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    1   % File: nlopt/lect9/lect9.tex [pure TeX code]
    2   % Last change: March 2, 2003
    3   %
    4   % Lecture No 9 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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   12   % the Euler fraktur font.
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   14   \input amssym
   15   \font\ninerm=cmr9
   16   \font\twelvesc=cmcsc10
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   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\sinc{\mathop{\rm sinc}\nolimits} % the sinc(x)=sin(x)/x function
   25   \def\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm
   26     \topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt
   27     \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
   28       \else\hfill\fi}
   29     \def\rightheadline{\tenrm{\it Lecture notes #1}
   30       \hfil{\it Nonlinear Optics 5A5513 (2003)}}
   31     \def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)}
   32       \hfil{\it Lecture notes #1}}
   33     \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps}
   34     \vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}}
   35     \hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1}
   36     \vskip 24pt\noindent}
   37   \def\section #1 {\medskip\goodbreak\noindent{\bf #1}
   38     \par\nobreak\smallskip\noindent}
   39   \def\subsection #1 {\smallskip\goodbreak\noindent{\it #1}
   40     \par\nobreak\smallskip\noindent}
   41   
   42   \lecture{9}
   43   In this lecture, we will focus on examples of electromagnetic wave
   44   propagation in nonlinear optical media, by applying the forms of Maxwell's
   45   equations as obtained in the eighth lecture to a set of particular
   46   nonlinear interactions as described by the previously formulated nonlinear
   47   susceptibility formalism.
   48   \medskip
   49   
   50   \noindent The outline for this lecture is:
   51   \item{$\bullet$}{General process for solving problems in nonlinear optics}
   52   \item{$\bullet$}{Second harmonic generation (SHG)}
   53   \item{$\bullet$}{Optical Kerr-effect}
   54   \medskip
   55   
   56   \section{General process for solving problems in nonlinear optics}
   57   The typical steps in the process of solving a theoretical problem in
   58   nonlinear optics typically involve:
   59   $$
   60     \bigg\{\matrix{
   61       {\rm define\ the\ optical\ interaction\ of\ interest}\cr
   62       {\rm (identifying\ the\ susceptibility)}\cr
   63     }\bigg\}
   64   $$
   65   $$\Downarrow$$
   66   $$
   67     \bigg\{\matrix{
   68       {\rm define\ in\ which\ medium\ the\ interaction\ take\ place}\cr
   69       {\rm (identify\ crystallographic\ point\ symmetry\ group)}\cr
   70     }\bigg\}
   71   $$
   72   $$\Downarrow$$
   73   $$
   74     \bigg\{\matrix{
   75       {\rm consider\ eventual\ additional\ symmetries\ and\ constraints}\cr
   76       {\rm (e.~g.~intrinsic,\ overall,\ or\ Kleinman\ symmetries)}\cr
   77     }\bigg\}
   78   $$
   79   $$\Downarrow$$
   80   $$
   81     \bigg\{\matrix{
   82       {\rm construct\ the\ polarization\ density}\cr
   83       {\rm (the\ Butcher\ and\ Cotter\ convention)}\cr
   84     }\bigg\}
   85   $$
   86   $$\Downarrow$$
   87   $$
   88     \bigg\{\matrix{
   89       {\rm formulate\ the\ proper\ wave\ equation}\cr
   90       {\rm (e.~g.~taking\ dispersion\ or\ diffraction\ into\ account)}\cr
   91     }\bigg\}
   92   $$
   93   $$\Downarrow$$
   94   $$
   95     \bigg\{\matrix{
   96       {\rm formulate\ the\ proper\ boundary}\cr
   97       {\rm conditions\ for\ the\ wave\ equation}\cr
   98     }\bigg\}
   99   $$
  100   $$\Downarrow$$
  101   $$
  102     \bigg\{\matrix{
  103       {\rm solve\ the\ wave\ equation\ under}\cr
  104       {\rm the\ boundary\ conditions}\cr
  105     }\bigg\}
  106   $$
  107   \vfill\eject
  108   
  109   \def\exercise#1#2{\medskip\goodbreak
  110      \noindent{\bf Exercise #1.}\hskip 4pt({\it #2})\hskip 4pt}
  111   \def\subexercise#1{\goodbreak
  112      {\bf #1.}\hskip 4pt}
  113   
  114   \section{Formulation of the exercises in this lecture}
  115   In order to illustrate the scheme as previously outlined, the following
  116   exercises serve as to give the connection between the susceptibilities,
  117   as extensively analysed from a quantum-mechanical basis in earlier
  118   lectures of this course, and the wave equation, derived from Maxwell's
  119   equations of motion for electromagnetic fields.
  120   
  121   \exercise{1}{Second harmonic generation in negative uniaxial media}
  122   Consider a continuous pump wave at angular frequency $\omega$, initially
  123   polarized in the $y$-direction and propagating in the positive $x$-direction
  124   of a negative uniaxial crystal of crystallographic point symmetry group $3m$.
  125   (Examples of crystals belonging to this class:
  126    beta-${\rm Ba}{\rm B}_2{\rm O}_4$/BBO, ${\rm Li}{\rm Nb}{\rm O}_3$.)
  127   
  128   \subexercise{1a} Formulate the polarization density of the medium for
  129   the pump and second harmonic wave.
  130   
  131   \subexercise{1b} Formulate the system of equations of motion for the
  132   electromagnetic fields.
  133   
  134   \subexercise{1c} Assuming no second harmonic signal present at the input,
  135   solve the equations of motion for the second harmonic field, using the
  136   non-depleted pump approximation, and derive an expression for the conversion
  137   efficiency of the second harmonic generation.
  138   
  139   \exercise{2}{Optical Kerr-effect -- continuous wave case}
  140   In this setup, a monochromatic optical wave is propagating in the positive
  141   $z$-direction of an isotropic optical Kerr-medium.
  142   
  143   \subexercise{2a} Formulate the polarization density of the medium
  144   for a wave polarized in the $xy$-plane.
  145   
  146   \subexercise{2b} Formulate the polarization density of the medium
  147   for a wave polarized in the $x$-direction.
  148   
  149   \subexercise{2c} Formulate the wave equation for continuous wave propagation
  150   in optical Kerr-media. The continuous wave is $x$-polarized and propagates
  151   in the positive $z$-direction.
  152   
  153   \subexercise{2d} For lossless media, solve the wave equation and give
  154   an expression for the nonlinear, intensity-dependent refractive index
  155   $n=n_0+n_2|{\bf E}_{\omega}|^2$.
