alpha_1\alpha_2\cdots\alpha_n}
  203       (-\omega_{\sigma};\omega_1,\omega_2,\ldots,\omega_n)
  204   $$
  205   is invariant under any of the $n!$ pairwise permutations of
  206   $(\alpha_1,\omega_1)$, $(\alpha_2,\omega_2)$, $\ldots$, $(\alpha_n,\omega_n)$.
  207   
  208   \section{Monochromatic fields}
  209   It should at this stage be emphasized that even though the
  210   electric field via the Fourier integral identity can be seen
  211   as a superposition of infinitely many infinitesimally narrow
  212   band monochromatic components, the superposition principle
  213   of linear optics, which states that the wave equation may be
  214   independently solved for each frequency component of the light,
  215   generally does {\sl not} hold in nonlinear optics.
  216   
  217   For monochromatic light, the electric field can be written as a
  218   superposition of a set of distinct terms in time domain as
  219   $$
  220     {\bf E}({\bf r},t)=\sum_k \Re[{\bf E}_{\omega_k}\exp(-i\omega_k t)],
  221   $$
  222   with the convention that the involved angular frequencies all are taken as
  223   positive, $\omega_k\ge 0$, and the electric field in the frequency domain
  224   simply becomes a superposition of delta peaks in the spectrum,
  225   $$
  226     \eqalign{
  227       {\bf E}({\bf r},\omega)
  228         &={{1}\over{2\pi}}\int^{\infty}_{-\infty}{\bf E}({\bf r},\tau)
  229           \exp(i\omega\tau)\,d\tau\cr
  230         &=\{{\rm express\ as\ monochromatic\ field}\}\cr
  231         &={{1}\over{2\pi}}\sum_k \int^{\infty}_{-\infty}
  232           \Re[{\bf E}_{\omega_k}\exp(-i\omega_k \tau)]
  233           \exp(i\omega\tau)\,d\tau\cr
  234         &=\{{\rm by\ definition}\}\cr
  235         &={{1}\over{4\pi}}\sum_k \int^{\infty}_{-\infty}
  236           [{\bf E}_{\omega_k}\exp(i(\omega-\omega_k)\tau)
  237           +{\bf E}^*_{\omega_k}\exp(i(\omega+\omega_k)\tau)]\,d\tau\cr
  238         &={{1}\over{2}}\sum_k \bigg[
  239             {\bf E}_{\omega_k}\underbrace{{{1}\over{2\pi}}\int^{\infty}_{-\infty}
  240             \exp(i(\omega-\omega_k)\tau)\,d\tau}_{
  241               \equiv\delta(\omega-\omega_k)}
  242            +{\bf E}^*_{\omega_k}
  243             \underbrace{{{1}\over{2\pi}}\int^{\infty}_{-\infty}
  244             \exp(i(\omega+\omega_k)\tau)\,d\tau}_{
  245               \equiv\delta(\omega+\omega_k)}\bigg]\cr
  246         &=\{{\rm definition\ of\ the\ delta\ function}\}\cr
  247         &={{1}\over{2}}\sum_k [{\bf E}_{\omega_k}\delta(\omega-\omega_k)
  248            +{\bf E}^*_{\omega_k}\delta(\omega+\omega_k)].\cr
  249   }
  250   $$
  251   By inserting this form of the electric field (taken in the frequency domain)
  252   into the polarization density, one obtains the polarization density
  253   in the monochromatic form
  254   $$
  255     {\bf P}^{(n)}({\bf r},t)=\sum_{\omega_{\sigma}\ge 0}
  256        \Re[{\bf P}^{(n)}_{\omega_{\sigma}}\exp(-i\omega_{\sigma} t)],
  257   $$
  258   with complex-valued Cartesian components at angular frequency
  259   $\omega_{\sigma}$ given as
  260   $$
  261     \eqalign{
  262       ({\bf P}^{(n)}_{\omega_{\sigma}})_{\mu}
  263          =2\varepsilon_0\sum_{\alpha_1}\cdots\sum_{\alpha_n}&
  264           \chi^{(n)}_{\mu\alpha_1\alpha_2\cdots\alpha_n}
  265             (-\omega_{\sigma};\omega_1,\omega_2,\ldots,\omega_n)
  266             (E_{\omega_1})_{\alpha_1}
  267             (E_{\omega_2})_{\alpha_2}
  268             \cdots(E_{\omega_n})_{\alpha_n}\cr
  269         &+\chi^{(n)}_{\mu\alpha_1\alpha_2\cdots\alpha_n}
  270             (-\omega_{\sigma};\omega_2,\omega_1,\ldots,\omega_n)
  271             (E_{\omega_2})_{\alpha_1}
  272             (E_{\omega_1})_{\alpha_2}
  273             \cdots(E_{\omega_n})_{\alpha_n}\cr
  274         &+{\rm all\ other\ distinguishable\ terms}\cr
  275     }
  276     \eqno{(8)}
  277   $$
  278   where, as previously, $\omega_{\sigma}=\omega_1+\omega_2+\ldots+\omega_n$.
