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    1   % File: nlopt/lect2/lect2.tex [pure TeX code]
    2   % Last change: January 10, 2003
    3   %
    4   % Lecture No 2 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   \font\ninerm=cmr9
   11   \font\twelvesc=cmcsc10
   12   \def\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm
   13     \topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt
   14     \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
   15       \else\hfill\fi}
   16     \def\rightheadline{\tenrm{\it Lecture notes #1}
   17       \hfil{\it Nonlinear Optics 5A5513 (2003)}}
   18     \def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)}
   19       \hfil{\it Lecture notes #1}}
   20     \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps}
   21     \vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}}
   22     \hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1}
   23     \vskip 24pt\noindent}
   24   \def\section #1 {\medskip\goodbreak\noindent{\bf #1}
   25     \par\nobreak\smallskip\noindent}
   26   \def\subsection #1 {\smallskip\goodbreak\noindent{\it #1}
   27     \par\nobreak\smallskip\noindent}
   28   
   29   \lecture{2}
   30   \section{Nonlinear polarization density}
   31   From the introductory perturbation analysis of the all-classical anharmonic
   32   oscillator in the previous lecture, we now {\sl a priori} know that
   33   it is possible to express the electric polarization density as a power
   34   series in the electric field of the optical wave.
   35   
   36   Loosely formulated, the electric polarization density in complex notation
   37   can be taken as the series
   38   $$
   39     \eqalign{
   40     P_{\mu}(\omega_{\sigma})=\varepsilon_0[
   41       \underbrace{\chi^{(1)}_{\mu\alpha}(-\omega_{\sigma};\omega_{\sigma})
   42         E_{\alpha}(\omega_{\sigma})}_{
   43           \sim P^{(1)}_{\mu}(\omega_{\sigma})}
   44        &+\underbrace{\chi^{(2)}_{\mu\alpha\beta}
   45         (-\omega_{\sigma};\omega_1,\omega_2)
   46         E_{\alpha}(\omega_1)E_{\beta}(\omega_2)}_{
   47           \sim P^{(2)}_{\mu}(\omega_{\sigma}),
   48           \ \omega_{\sigma}=\omega_1+\omega_2}\cr
   49       &+\underbrace{\chi^{(3)}_{\mu\alpha\beta\gamma}
   50         (-\omega_{\sigma};\omega_1,\omega_2,\omega_3)
   51         E_{\alpha}(\omega_1)E_{\beta}(\omega_2)E_{\gamma}(\omega_3)}_{
   52           \sim P^{(3)}_{\mu}(\omega_{\sigma}),
   53           \ \omega_{\sigma}=\omega_1+\omega_2+\omega_3}+\ldots\,]\cr
   54   }
   55   $$
   56   where we adopted Einstein's convention of summation for terms with repeated
   57   subscripts. (A more formal formulation of the polarization density will be
   58   described later.)
   59   
   60   \section{Symmetries in nonlinear optics}
   61   There are essentially four classes of symmetries that we will
   62   encounter in this course:
   63   \smallskip
   64   
   65   \item{$[1]$}{Intrinsic permutation symmetry:
   66   $$
   67     \eqalign{
   68       \chi^{(3)}_{\mu\alpha\beta\gamma}
   69        (-\omega_{\sigma};\omega_1,\omega_2,\omega_3)
   70         &=\chi^{(3)}_{\mu\beta\alpha\gamma}
   71           (-\omega_{\sigma};\omega_2,\omega_1,\omega_3)\cr
   72         &=\chi^{(3)}_{\mu\beta\gamma\alpha}
   73           (-\omega_{\sigma};\omega_2,\omega_3,\omega_1)\cr
   74         &=\chi^{(3)}_{\mu\gamma\beta\alpha}
   75           (-\omega_{\sigma};\omega_3,\omega_2,\omega_1),\cr
   76     }
   77   $$
   78   that is to say, invariance under the $n!$ possible permutations of
   79   $(\alpha_k,\omega_k)$, $k=1,2,\ldots,n$. This principle applies
   80   generally to resonant as well as nonresonant media. ({\sl Described
   81   in this lecture.})}
   82   \item{$[2]$}{Overall permutation symmetry:
   83   $$
   84     \eqalign{
   85       \chi^{(3)}_{\mu\alpha\beta\gamma}
   86        (-\omega_{\sigma};\omega_1,\omega_2,\omega_3)
   87         &=\chi^{(3)}_{\alpha\mu\beta\gamma}
   88           (\omega_1;-\omega_{\sigma},\omega_2,\omega_3)\cr
   89         &=\chi^{(3)}_{\alpha\beta\mu\gamma}
   90           (\omega_1;\omega_2,-\omega_{\sigma},\omega_3)\cr
   91         &=\chi^{(3)}_{\alpha\beta\gamma\mu}
   92           (\omega_1;\omega_2,\omega_3,-\omega_{\sigma}),\cr
   93     }
   94   $$
   95   that is to say, invariance under the $(n+1)!$ possible permutations of
   96   $(\mu,-\omega_{\sigma})$, $(\alpha_k,\omega_k)$, $k=1,2,\ldots,n$.