  156   \vfill\eject
  157   
  158   \section{Second harmonic generation}
  159   \subsection{The optical interaction}
  160   In the case of second harmonic generation (SHG), two photons at angular
  161   frequency~$\omega$ combine to a photon at twice the angular frequency,
  162   $$
  163     \hbar\omega+\hbar\omega\to\hbar(2\omega).
  164   $$
  165   This interaction is for the second harmonic wave (at angular
  166   frequency~$\omega$)described by the second order susceptibility
  167   $$
  168     \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega,\omega),
  169   $$
  170   where $\omega_{\sigma}=2\omega$ is the generated second harmonic frequency
  171   of the light.
  172   
  173   \subsection{Symmetries of the medium}
  174   In this example we consider second harmonic generation in trigonal media
  175   of crystallographic point symmetry group $3m$.
  176   (Example: ${\rm Li}{\rm Nb}{\rm O}_3$)
  177   \bigskip
  178   \centerline{\epsfxsize=150mm\epsfbox{../images/shgsetup/shgsetup.1}}
  179   \medskip
  180   \centerline{Figure 1. The setup for optical second harmonic generation in
  181   ${\rm Li}{\rm Nb}{\rm O}_3$.}
  182   \medskip
  183   For this point symmetry group, the nonzero tensor elements of the first
  184   order susceptibility are (for example according to Table A3.1 in
  185   {\sl The Elements of Nonlinear Optics})
  186   $$
  187     \chi^{(1)}_{xx}=\chi^{(1)}_{yy},\qquad\chi^{(1)}_{zz},
  188   $$
  189   which gives the ordinary refractive indices
  190   $$
  191     n_x(\omega)=n_y(\omega)=[1+\chi^{(1)}_{xx}(-\omega;\omega)]^{1/2}
  192       \equiv n_{\rm O}(\omega)
  193   $$
  194   for waves components polarized in the $x$- or $y$-directions, and
  195   the extraordinary refractive index
  196   $$
  197     n_z(\omega)=[1+\chi^{(1)}_{zz}(-\omega;\omega)]^{1/2}
  198       \equiv n_{\rm E}(\omega)
  199   $$
  200   for the wave component polarized in the $z$-direction. Since we here are
  201   considering a negatively uniaxial crystal (see Butcher and Cotter, p.~214),
  202   these refractive indices satisfy the inequality
  203   $$n_{\rm E}(\omega)\le n_{\rm O}(\omega).$$
  204   
  205   The nonzero tensor elements of the second order susceptibility are
  206   (for example according to Table A3.2 in {\sl The Elements of Nonlinear Optics})
  207   $$
  208     \eqalign{
  209       \chi^{(2)}_{yxx}&=\chi^{(2)}_{xyx}
  210         =\chi^{(2)}_{xxy}=-\chi^{(2)}_{yyy},\qquad
  211       \chi^{(2)}_{zzz},\cr
  212       \chi^{(2)}_{zxx}&=\chi^{(2)}_{zyy},\qquad
  213       \chi^{(2)}_{yyz}=\chi^{(2)}_{xxz},\qquad
  214       \chi^{(2)}_{yzy}=\chi^{(2)}_{xzx},\cr
  215     }\eqno{(1)}
  216   $$
  217   
  218   \subsection{Additional symmetries}
  219   Intrinsic permutation symmetry for the case of second harmonic generation
  220   gives
  221   $$
  222     \chi^{(2)}_{xxz}(-2\omega;\omega,\omega)
  223       =\chi^{(2)}_{xzx}(-2\omega;\omega,\omega)
  224       =\chi^{(2)}_{yzy}(-2\omega;\omega,\omega)
  225       =\chi^{(2)}_{yyz}(-2\omega;\omega,\omega),
  226   $$
  227   which reduces the second order susceptibility in Eq.~(1) to a set
  228   of 11 tensor elements, of which only 4 are independent.
  229   (We recall that the intrinsic permutation symmetry is always applicable,
  230   as being a consequence of the symmetrization described in lectures two
  231   and five.)
  232   Whenever Kleinman symmetry holds, the susceptibility is in addition symmetric
  233   under any permutation of the indices, which hence gives the additional relation
  234   $$
  235     \chi^{(2)}_{zxx}(-2\omega;\omega,\omega)
  236       =\chi^{(2)}_{xzx}(-2\omega;\omega,\omega)
  237       =\chi^{(2)}_{xxz}(-2\omega;\omega,\omega),
  238   $$
  239   i.~e.~reducing the second order susceptibility to a set of 11 tensor
  240   elements, of which only 3 are independent.
  241   
  242   To summarize, the set of nonzero tensor elements describing second
  243   harmonic generation under Kleinman symmetry is
  244   $$
  245     \eqalign{
  246       \chi^{(2)}_{yxx}&=\chi^{(2)}_{xyx}
  247                        =\chi^{(2)}_{xxy}=-\chi^{(2)}_{yyy},\qquad
  248       \chi^{(2)}_{zzz},\cr
  249       \chi^{(2)}_{zxx}&=\chi^{(2)}_{zyy}=
  250       \chi^{(2)}_{yyz}=\chi^{(2)}_{xxz}=
  251       \chi^{(2)}_{yzy}=\chi^{(2)}_{xzx}.\cr
  252     }\eqno{(2)}
  253   $$
  254   
  255   For the pump field at angular frequency $\omega$, the relevant susceptibility
  256   describing the interaction with the second harmonic wave
  257   is\footnote{${}^1$}{Keep in mind that in the convention of Butcher and Cotter,
  258   the frequency arguments to the right of the semicolon may be writen in
  259   arbitrary order, hence we may in an equal description instead use
  260   $$\chi^{(2)}_{xxz}(-\omega;-\omega,2\omega)$$ for the description of
  261   the second order interaction between light and matter.}
  262   $$
  263     \chi^{(2)}_{\mu\alpha\beta}(-\omega;2\omega,-\omega).
  264   $$
  265   For an arbitrary frequency argument, this is the proper form of the
  266   susceptibility to use for the fundamental field, and this form generally
  267   differ from that of the susceptibilities for the second harmonic field.
  268   However, whenever Kleinman symmetry holds, the susceptibility for the
  269   fundamental field can be cast into the same parameters as for the second
  270   harmonic field, since
  271   $$
  272     \eqalign{
  273       \chi^{(2)}_{\mu\alpha\beta}(-\omega;2\omega,-\omega)
  274        &=\big\{{\rm Apply\ overall\ permutation\ symmetry}\big\}\cr
  275        &=\chi^{(2)}_{\alpha\mu\beta}(2\omega;-\omega,-\omega)\cr
  276        &=\big\{{\rm Apply\ Kleinman\ symmetry}\big\}\cr
  277        &=\chi^{(2)}_{\mu\alpha\beta}(2\omega;-\omega,-\omega)\cr
  278        &=\big\{{\rm Apply\ reality\ condition\ [B.\,\&C.\,Eq.\,(2.43)]}\big\}\cr
  279        &=[\chi^{(2)}_{\mu\alpha\beta}(-2\omega;\omega,\omega)]^*\cr
  280        &=\chi^{(2)}_{\mu\alpha\beta}(-2\omega;\omega,\omega).\cr
  281     }
  282   $$
  283   Hence the second order interaction is described by the same set of tensor
  284   elements for the fundamental as well as the second harmonic optical wave
  285   whenever Kleinman symmetry applies.