  279   In the right hand side of Eq.~(8), the summation is performed over
  280   all distinguishble terms, that is to say over all the possible
  281   combinations of $\omega_1,\omega_2,\ldots,\omega_n$ that give
  282   rise to the particular $\omega_{\sigma}$.
  283   Within this respect, a certain frequency and its negative counterpart
  284   are to be considered as distinct frequencies when appearing in the
  285   set. In general, there are several possible combinations that
  286   give rise to a certain $\omega_{\sigma}$; for example
  287   $(\omega,\omega,-\omega)$, $(\omega,-\omega,\omega)$, and
  288   $(-\omega,\omega,\omega)$ form the set of distinct combinations
  289   of optical frequencies that give rise to optical Kerr-effect
  290   (a field dependent contribution to the polarization density
  291   at $\omega_{\sigma}=\omega+\omega-\omega=\omega$).
  292   
  293   A general conclusion of the form of Eq.~(8), keeping the intrinsic
  294   permutation symmetry in mind, is that only one term needs to be
  295   written, and the number of times this term appears in the expression
  296   for the polarization density should consequently be equal to
  297   the number of distinguishable combinations of
  298   $\omega_1,\omega_2,\ldots,\omega_n$.
  299   
  300   \section{Convention for description of nonlinear optical polarization}
  301   As a ``recipe'' in theoretical nonlinear optics, Butcher and Cotter
  302   provide a very useful convention which is well worth to hold on to.
  303   For a superposition of monochromatic waves, and by invoking the general
  304   property of the intrinsic permutation symmetry, the monochromatic
  305   form of the $n$th order polarization density can be written as
  306   $$
  307     (P^{(n)}_{\omega_{\sigma}})_{\mu}
  308       =\varepsilon_0\sum_{\alpha_1}\cdots\sum_{\alpha_n}\sum_{\omega}
  309        K(-\omega_{\sigma};\omega_1,\ldots,\omega_n)
  310        \chi^{(n)}_{\mu\alpha_1\cdots\alpha_n}
  311        (-\omega_{\sigma};\omega_1,\ldots,\omega_n)
  312        (E_{\omega_1})_{\alpha_1}\cdots(E_{\omega_n})_{\alpha_n}.
  313     \eqno{(9)}
  314   $$
  315   The first summations in Eq.~(9), over $\alpha_1,\ldots,\alpha_n$,
  316   is simply an explicit way of stating that the Einstein convention
  317   of summation over repeated indices holds.
  318   The summation sign $\sum_{\omega}$, however, serves as a reminder
  319   that the expression that follows is to be summed over
  320   {\sl all distinct sets of $\omega_1,\ldots,\omega_n$}.
  321   Because of the intrinsic permutation symmetry, the frequency arguments
  322   appearing in Eq.~(9) may be written in arbitrary order.
  323   
  324   By ``all distinct sets of $\omega_1,\ldots,\omega_n$'', we here mean
  325   that the summation is to be performed, as for example in the case of
  326   optical Kerr-effect, over the single set of nonlinear susceptibilities
  327   that contribute to a certain angular frequency as
  328   $(-\omega;\omega,\omega,-\omega)$ {\sl or} $(-\omega;\omega,-\omega,\omega)$
  329   {\sl or} $(-\omega;-\omega,\omega,\omega)$.
  330   In this example, each of the combinations are considered as {\sl distinct},
  331   and it is left as an arbitary choice which one of these sets that are
  332   most convenient to use (this is simply a matter of choosing notation,
  333   and does not by any means change the description of the interaction).