   97   This principle applies to nonresonant media, where all optical frequencies
   98   appearing in the formula for the susceptibility are removed far from the
   99   transition frequencies of the medium. ({\sl Described in lecture~4.})}
  100   \item{$[3]$}{Kleinman symmetry:
  101   $$
  102     \eqalign{
  103       \chi^{(3)}_{\mu\alpha\beta\gamma}
  104        (-\omega_{\sigma};\omega_1,\omega_2,\omega_3)
  105         &=\chi^{(3)}_{\alpha\mu\beta\gamma}
  106           (-\omega_{\sigma};\omega_1,\omega_2,\omega_3)\cr
  107         &=\chi^{(3)}_{\alpha\beta\mu\gamma}
  108           (-\omega_{\sigma};\omega_1,\omega_2,\omega_3)\cr
  109         &=\chi^{(3)}_{\alpha\beta\gamma\mu}
  110           (-\omega_{\sigma};\omega_1,\omega_2,\omega_3),\cr
  111     }
  112   $$
  113   that is to say, invariance under the $(n+1)!$ possible permutations of
  114   the subscripts $\mu,\alpha_1,\ldots,\alpha_n$.
  115   This principle is a consequence of the overall permutation symmetry, and
  116   applies in the low-frequency limit of nonresonant media.}
  117   \item{$[4]$}{Spatial symmetries, given by the point symmetry class
  118   of the medium. ({\sl Described in lecture~6.})}
  119   \smallskip
  120   
  121   \section{Conditions for observing nonlinear optical interactions}
  122   Loosely formulated, a nonlinear response between light and matter
  123   depends on one of two key indgredients: either there is a resonance
  124   between the light wave and some natural oscillation mode of the
  125   medium, or the light is sufficiently intense.
  126   {\sl Direct resonance} can occur in isolated intervals of the electromagnetic
  127   spectrum at
  128   \item{$\bullet$}{ultraviolet and visible frequencies ($10^{15}$ ${\rm s}^{-1}$)
  129     where the oscillator corresponds to an electronic transition of the medium,}
  130   \item{$\bullet$}{infrared ($10^{13}$ ${\rm s}^{-1}$), where the medium has
  131     vibrational modes, and}
  132   \item{$\bullet$}{the far infrared-microwave range ($10^{11}$ ${\rm s}^{-1}$),
  133     where there are rotational modes.}
  134   These interactions are also called {\sl one-photon processes}, and
  135   are schematically illustrated in Fig.~3.
  136   \medskip
  137   \centerline{\epsfxsize=70mm\epsfbox{../images/onephot/onephot.1}}
  138   \medskip
  139   \centerline{Figure 3. Transition scheme of the one-photon process.}
  140   \medskip
  141   \noindent
  142   The lower frequency modes can be excited at optical frequencies
  143   ($10^{15}$ ${\rm s}^{-1}$) through indirect resonant processes in which
  144   the difference in frequencies and wave vectors of two light waves,
  145   called the pump and Stokes wave, respectively, matches the frequency
  146   and wave vector of one of these lower frequency modes.