  286   \vfill\eject
  287   
  288   \subsection{The polarization density}
  289   Following the convention of Butcher and Cotter,\footnote{${}^2$}{See course
  290   material on the Butcher and Cotter convention handed out during the third
  291   lecture. Notice that for the first order polarization density, one at optical
  292   frequencies {\sl always} has the trivial degeneracy factor
  293   $$K(-2\omega;\omega)=2^{l+m-n}p=2^{1+0-1}\times 1=1.$$}
  294   the degeneracy factor for the second harmonic signal at $2\omega$ is
  295   $$
  296     K(-2\omega;\omega,\omega)=2^{l+m-n}p,
  297   $$
  298   where
  299   $$
  300     \eqalign{
  301       p&=\{{\rm the\ number\ of\ {\sl distinct}\ permutations\ of}
  302          \ \omega,\omega\}=1,\cr
  303       n&=\{{\rm the\ order\ of\ the\ nonlinearity}\}=2,\cr
  304       m&=\{{\rm the\ number\ of\ angular\ frequencies}\ \omega_k
  305          \ {\rm that\ are\ zero}\}=0,\cr
  306       l&=\bigg\lbrace\matrix{1,\qquad{\rm if}\ 2\omega\ne 0,\cr
  307                              0,\qquad{\rm otherwise}.}\bigg\rbrace=1,\cr
  308     }
  309   $$
  310   i.~e.
  311   $$
  312     K(-2\omega;\omega,\omega)=2^{1+0-2}\times 1=1/2.
  313   $$
  314   For the fundamental optical field at $\omega$, one might be mislead to assume
  315   that since the second order interaction for this field is described by an
  316   identical set of tensor elements as for the second harmonic wave, the
  317   degeneracy factor must also be identical to the previously derived one.
  318   This is, however, {\sl a very wrong assumption}, and one can easily verify
  319   that the proper degeneracy factor for the fundamental field instead is
  320   given as
  321   $$
  322     K(-\omega;2\omega,-\omega)=2^{l+m-n}p,
  323   $$
  324   where
  325   $$
  326     \eqalign{
  327       p&=\{{\rm the\ number\ of\ {\sl distinct}\ permutations\ of}
  328          \ 2\omega,-\omega\}=2,\cr
  329       n&=\{{\rm the\ order\ of\ the\ nonlinearity}\}=2,\cr
  330       m&=\{{\rm the\ number\ of\ angular\ frequencies}\ \omega_k
  331          \ {\rm that\ are\ zero}\}=0,\cr
  332       l&=\bigg\lbrace\matrix{1,\qquad{\rm if}\ \omega\ne 0,\cr
  333                              0,\qquad{\rm otherwise}.}\bigg\rbrace=1,\cr
  334     }
  335   $$
  336   i.~e.
  337   $$
  338     K(-\omega;2\omega,-\omega)=2^{1+0-2}\times 2=1.
  339   $$
  340   
  341   The general second harmonic polarization density of the medium is hence
  342   given as
  343   $$
  344     \eqalign{
  345       [{\bf P}^{({\rm NL})}_{2\omega}]_z&=[{\bf P}^{(2)}_{2\omega}]_z
  346          =\varepsilon_0 \underbrace{K(-2\omega;\omega,\omega)
  347            \chi^{(2)}_{z\alpha\beta}(-2\omega;\omega,\omega)}_{
  348              ={{1}\over{2}}\chi^{(2)}_{z\alpha\beta}(-2\omega;\omega,\omega)}
  349            E^{\alpha}_{\omega}E^{\beta}_{\omega}\cr
  350         &=(\varepsilon_0/2)[\chi^{(2)}_{zxx} E^x_{\omega} E^{x}_{\omega}
  351           +\chi^{(2)}_{zyy} E^y_{\omega} E^{y}_{\omega}
  352           +\chi^{(2)}_{zzz} E^z_{\omega} E^{z}_{\omega}]\cr
  353         &=(\varepsilon_0/2)[\chi^{(2)}_{zxx}
  354             (E^x_{\omega} E^{x}_{\omega}+E^y_{\omega} E^{y}_{\omega})
  355           +\chi^{(2)}_{zzz} E^z_{\omega} E^{z}_{\omega}],\cr
  356       [{\bf P}^{({\rm NL})}_{2\omega}]_y
  357         &=(\varepsilon_0/2)[\chi^{(2)}_{yxx} E^x_{\omega} E^{x}_{\omega}
  358           +\chi^{(2)}_{yyy} E^y_{\omega} E^{y}_{\omega}
  359           +\chi^{(2)}_{yyz} E^y_{\omega} E^{z}_{\omega}
  360           +\chi^{(2)}_{yzy} E^z_{\omega} E^{y}_{\omega}]\cr
  361         &=(\varepsilon_0/2)[\chi^{(2)}_{yxx}
  362             (E^x_{\omega} E^{x}_{\omega}-E^y_{\omega} E^{y}_{\omega})
  363           +\chi^{(2)}_{zxx}
  364             (E^y_{\omega} E^{z}_{\omega}+E^z_{\omega} E^{y}_{\omega})],\cr
  365       [{\bf P}^{({\rm NL})}_{2\omega}]_x
  366         &=(\varepsilon_0/2)[\chi^{(2)}_{xxy} E^x_{\omega} E^{y}_{\omega}
  367           +\chi^{(2)}_{xyx} E^y_{\omega} E^{x}_{\omega}
  368           +\chi^{(2)}_{xxz} E^x_{\omega} E^{z}_{\omega}
  369           +\chi^{(2)}_{xzx} E^z_{\omega} E^{x}_{\omega}]\cr
  370         &=(\varepsilon_0/2)[\chi^{(2)}_{yxx}
  371             (E^x_{\omega} E^{y}_{\omega}+E^y_{\omega} E^{x}_{\omega})
  372           +\chi^{(2)}_{zxx}