  334   
  335   In Eq.~(9), the degeneracy factor $K$ is formally described as
  336   $$K(-\omega_{\sigma};\omega_1,\ldots,\omega_n)=2^{l+m-n}p$$
  337   where
  338   $$
  339     \eqalign{
  340       p&={\rm the\ number\ of\ {\sl distinct}\ permutations\ of}
  341          \ \omega_1,\omega_2,\ldots,\omega_1,\cr
  342       n&={\rm the\ order\ of\ the\ nonlinearity},\cr
  343       m&={\rm the\ number\ of\ angular\ frequencies}\ \omega_k
  344          \ {\rm that\ are\ zero,\ and}\cr
  345       l&=\bigg\lbrace\matrix{1,\qquad{\rm if}\ \omega_{\sigma}\ne 0,\cr
  346                              0,\qquad{\rm otherwise}.}\cr
  347     }
  348   $$
  349   In other words, $m$ is the number of DC electric fields present,
  350   and $l=0$ if the nonlinearity we are analyzing gives a static, DC,
  351   polarization density, such as in the previously (in the spring model)
  352   described case of optical rectification in the presence of second
  353   harmonic fields (SHG).
  354   
  355   A list of frequently encountered nonlinear phenomena in nonlinear
  356   optics, including the degeneracy factors as conforming to the above
  357   convention, is given in Butcher and Cotters book, Table 2.1, on page 26.
  358   
  359   \section{Note on the complex representation of the optical field}
  360   Since the observable electric field of the light, in Butcher and
  361   Cotters notation taken as
  362   $$
  363     {\bf E}({\bf r},t)={{1}\over{2}}\sum_{\omega_k\ge 0}
  364     [{\bf E}_{\omega_k}\exp(-i\omega_k t)+{\bf E}^*_{\omega_k}\exp(i\omega_k t)],
  365   $$
  366   is a real-valued quantity, it follows that negative frequencies in the
  367   complex notation should be interpreted as the complex conjugate of the
  368   respective field component, or
  369   $$
  370     {\bf E}_{-\omega_k}={\bf E}^*_{\omega_k}.
  371   $$
  372   
  373   \section{Example: Optical Kerr-effect}
  374   Assume a monochromatic optical wave (containing forward and/or backward
  375   propagating components) polarized in the $xy$-plane,
  376   $$
  377     {\bf E}(z,t)=\Re[{\bf E}_{\omega}(z)\exp(-i\omega_t)]\in{\Bbb R}^3,
  378   $$
  379   with all spatial variation of the field contained in
  380   $$
  381     {\bf E}_{\omega}(z)={\bf e}_x E^x_{\omega}(z)
  382       +{\bf e}_y E^y_{\omega}(z)\in{\Bbb C}^3.
  383   $$
  384   Optical Kerr-effect is in isotropic media described by the third order
  385   susceptibility
  386   $$
  387     \chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega),
  388   $$
  389   with nonzero components of interest for the $xy$-polarized beam given in
  390   Appendix 3.3 of Butcher and Cotters book as
  391   $$
  392     \chi^{(3)}_{xxxx}=\chi^{(3)}_{yyyy},\quad
  393     \chi^{(3)}_{xxyy}=\chi^{(3)}_{yyxx}
  394     =\bigg\{\matrix{{\rm intr.\ perm.\ symm.}\cr
  395                (\alpha,\omega)\rightleftharpoons(\beta,\omega)\cr}\bigg\}=
  396     \chi^{(3)}_{xyxy}=\chi^{(3)}_{yxyx},\quad
  397     \chi^{(3)}_{xyyx}=\chi^{(3)}_{yxxy},
  398   $$
  399   with
  400   $$
  401     \chi^{(3)}_{xxxx}=\chi^{(3)}_{xxyy}+\chi^{(3)}_{xyxy}+\chi^{(3)}_{xyyx}.