  147   These three-frequency interactions are sometime called {\sl two-photon
  148   processes}. In the case where there ``lower frequency'' mode is an
  149   electronic transition or in the vibrational range (in which case the
  150   Stokes frequency can be of the same order of magnitude as that of
  151   the light wave), this process is called {\sl Raman scattering}.
  152   \medskip
  153   \centerline{\epsfxsize=70mm\epsfbox{../images/raman/sraman.1}
  154     \epsfxsize=70mm\epsfbox{../images/raman/asraman.1}}
  155   \medskip
  156   \centerline{Figure 4. Transition schemes of stimulated Raman Scattering.}
  157   \medskip
  158   \noindent
  159   The stimulated Raman scattering is essentially a two-photon process in which
  160   one photon at $\omega_1$ is absorbed and one photon at $\omega_2$ is emitted,
  161   while the material makes a transition from the initial state $|a\rangle$
  162   to the final state $|c\rangle$, as shown in Fig.~4.
  163   Energy conservation requires the Raman resonance frequency
  164   (electronic or vibrational) to satisfy
  165   $\hbar\Omega_{ca}=\hbar(\omega_1-\omega_2)$,
  166   and hence we may classify the Stokes and Anti-Stokes transitions as
  167   $$
  168     \eqalign{
  169       \hbar\Omega_{ca}>0\qquad&\Leftrightarrow
  170         \qquad{\textstyle{\rm Stokes\ Raman}}\cr
  171       \hbar\Omega_{ca}<0\qquad&\Leftrightarrow
  172         \qquad{\textstyle{\rm Anti-Stokes\ Raman}}\cr
  173     }
  174   $$
  175   When the lower frequency mode instead is an {\sl acoustic} mode of
  176   the material, the process is instead called {\sl Brillouin scattering}.
  177   
  178   As will be shown explicitly later on in the course, optical resonance
  179   with transitions of the material is an important tool for ``boosting up''
  180   the nonlinearities, with enhanced possibilities of applications.
  181   However, at single-photon resonance we have a strong absorption
  182   at the frequency of the optical field, and in most cases this
  183   is a non-desirable effect, since it decreases the optical intensity,
  184   even though it meanwhile also enhances the nonlinearity of the material.
  185   
  186   However, by instead exploiting the two-photon resonance, the desired
  187   nonlinearity can be significantly enhanced whilst at the same time
  188   the competing absorption process can be minimized by avoiding
  189   coincidences between optical frequencies and {\sl single-photon} resonances.
  190   
  191   One general drawback with the resonant enhancement is that the response
  192   time of the polarization density of the material is slowed down, affecting
  193   applications such as optical switching or modulation, where speed
  194   is of importance.
  195   
  196   \section{Phenomenological description of the susceptibility tensors}
  197   Before entering the full quantum-mechanical formalism, we will
  198   assume the medium to possess a temporal response described by
  199   {\sl time response functions} $R(t)$.
  200   
  201   The very first step in the analysis is to express the electric
  202   polarization density, which in classical electrodynamical terms
  203   is expressed as the sum over all $M$ electric charges $q_k$ in a small
  204   volume~$V$ centered at ${\bf r}$,
  205   $$
  206     {\bf P}({\bf r},t)={{1}\over{V}}\sum^{M}_{k=1} q_k {\bf r}_k(t),
  207   $$
  208   as a series expansion
  209   $$
  210     {\bf P}({\bf r},t)={\bf P}^{(0)}({\bf r},t)+{\bf P}^{(1)}({\bf r},t)
  211                         +{\bf P}^{(2)}({\bf r},t)+\ldots
  212                         +{\bf P}^{(n)}({\bf r},t)+\ldots,
  213   $$
  214   where ${\bf P}^{(1)}({\bf r},t)$ is linear in the electric field,
  215   ${\bf P}^{(2)}({\bf r},t)$ is quadratic in the electric field, etc.
  216   The field-independent ${\bf P}^{(0)}({\bf r},t)$ corresponds to
  217   eventually appearing static polarization of the medium.
  218   It should be emphasized that any of these terms may be linear as
  219   well as nonlinear functions of the electric field of the optical
  220   wave; i.~e.~an externally applied static electric field may
  221   together with the electric field of the light interact through,
  222   for example, ${\bf P}^{(3)}({\bf r},t)$ to induce an electric
  223   polarization that is linear in the optical field.