  373             (E^x_{\omega} E^{z}_{\omega}+E^z_{\omega} E^{x}_{\omega})],\cr
  374     }
  375   $$
  376   while the general polarization density at the angular frequency of the pump
  377   field becomes\footnote{${}^3$}{Keep in mind that a negative frequency
  378   argument to the right of the semicolon in the susceptibility is to be
  379   associated with the complex conjugate of the respective electric field;
  380   see Butcher and Cotter, section 2.3.2.}
  381   $$
  382     \eqalign{
  383       [{\bf P}^{({\rm NL})}_{\omega}]_z&=[{\bf P}^{(2)}_{\omega}]_z
  384          =\varepsilon_0 \underbrace{K(-\omega;2\omega,-\omega)
  385            \chi^{(2)}_{z\alpha\beta}(-\omega;2\omega,-\omega)}_{
  386              =\chi^{(2)}_{z\alpha\beta}(-2\omega;\omega,\omega)}
  387            E^{\alpha}_{2\omega}E^{\beta}_{-\omega}\cr
  388         &=\varepsilon_0[\chi^{(2)}_{zxx} E^x_{2\omega} E^{x*}_{\omega}
  389           +\chi^{(2)}_{zyy} E^y_{2\omega} E^{y*}_{\omega}
  390           +\chi^{(2)}_{zzz} E^z_{2\omega} E^{z*}_{\omega}]\cr
  391         &=\varepsilon_0[\chi^{(2)}_{zxx}
  392             (E^x_{2\omega} E^{x*}_{\omega}+E^y_{2\omega} E^{y*}_{\omega})
  393           +\chi^{(2)}_{zzz} E^z_{2\omega} E^{z*}_{\omega}],\cr
  394       [{\bf P}^{({\rm NL})}_{\omega}]_y
  395         &=\varepsilon_0[\chi^{(2)}_{yxx} E^x_{2\omega} E^{x*}_{\omega}
  396           +\chi^{(2)}_{yyy} E^y_{2\omega} E^{y*}_{\omega}
  397           +\chi^{(2)}_{yyz} E^y_{2\omega} E^{z*}_{\omega}
  398           +\chi^{(2)}_{yzy} E^z_{2\omega} E^{y*}_{\omega}]\cr
  399         &=\varepsilon_0[\chi^{(2)}_{yxx}
  400             (E^x_{2\omega} E^{x*}_{\omega}-E^y_{2\omega} E^{y*}_{\omega})
  401           +\chi^{(2)}_{zxx}
  402             (E^y_{2\omega} E^{z*}_{\omega}+E^z_{2\omega} E^{y*}_{\omega})],\cr
  403       [{\bf P}^{({\rm NL})}_{\omega}]_x
  404         &=\varepsilon_0[\chi^{(2)}_{xxy} E^x_{2\omega} E^{y*}_{\omega}
  405           +\chi^{(2)}_{xyx} E^y_{2\omega} E^{x*}_{\omega}
  406           +\chi^{(2)}_{xxz} E^x_{2\omega} E^{z*}_{\omega}
  407           +\chi^{(2)}_{xzx} E^z_{2\omega} E^{x*}_{\omega}]\cr
  408         &=\varepsilon_0[\chi^{(2)}_{yxx}
  409             (E^x_{2\omega} E^{y*}_{\omega}+E^y_{2\omega} E^{x*}_{\omega})
  410           +\chi^{(2)}_{zxx}
  411             (E^x_{2\omega} E^{z*}_{\omega}+E^z_{2\omega} E^{x*}_{\omega})].\cr
  412     }
  413   $$
  414   For a pump wave polarized in the $yz$-plane of the crystal frame, the
  415   polarization density of the medium hence becomes
  416   $$
  417     \eqalign{
  418       [{\bf P}^{({\rm NL})}_{2\omega}]_z
  419         &=(\varepsilon_0/2)[\chi^{(2)}_{zxx}
  420             E^y_{\omega} E^y_{\omega}
  421           +\chi^{(2)}_{zzz} E^z_{\omega} E^{z}_{\omega}],\cr
  422       [{\bf P}^{({\rm NL})}_{2\omega}]_y
  423         &=(\varepsilon_0/2)[-\chi^{(2)}_{yxx} E^y_{\omega} E^{y}_{\omega}
  424           +\chi^{(2)}_{zxx}
  425             (E^y_{\omega} E^{z}_{\omega}+E^z_{\omega} E^{y}_{\omega})],\cr
  426       [{\bf P}^{({\rm NL})}_{2\omega}]_x
  427         &=0,\cr
  428     }
  429   $$
  430   and
  431   $$
  432     \eqalign{
  433       [{\bf P}^{({\rm NL})}_{\omega}]_z
  434         &=\varepsilon_0[\chi^{(2)}_{zxx} E^y_{2\omega} E^{y*}_{\omega}
  435           +\chi^{(2)}_{zzz} E^z_{2\omega} E^{z*}_{\omega}],\cr
  436       [{\bf P}^{({\rm NL})}_{\omega}]_y
  437         &=\varepsilon_0[-\chi^{(2)}_{yxx}E^y_{2\omega} E^{y*}_{\omega}
  438           +\chi^{(2)}_{zxx}
  439             (E^y_{2\omega} E^{z*}_{\omega}+E^z_{2\omega} E^{y*}_{\omega})],\cr
  440       [{\bf P}^{({\rm NL})}_{\omega}]_x
  441         &=0.\cr
  442     }
  443   $$
  444   
  445   \subsection{The wave equation}
  446   Strictly speaking, the previously formulated polarization density
  447   gives a coupled system between the polarization states of both the
  448   fundamental and second harmonic waves, since both the $y$- and $z$-components
  449   of the polarization densities at $\omega$ and $2\omega$ contain components
  450   of all other field components.
  451   However, for simplicity we will here restrict the continued analysis to the
  452   case of a $y$-polarized input pump wave, which through the
  453   $\chi^{(2)}_{zyy}=\chi^{(2)}_{zxx}$ elements give rise to a $z$-polarized
  454   second harmonic frequency component at $2\omega$.