  402   $$
  403   The degeneracy factor $K(-\omega;\omega,\omega,-\omega)$ is calculated as
  404   $$
  405     K(-\omega;\omega,\omega,-\omega)=2^{l+m-n}p=2^{1+0-3}3=3/4.
  406   $$
  407   From this set of nonzero susceptibilities, and using the calculated
  408   value of the degeneracy factor in the convention of Butcher and Cotter,
  409   we hence have the third order electric polarization density at
  410   $\omega_{\sigma}=\omega$ given as ${\bf P}^{(n)}({\bf r},t)=
  411   \Re[{\bf P}^{(n)}_{\omega}\exp(-i\omega t)]$, with
  412   $$
  413     \eqalign{
  414       {\bf P}^{(3)}_{\omega}
  415       &=\sum_{\mu}{\bf e}_{\mu}(P^{(3)}_{\omega})_{\mu}\cr
  416       &=\{{\rm Using\ the\ convention\ of\ Butcher\ and\ Cotter}\}\cr
  417       &=\sum_{\mu}{\bf e}_{\mu}
  418         \bigg[\varepsilon_0{{3}\over{4}}\sum_{\alpha}\sum_{\beta}\sum_{\gamma}
  419          \chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega)
  420          (E_{\omega})_{\alpha}(E_{\omega})_{\beta}(E_{-\omega})_{\gamma}\bigg]\cr
  421       &=\{{\rm Evaluate\ the\ sums\ over\ } (x,y,z)
  422           {\rm\ for\ field\ polarized\ in\ the\ }xy{\rm\ plane}\}\cr
  423       &=\varepsilon_0{{3}\over{4}}\{
  424         {\bf e}_x[
  425           \chi^{(3)}_{xxxx} E^x_{\omega} E^x_{\omega} E^x_{-\omega}
  426           +\chi^{(3)}_{xyyx} E^y_{\omega} E^y_{\omega} E^x_{-\omega}
  427           +\chi^{(3)}_{xyxy} E^y_{\omega} E^x_{\omega} E^y_{-\omega}
  428           +\chi^{(3)}_{xxyy} E^x_{\omega} E^y_{\omega} E^y_{-\omega}]\cr
  429        &\qquad\quad
  430        +{\bf e}_y[
  431           \chi^{(3)}_{yyyy} E^y_{\omega} E^y_{\omega} E^y_{-\omega}
  432           +\chi^{(3)}_{yxxy} E^x_{\omega} E^x_{\omega} E^y_{-\omega}
  433           +\chi^{(3)}_{yxyx} E^x_{\omega} E^y_{\omega} E^x_{-\omega}
  434           +\chi^{(3)}_{yyxx} E^y_{\omega} E^x_{\omega} E^x_{-\omega}]\}\cr
  435       &=\{{\rm Make\ use\ of\ }{\bf E}_{-\omega}={\bf E}^*_{\omega}
  436           {\rm\ and\ relations\ }\chi^{(3)}_{xxyy}=\chi^{(3)}_{yyxx},
  437           {\rm\ etc.}\}\cr
  438       &=\varepsilon_0{{3}\over{4}}\{
  439         {\bf e}_x[
  440           \chi^{(3)}_{xxxx} E^x_{\omega} |E^x_{\omega}|^2
  441           +\chi^{(3)}_{xyyx} E^y_{\omega}{}^2 E^{x*}_{\omega}
  442           +\chi^{(3)}_{xyxy} |E^y_{\omega}|^2 E^x_{\omega}
  443           +\chi^{(3)}_{xxyy} E^x_{\omega} |E^y_{\omega}|^2]\cr
  444        &\qquad\quad
  445        +{\bf e}_y[
  446           \chi^{(3)}_{xxxx} E^y_{\omega} |E^y_{\omega}|^2
  447           +\chi^{(3)}_{xyyx} E^x_{\omega}{}^2 E^{y*}_{\omega}
  448           +\chi^{(3)}_{xyxy} |E^x_{\omega}|^2 E^y_{\omega}
  449           +\chi^{(3)}_{xxyy} E^y_{\omega} |E^x_{\omega}|^2]\}\cr
  450       &=\{{\rm Make\ use\ of\ intrinsic\ permutation\ symmetry}\}\cr
  451       &=\varepsilon_0{{3}\over{4}}\{
  452         {\bf e}_x[
  453           (\chi^{(3)}_{xxxx} |E^x_{\omega}|^2
  454             +2\chi^{(3)}_{xxyy} |E^y_{\omega}|^2) E^x_{\omega}
  455           +(\chi^{(3)}_{xxxx}-2\chi^{(3)}_{xxyy})
  456            E^y_{\omega}{}^2 E^{x*}_{\omega}\cr
  457        &\qquad\quad
  458         {\bf e}_y[
  459           (\chi^{(3)}_{xxxx} |E^y_{\omega}|^2
  460             +2\chi^{(3)}_{xxyy} |E^x_{\omega}|^2) E^y_{\omega}
  461           +(\chi^{(3)}_{xxxx}-2\chi^{(3)}_{xxyy})
  462            E^x_{\omega}{}^2 E^{y*}_{\omega}.\cr
  463     }
  464   $$
  465   For the optical field being linearly polarized, say in the $x$-direction,
  466   the expression for the polarization density is significantly simplified,
  467   to yield
  468   $$
  469     {\bf P}^{(3)}_{\omega}=\varepsilon_0(3/4){\bf e}_x
  470       \chi^{(3)}_{xxxx} |E^x_{\omega}|^2 E^x_{\omega},
  471   $$
  472   i.~e.~taking a form that can be interpreted as an intensity-dependent
  473   ($\sim|E^x_{\omega}|^2$) contribution to the refractive index
  474   (cf.~Butcher and Cotter \S 6.3.1).
  475   \bye
  476   

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Last modified Wednesday 16 Nov 2011