  224   
  225   \medskip
  226   \centerline{\epsfxsize=85mm\epsfbox{../images/respfunc/respfunc.1}}
  227   \centerline{Figure 9. Schematic form of a possible response function
  228     in time domain.}
  229   \medskip
  230   
  231   In the analysis that now is to follow, we will a priori assume
  232   that it is possible to express the polarization density as a
  233   perturbation series. Later on, we will show how each of the
  234   terms can be derived from a more stringent basis, where it
  235   also will be stated which conditions that must hold in order
  236   to apply perturbation theory.
  237   
  238   \section{Linear polarization response function}
  239   Since ${\bf P}^{(1)}({\bf r},t)$ is taken as linear in the electric field,
  240   we may express the linear polarization density of the medium as
  241   being related to the optical field as
  242   $$
  243     P^{(1)}_{\mu}({\bf r},t)=\varepsilon_0
  244       \int^{\infty}_{-\infty} T^{(1)}_{\mu\alpha}(t;\tau)
  245         E_{\alpha}({\bf r},\tau)\,d\tau,
  246   \eqno{(7)}
  247   $$
  248   where $T^{(1)}_{\mu\alpha}(t;\tau)$ is a rank-two tensor that
  249   weights all contributions in time from the electric field of
  250   the light.
  251   A few required properties of $T^{(1)}_{\mu\alpha}(t;\tau)$ can
  252   immediately be stated:
  253   \smallskip
  254   
  255   \item{$\bullet$}{{\sl Causality.} We require that no optically
  256     induced contribution can occur {\sl before} the field is applied,
  257     i.~e.~$T^{(1)}_{\mu\alpha}(t;\tau)=0$ for $t\le\tau$.}
  258   \smallskip
  259   
  260   \item{$\bullet$}{{\sl Time invariance.} Under most circumstances,
  261     we may in addition assume that the material parameters are
  262     constant in time, such that $P^{(1)}_{\mu}({\bf r},t')$
  263     is identical to the polarisation as induced by the time-displaced
  264     electric field $E_{\alpha}({\bf r},t')$.}
  265   \smallskip
  266   \noindent The second of these properties is essentially a manifestation
  267   of an adiabatically following change of the carrier wave of the
  268   optical field.
  269   
  270   By using Eq.~(7), the time invariance of the constitutive relation
  271   gives that
  272   $$
  273     \eqalign{
  274       P^{(1)}_{\mu}({\bf r},t+t_0)
  275         &=\varepsilon_0\int^{\infty}_{-\infty}
  276            T^{(1)}_{\mu\alpha}(t+t_0;\tau) E_{\alpha}({\bf r},\tau)\,d\tau\cr
  277         &=\big\{{\rm Should\ equal\ to\ polarization\ induced\ by}
  278                 \ E_{\alpha}({\bf r},\tau+t_0)\big\}\cr
  279         &=\varepsilon_0\int^{\infty}_{-\infty}
  280            T^{(1)}_{\mu\alpha}(t;\tau) E_{\alpha}({\bf r},\tau+t_0)\,d\tau\cr
  281         &=\big\{\tau'=\tau+t_0\big\}\cr
  282         &=\varepsilon_0\int^{\infty}_{-\infty}
  283            T^{(1)}_{\mu\alpha}(t;\tau'-t_0) E_{\alpha}({\bf r},\tau')\,d\tau',\cr
  284     }
  285   $$
  286   from which we, by changing the ``dummy'' variable of integration
  287   back to $\tau$, obtain the relation
  288   $$
  289     T^{(1)}_{\mu\alpha}(t+t_0;\tau)=T^{(1)}_{\mu\alpha}(t;\tau-t_0).
  290   $$
  291   In particular, by setting $t=0$ and replacing the arbitrary time
  292   displacement $t_0$ by $t$, one finds that the response of the
  293   medium depends only of the time difference $\tau-t$,
  294   $$
  295     \eqalign{
  296       T^{(1)}_{\mu\alpha}(t;\tau)
  297        &=T^{(1)}_{\mu\alpha}(0;\tau-t)\cr
  298        &=R^{(1)}_{\mu\alpha}(\tau-t)\cr
  299     }
  300   $$
  301   where we defined the linear polarization response function
  302   $R^{(1)}_{\mu\alpha}(\tau-t)$, being a rank-two tensor depending
  303   only on the time difference $\tau-t$.