  455   
  456   The electric fields of the fundamental and second harmonic optical waves
  457   are for the forward propagating configuration expressed in their
  458   {\sl spatial envelopes} ${\bf A}_{\omega}$ and ${\bf A}_{2\omega}$ as
  459   $$
  460     \eqalign{
  461       {\bf E}_{\omega}(x)&={\bf e}_y A^y_{\omega}(x)\exp(ik_{\omega_y} x),
  462         \qquad k_{\omega_y}\equiv\omega n_{\omega_y}/c
  463                            \equiv\omega n_{\rm O}(\omega)/c\cr
  464       {\bf E}_{2\omega}(x)&={\bf e}_z A^z_{2\omega}(x)\exp(ik_{2\omega_z} x),
  465         \qquad k_{2\omega_z}\equiv 2\omega n_{2\omega_z}/c
  466                            \equiv 2\omega n_{\rm E}(2\omega)/c\cr
  467     }
  468   $$
  469   Using the above separation of the natural, spatial oscillation of the
  470   light, in the infinite plane wave approximation and by using the slowly
  471   varying envelope approximation, the wave equation for the envelope of the
  472   second harmonic optical field becomes (see Eq.~(6) in the notes from
  473   lecture eight)
  474   $$
  475     \eqalign{
  476       {{\partial A^z_{2\omega}}\over{\partial x}}
  477         &=i{{\mu_0(2\omega)^2}\over{2k_{2\omega_z}}}
  478           [{\bf P}^{({\rm NL})}_{2\omega}]_z
  479             \exp(-ik_{2\omega_z}x)\cr
  480         &=i{{\mu_0(2\omega)^2}\over{2(2\omega n_{2\omega_z}/c)}}
  481             \underbrace{
  482             {{\varepsilon_0}\over{2}}
  483             \chi^{(2)}_{zxx} A^y_{\omega}{}^2\exp(2ik_{\omega_y}x)}_{
  484                =[{\bf P}^{({\rm NL})}_{2\omega}]_z}
  485             \exp(-ik_{2\omega_z}x)\cr
  486         &=i{{\omega\chi^{(2)}_{zxx}}\over{2 n_{2\omega_z} c}}
  487             A^y_{\omega}{}^2\exp[i(2k_{\omega_y}-k_{2\omega_z})x],\cr
  488     }
  489   $$
  490   while for the fundamental wave,
  491   $$
  492     \eqalign{
  493       {{\partial A^y_{\omega}}\over{\partial x}}
  494         &=i{{\mu_0 \omega^2}\over{2k_{\omega_y}}}
  495           [{\bf P}^{({\rm NL})}_{\omega}]_y
  496             \exp(-ik_{\omega_y}x)\cr
  497         &=i{{\mu_0 \omega^2}\over{2(\omega n_{\omega_y}/c)}}
  498             \underbrace{\varepsilon_0\chi^{(2)}_{zxx}
  499               A^z_{2\omega}\exp(ik_{2\omega_z}x)
  500               A^{y*}_{\omega}\exp(-ik_{\omega_y}x)}_{
  501                =[{\bf P}^{({\rm NL})}_{\omega}]_y}
  502             \exp(-ik_{\omega_y}x)\cr
  503         &=i{{\omega\chi^{(2)}_{zxx}}\over{2 n_{\omega_y} c}}
  504             A^z_{2\omega}A^{y*}_{\omega}
  505             \exp[-i(2k_{\omega_y}-k_{2\omega_z})x].\cr
  506     }
  507   $$
  508   These equations can hence be summarized by the coupled system
  509   $$
  510     \eqalignno{
  511       {{\partial A^z_{2\omega}}\over{\partial x}}
  512         &=i{{\omega\chi^{(2)}_{zxx}}\over{2 n_{2\omega_z} c}}
  513             A^y_{\omega}{}^2\exp(i\Delta k x),&({\rm 3a})\cr
  514       {{\partial A^y_{\omega}}\over{\partial x}}
  515         &=i{{\omega\chi^{(2)}_{zxx}}\over{2 n_{\omega_y} c}}
  516             A^z_{2\omega}A^{y*}_{\omega}
  517             \exp(-i\Delta k x).&({\rm 3b})\cr
  518     }
  519   $$
  520   where
  521   $$
  522     \eqalign{
  523       \Delta k
  524         &=2k_{\omega_y}-k_{2\omega_z}\cr
  525         &=2\omega n_{\omega_y}/c-2\omega n_{2\omega_z}/c\cr
  526         &=(2\omega/c)(n_{\omega_y}-\omega n_{2\omega_z})\cr
  527     }
  528   $$
  529   is the so-called phase mismatch between the pump and second harmonic wave.
  530   
  531   \subsection{Boundary conditions}
  532   Here the boundary conditions are simply that no second harmonic signal
  533   is present at the input,
  534   $$A^z_{2\omega}(0)=0,$$
  535   together with a known input field at the fundamental frequency,
  536   $$A^y_{\omega}(0)=\{{\rm known}\}.$$
  537   
  538   \subsection{Solving the wave equation}
  539   Considering a nonzero $\Delta k$, the conversion efficiency is regularly
  540   quite small, and one may approximately take the spatial distribution of
  541   the pump wave to be constant, $A^y_{\omega}(x)\approx A^y_{\omega}(0)$.
  542   Using this approximation\footnote{${}^4$}{For an outline of the method of
  543   solving the coupled system~(1) exactly in terms of Jacobian elliptic
  544   functions (thus allowing for a depleted pump as well), see J.~A.~Armstrong,
  545   N.~Bloembergen, J.~Ducuing, and P.~S.~Pershan, Phys.~Rev.~{\bf 127},
  546   {1918--1939}, (1962).}, and by applying the initial condition
  547   $A^z_{2\omega}(0)=0$ of the second harmonic signal, one finds
  548   $$
  549     \eqalign{
  550       A^z_{2\omega}(L)
  551         &=\int^L_0{{\partial A^z_{2\omega}(z)}\over{\partial x}}\,dx\cr
  552         &=\int^L_0 i{{\omega\chi^{(2)}_{zxx}}\over{2 n_{{2\omega}_z}} c}
  553           A^y_{\omega}{}^2(0)\exp(i\Delta k x)\,dx\cr
  554         &={{\omega\chi^{(2)}_{zxx}}\over{2 n_{{2\omega}_z}} c}
  555           A^y_{\omega}{}^2(0){{1}\over{\Delta k}}[\exp(i\Delta k L)-1]\cr
  556         &=\big\{{\rm Use\ }[\exp(i\Delta k L)-1]/\Delta k
  557              =iL\exp(i\Delta k L/2)\sinc(\Delta k L/2)\big\}\cr
  558         &=i{{\omega\chi^{(2)}_{zxx} L}\over{2 n_{{2\omega}_z}} c}
  559           A^y_{\omega}{}^2(0)\exp(i\Delta k L/2)\sinc(\Delta k L/2).\cr
  560     }
  561   $$
  562   \vfill\eject
  563   
  564   %%\bigskip
  565   \centerline{\epsfxsize=110mm\epsfbox{type1.eps}}
  566   \medskip
  567   \noindent{Figure 2. Conversion efficiency $I_{2\omega}(L)/I_{\omega}(0)$
  568   as function of normalized crystal length $\Delta k L/2$.
  569   The conversion efficiency is in the phase mismatched case ($\Delta k\ne 0$)
  570   a periodic function, with period $2L_{\rm c}$, with
  571   $L_{\rm c}=\pi/\Delta k$ being the {\sl coherence length}.}
  572   \medskip
  573   \noindent
  574   In terms if the light intensities of the waves, one after a propagation
  575   distance $x=L$ hence has the second harmonic signal with intensity
  576   $I_{2\omega}(L)$ expressed in terms of the input intensity $I_{\omega}$ as
  577   $$
  578     \eqalign{
  579       I_{2\omega}(L)
  580         &={{1}\over{2}}\varepsilon_0 c n_{2\omega_z}|A^z_{2\omega}(L)|^2\cr
  581         &={{1}\over{2}}\varepsilon_0 c n_{2\omega_z}
  582           \Big|i{{\omega\chi^{(2)}_{zxx} L}\over{2 n_{{2\omega}_z}} c}
  583           A^y_{\omega}{}^2(0)\exp(i\Delta k L/2)\sinc(\Delta k L/2)\Big|^2\cr
  584         &=\varepsilon_0
  585           {{\omega^2 L^2}\over{8 n_{{2\omega}_z}} c}
  586           |\chi^{(2)}_{zxx}|^2
  587           |A^y_{\omega}(0)|^4\sinc^2(\Delta k L/2)\cr
  588         &=\bigg\{{\rm Use\ }|A^y_{\omega}(0)|^2
  589            ={{2 I_{\omega}(0)}\over{\varepsilon_0 c n_{\omega_y}}}\bigg\}\cr
  590         &=\varepsilon_0
  591           {{\omega^2 L^2}\over{8 n_{{2\omega}_z}} c}
  592           |\chi^{(2)}_{zxx}|^2
  593           {{4 I^2_{\omega}(0)}\over{\varepsilon^2_0 c^2 n^2_{\omega_y}}}
  594           \sinc^2(\Delta k L/2)\cr
  595         &={{\omega^2 L^2}\over{2 \varepsilon_0 c^3}}
  596           {{|\chi^{(2)}_{zxx}(-2\omega;\omega,\omega)|^2}
  597             \over{n_{{2\omega}_z} n^2_{\omega_y}}}
  598           I^2_{\omega}(0)\sinc^2(\Delta k L/2),\cr
  599     }
  600   $$
  601   i.~e.~with the conversion efficiency
  602   $$
  603     {{I_{2\omega}(L)}\over{I_{\omega}(0)}}
  604       ={{\omega^2 L^2}\over{2 \varepsilon_0 c^3}}
  605        {{|\chi^{(2)}_{zxx}(-2\omega;\omega,\omega)|^2}
  606         \over{n_{{2\omega}_z} n^2_{\omega_y}}}
  607        I_{\omega}(0)\sinc^2(\Delta k L/2).
  608   $$
  609   \vfill\eject
  610   
  611   %\bigskip
  612   \centerline{\epsfxsize=110mm\epsfbox{neguniax.eps}}
  613   \medskip
  614   \noindent{Figure 3. Ordinary and extraordinary refractive indices
  615   of a negative uniaxial crystal as function of vacuum wavelength of
  616   the light, in the case of normal dispersion. Phase matching between
  617   the pump and second harmonic wave is obtained whenever
  618   $n_{\omega_y}\equiv n_{\rm O}(\omega)=n_{\rm E}(2\omega)\equiv
  619   n_{2\omega_z}$.}
  620   \medskip
  621   
  622   \bigskip
  623   \centerline{\epsfxsize=110mm\epsfbox{qpm.eps}}
  624   \medskip
  625   \noindent{Figure 4. Conversion efficiency $I_{2\omega}(L)/I_{\omega}(0)$
  626   as function of normalized crystal length $\Delta k L/2$ when the material
  627   properties are periodically reversed, with a half-period of $L_{\rm c}$.}
  628   \medskip
  629   \vfill\eject
  630   
  631   \section{Optical Kerr-effect - Field corrected refractive index}
  632   As a start, we assume a monochromatic optical wave (containing forward
  633   and/or backward propagating components) polarized in the $xy$-plane,
  634   $$
  635     {\bf E}(z,t)=\Re[{\bf E}_{\omega}(z)\exp(-i\omega t)]\in{\Bbb R}^3,
  636   $$
  637   with all spatial variation of the field contained in
  638   $$
  639     {\bf E}_{\omega}(z)={\bf e}_x E^x_{\omega}(z)
  640       +{\bf e}_y E^y_{\omega}(z)\in{\Bbb C}^3.
  641   $$
  642   
  643   \subsection{The optical interaction}
  644   Optical Kerr-effect is in isotropic media described by the third order
  645   susceptibility\footnote{${}^5$}{Again, keep in mind that in the convention
  646   of Butcher and Cotter, the frequency arguments to the right of the semicolon
  647   may be writen in arbitrary order, hence we may in an equal description
  648   instead use
  649   $$\chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,-\omega,\omega)$$
  650   or
  651   $$\chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;-\omega,\omega,\omega)$$
  652   for this description of the third order interaction between light and matter.}
  653   $$
  654     \chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega).
  655   $$
  656   
  657   \subsection{Symmetries of the medium}
  658   The general set of nonzero components of $\chi^{(3)}_{\mu\alpha\beta\gamma}$
  659   for isotropic media are from Appendix A3.3 of Butcher and Cotters book
  660   given as
  661   $$
  662     \eqalign{
  663     \chi^{(3)}_{xxxx}&=\chi^{(3)}_{yyyy}=\chi^{(3)}_{zzzz},\cr
  664     \chi^{(3)}_{yyzz}&=\chi^{(3)}_{zzyy}
  665       =\chi^{(3)}_{zzxx}=\chi^{(3)}_{xxzz}
  666       =\chi^{(3)}_{xxyy}=\chi^{(3)}_{yyxx}\cr
  667     \chi^{(3)}_{yzyz}&=\chi^{(3)}_{zyzy}
  668       =\chi^{(3)}_{zxzx}=\chi^{(3)}_{xzxz}
  669       =\chi^{(3)}_{xyxy}=\chi^{(3)}_{yxyx}\cr
  670     \chi^{(3)}_{yzzy}&=\chi^{(3)}_{zyyz}
  671       =\chi^{(3)}_{zxxz}=\chi^{(3)}_{xzzx}
  672       =\chi^{(3)}_{xyyx}=\chi^{(3)}_{yxxy}\cr
  673     }\eqno{(4)}
  674   $$
  675   with
  676   $$
  677     \chi^{(3)}_{xxxx}=\chi^{(3)}_{xxyy}+\chi^{(3)}_{xyxy}+\chi^{(3)}_{xyyx},
  678   $$
  679   i.~e.~a general set of 21 elements, of which only 3 are independent.
  680   
  681   \subsection{Additional symmetries}
  682   By applying the intrinsic permutation symmetry in the middle indices
  683   for optical Kerr-effect, one generally has
  684   $$
  685     \chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega)
  686       =\chi^{(3)}_{\mu\beta\alpha\gamma}(-\omega;\omega,\omega,-\omega),
  687   $$
  688   which hence slightly reduce the set~(4) to still 21 nonzero elements,
  689   but of which now only two are independent.
  690   For a beam polarized in the $xy$-plane, the elements of interest are
  691   only those which only contain $x$ or $y$ in the indices, i.~e.~the subset
  692   $$
  693     \chi^{(3)}_{xxxx}=\chi^{(3)}_{yyyy},\quad
  694     \chi^{(3)}_{xxyy}=\chi^{(3)}_{yyxx}
  695     =\bigg\{\matrix{{\rm intr.\ perm.\ symm.}\cr
  696                (\alpha,\omega)\rightleftharpoons(\beta,\omega)\cr}\bigg\}=
  697     \chi^{(3)}_{xyxy}=\chi^{(3)}_{yxyx},\quad
  698     \chi^{(3)}_{xyyx}=\chi^{(3)}_{yxxy},
  699   $$
  700   with
  701   $$
  702     \chi^{(3)}_{xxxx}=\chi^{(3)}_{xxyy}+\chi^{(3)}_{xyxy}+\chi^{(3)}_{xyyx},
  703   $$
  704   i.~e.~a set of eight elements, of which only two are independent.
  705   
  706   \subsection{The polarization density}
  707   The degeneracy factor $K(-\omega;\omega,\omega,-\omega)$ is calculated as
  708   $$
  709     K(-\omega;\omega,\omega,-\omega)=2^{l+m-n}p=2^{1+0-3}3=3/4.
  710   $$
  711   From the reduced set of nonzero susceptibilities for the beam polarized in
  712   the $xy$-plane, and by using the calculated
  713   value of the degeneracy factor in the convention of Butcher and Cotter,
  714   we hence have the third order electric polarization density at
  715   $\omega_{\sigma}=\omega$ given as ${\bf P}^{(n)}({\bf r},t)=
  716   \Re[{\bf P}^{(n)}_{\omega}\exp(-i\omega t)]$, with
  717   $$
  718     \eqalign{
  719       {\bf P}^{(3)}_{\omega}
  720       &=\sum_{\mu}{\bf e}_{\mu}(P^{(3)}_{\omega})_{\mu}\cr
  721       &=\{{\rm Using\ the\ convention\ of\ Butcher\ and\ Cotter}\}\cr
  722       &=\sum_{\mu}{\bf e}_{\mu}
  723         \bigg[\varepsilon_0{{3}\over{4}}\sum_{\alpha}\sum_{\beta}\sum_{\gamma}
  724          \chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega)
  725          (E_{\omega})_{\alpha}(E_{\omega})_{\beta}(E_{-\omega})_{\gamma}\bigg]\cr
  726       &=\{{\rm Evaluate\ the\ sums\ over\ } (x,y,z)
  727           {\rm\ for\ field\ polarized\ in\ the\ }xy{\rm\ plane}\}\cr
  728       &=\varepsilon_0{{3}\over{4}}\{
  729         {\bf e}_x[
  730           \chi^{(3)}_{xxxx} E^x_{\omega} E^x_{\omega} E^x_{-\omega}
  731           +\chi^{(3)}_{xyyx} E^y_{\omega} E^y_{\omega} E^x_{-\omega}
  732           +\chi^{(3)}_{xyxy} E^y_{\omega} E^x_{\omega} E^y_{-\omega}
  733           +\chi^{(3)}_{xxyy} E^x_{\omega} E^y_{\omega} E^y_{-\omega}]\cr
  734        &\qquad\quad
  735        +{\bf e}_y[
  736           \chi^{(3)}_{yyyy} E^y_{\omega} E^y_{\omega} E^y_{-\omega}
  737           +\chi^{(3)}_{yxxy} E^x_{\omega} E^x_{\omega} E^y_{-\omega}
  738           +\chi^{(3)}_{yxyx} E^x_{\omega} E^y_{\omega} E^x_{-\omega}
  739           +\chi^{(3)}_{yyxx} E^y_{\omega} E^x_{\omega} E^x_{-\omega}]\}\cr
  740       &=\{{\rm Make\ use\ of\ }{\bf E}_{-\omega}={\bf E}^*_{\omega}
  741           {\rm\ and\ relations\ }\chi^{(3)}_{xxyy}=\chi^{(3)}_{yyxx},
  742           {\rm\ etc.}\}\cr
  743       &=\varepsilon_0{{3}\over{4}}\{
  744         {\bf e}_x[
  745           \chi^{(3)}_{xxxx} E^x_{\omega} |E^x_{\omega}|^2
  746           +\chi^{(3)}_{xyyx} E^y_{\omega}{}^2 E^{x*}_{\omega}
  747           +\chi^{(3)}_{xyxy} |E^y_{\omega}|^2 E^x_{\omega}
  748           +\chi^{(3)}_{xxyy} E^x_{\omega} |E^y_{\omega}|^2]\cr
  749        &\qquad\quad
  750        +{\bf e}_y[
  751           \chi^{(3)}_{xxxx} E^y_{\omega} |E^y_{\omega}|^2
  752           +\chi^{(3)}_{xyyx} E^x_{\omega}{}^2 E^{y*}_{\omega}
  753           +\chi^{(3)}_{xyxy} |E^x_{\omega}|^2 E^y_{\omega}
  754           +\chi^{(3)}_{xxyy} E^y_{\omega} |E^x_{\omega}|^2]\}\cr
  755       &=\{{\rm Make\ use\ of\ the\ intrinsic\ permutation\ symmetry}\}\cr
  756       &=\varepsilon_0{{3}\over{4}}\{
  757         {\bf e}_x[
  758           (\chi^{(3)}_{xxxx} |E^x_{\omega}|^2
  759             +2\chi^{(3)}_{xxyy} |E^y_{\omega}|^2) E^x_{\omega}
  760           +(\chi^{(3)}_{xxxx}-2\chi^{(3)}_{xxyy})
  761            E^y_{\omega}{}^2 E^{x*}_{\omega}\cr
  762        &\qquad\quad
  763         {\bf e}_y[
  764           (\chi^{(3)}_{xxxx} |E^y_{\omega}|^2
  765             +2\chi^{(3)}_{xxyy} |E^x_{\omega}|^2) E^y_{\omega}
  766           +(\chi^{(3)}_{xxxx}-2\chi^{(3)}_{xxyy})
  767            E^x_{\omega}{}^2 E^{y*}_{\omega}.\cr
  768     }
  769   $$
  770   For the optical field being linearly polarized, say in the $x$-direction,
  771   the expression for the polarization density is significantly simplified,
  772   to yield
  773   $$
  774     {\bf P}^{(3)}_{\omega}=\varepsilon_0(3/4){\bf e}_x
  775       \chi^{(3)}_{xxxx} |E^x_{\omega}|^2 E^x_{\omega},
  776   $$
  777   i.~e.~taking a form that can be interpreted as an intensity-dependent
  778   ($\sim|E^x_{\omega}|^2$) contribution to the refractive index
  779   (cf.~Butcher and Cotter \S 6.3.1).
  780   
  781   \subsection{The wave equation -- Time independent case}
  782   In this example, we consider continuous wave
  783   propagation\footnote{${}^6$}{That is to say, a time independent problem
  784   with the temporal envelope of the electrical field being constant in time.}
  785   in optical Kerr-media, using
  786   light polarized in the $x$-direction and propagating along the positive
  787   direction of the $z$-axis,
  788   $$
  789     {\bf E}({\bf r},t)=\Re[{\bf E}_{\omega}(z)\exp(-i\omega t)],
  790     \qquad{\bf E}_{\omega}(z)={\bf A}_{\omega}(z)\exp(ikz)
  791                              ={\bf e}_x A^x_{\omega}(z)\exp(ikz),
  792   $$
  793   where, as previously, $k=\omega n_0/c$.
  794   From material handed out during the third lecture (notes on the Butcher
  795   and Cotter convention), the nonlinear polarization density for $x$-polarized
  796   light is given as ${\bf P}^{({\rm NL})}_{\omega}={\bf P}^{(3)}_{\omega}$, with
  797   $$
  798     \eqalign{
  799       {\bf P}^{(3)}_{\omega}
  800         &=\varepsilon_0(3/4){\bf e}_x\chi^{(3)}_{xxxx}
  801           (-\omega;\omega,\omega,-\omega)
  802           |E^x_{\omega}|^2 E^x_{\omega}\cr
  803         &=\varepsilon_0(3/4)\chi^{(3)}_{xxxx}
  804           |{\bf E}_{\omega}|^2 {\bf E}_{\omega}\cr
  805         &=\varepsilon_0(3/4)\chi^{(3)}_{xxxx}
  806           |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\exp(ikz),\cr
  807     }
  808   $$
  809   and the time independent wave equation for the field envelope
  810   ${\bf A}_{\omega}$, using Eq.~(6), becomes
  811   $$
  812     \eqalign{
  813         {{\partial}\over{\partial z}}{\bf A}_{\omega}
  814            &=i{{\mu_0\omega^2}\over{2k}}
  815              \underbrace{\varepsilon_0(3/4)\chi^{(3)}_{xxxx}
  816              |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\exp(ikz)}_{
  817                ={\bf P}^{({\rm NL})}_{\omega}(z)}\exp(-ikz)\cr
  818            &=i{{3\omega^2}\over{8c^2k}}\chi^{(3)}_{xxxx}
  819              |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\cr
  820            &=\{{\rm since\ }k=\omega n_0(\omega)/c\}\cr
  821            &=i{{3\omega}\over{8cn_0}}
  822              \chi^{(3)}_{xxxx}
  823              |{\bf A}_{\omega}|^2 {\bf A}_{\omega},\cr
  824     }
  825   $$
  826   or, equivalently, in its scalar form
  827   $$
  828     {{\partial}\over{\partial z}}A^x_{\omega}
  829       =i{{3\omega}\over{8cn_0}}\chi^{(3)}_{xxxx}
  830          |A^x_{\omega}|^2 A^x_{\omega}.
  831   $$
  832   
  833   \subsection{Boundary conditions -- Time independent case}
  834   For this special case of unidirectional wave propagation, the
  835   boundary condition is simply a known optical field at the input,
  836   $$A^x_{\omega}(0)=\{{\rm known}\}.$$
  837   
  838   \subsection{Solving the wave equation -- Time independent case}
  839   If the medium of interest now is analyzed at an angular frequency far
  840   from any resonance, we may look for solutions to this equation with
  841   $|{\bf A}_{\omega}(z)|$ being constant (for a lossless medium).
  842   For such a case it is straightforward to integrate the final wave
  843   equation to yield the general solution
  844   $$
  845     {\bf A}_{\omega}(z)={\bf A}_{\omega}(0)
  846       \exp[i{{3\omega}\over{8cn_0}}\chi^{(3)}_{xxxx}|{\bf A}_{\omega}(0)|^2 z],
  847   $$
  848   or, again equivalently, in the scalar form
  849   $$
  850     A^x_{\omega}(z)=
  851       A^x_{\omega}(0)
  852       \exp[i{{3\omega}\over{8cn_0}}\chi^{(3)}_{xxxx}|A^x_{\omega}(0)|^2 z],
  853   $$
  854   which hence gives the solution for the real-valued electric field
  855   ${\bf E}({\bf r},t)$ as
  856   $$
  857     \eqalign{
  858       {\bf E}({\bf r},t)&=\Re[{\bf E}_{\omega}(z)\exp(-i\omega t)]\cr
  859         &=\Re\{{\bf A}_{\omega}(z)\exp[i(kz-\omega t)]\}\cr
  860         &=\Re\{{\bf A}_{\omega}(0)
  861             \exp[i(\underbrace{{{\omega n_0}\over{c}}z
  862              +{{3\omega}\over{8cn_0}}\chi^{(3)}_{xxxx}|A^x_{\omega}(0)|^2 z}_{
  863                 \equiv k_{\rm eff}z}
  864                  -\omega t)]\}.\cr
  865     }
  866   $$
  867   From this solution, one immediately finds that the wave propagates
  868   with an effective propagation constant
  869   $$
  870     k_{\rm eff}={{\omega}\over{c}}
  871       [n_0+{{3}\over{8n_0}}\chi^{(3)}_{xxxx}|A^x_{\omega}(0)|^2],
  872   $$
  873   that is to say, experiencing the intensity dependent refractive index
  874   $$
  875     \eqalign{
  876       n_{\rm eff}
  877         &=n_0+{{3}\over{8n_0}}\chi^{(3)}_{xxxx}|A^x_{\omega}(0)|^2\cr
  878         &=n_0+ n_2 |A^x_{\omega}(0)|^2,\cr
  879     }
  880   $$
  881   with
  882   $$n_2={{3}\over{8n_0}}\chi^{(3)}_{xxxx}.$$
  883   
  884   \bye
  885   

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