  304   
  305   To summarize, the linear contribution to the electric polarization
  306   density is given in terms of the linear polarization response
  307   function as
  308   $$
  309     \eqalign{
  310       P^{(1)}_{\mu}({\bf r},t)
  311         &=\varepsilon_0\int^{\infty}_{-\infty}
  312           R^{(1)}_{\mu\alpha}(t-\tau) E_{\alpha}({\bf r},\tau)\,d\tau,\cr
  313         &=\varepsilon_0\int^{\infty}_{-\infty}
  314           R^{(1)}_{\mu\alpha}(\tau') E_{\alpha}({\bf r},t-\tau')\,d\tau',\cr
  315       }\eqno{(8)}
  316   $$
  317   and the causality condition for the linear polarization
  318   response function requires that
  319   $$
  320     R^{(1)}_{\mu\alpha}(\tau-t)=0,\qquad t\le\tau,
  321   $$
  322   and in addition, since the relation~(8) is to hold for arbitrary
  323   time evolution of the electrical field, and since the polarization
  324   density $P^{(1)}_{\mu}({\bf r},t)$ and the electric field
  325   $E^{(1)}_{\alpha}({\bf r},t)$ both are real-valued quantities,
  326   we also require the polarization response function
  327   $R^{(1)}_{\mu\alpha}(\tau-t)$ to be a real-valued function.
  328   
  329   \section{Quadratic polarization response function}
  330   The second-order, quadratic polarization density can in similar to the
  331   linear one be phenomenologically be written as the sum of all
  332   infinitesimal previous contributions in time.
  333   In this case, we must though include the possibility that not all
  334   contributions origin in the same time scale, and hence the proper
  335   formulation of the second order polarization density is as a
  336   two-dimensional integral,
  337   $$
  338     P^{(2)}_{\mu}({\bf r},t)=\varepsilon_0
  339       \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  340         T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
  341         E_{\alpha}({\bf r},\tau_1) E_{\beta}({\bf r},\tau_2)
  342       \,d\tau_1\,d\tau_2.
  343   \eqno{(9)}
  344   $$
  345   The tensor $T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)$ uniquely
  346   determines the quadratic, second-order polarization of the medium.
  347   However, the tensor $T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)$
  348   is not itself unique. To see this, we may express the tensor
  349   as a sum of a symmetric part and an antisymmetric part,
  350   $$
  351     \eqalign{
  352       T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
  353         &={{1}\over{2}}
  354          \underbrace{
  355            \bigg[T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
  356             +T^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)\bigg]
  357          }_{\rm symmetric}
  358          +{{1}\over{2}}
  359          \underbrace{
  360            \bigg[T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
  361              -T^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)\bigg]
  362          }_{\rm antisymmetric}\cr
  363         &=\underbrace{S^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)}_{
  364             =S^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)}
  365             +\underbrace{A^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)}_{
  366             =-A^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)}\cr
  367     }
  368   $$
  369   As this form of the response function is inserted into the
  370   original expression (9) for the second order polarization density,
  371   we immediately find that it is left invariant under the interchange
  372   of the dummy variables $(\alpha,\tau_1)$ and $(\beta,\tau_2)$
  373   (since the integration is performed from minus to plus infinity
  374   in time). In particular, it from this follows that
  375   
  376   
  377   $$
  378     \eqalign{
  379       P^{(2)}_{\mu}({\bf r},t)&=\varepsilon_0
  380         \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  381           [S^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
  382               +A^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)]
  383           E_{\alpha}({\bf r},\tau_1) E_{\beta}({\bf r},\tau_2)
  384         \,d\tau_1\,d\tau_2\cr
  385       &=\varepsilon_0
  386         \underbrace{\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  387           S^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
  388           E_{\alpha}({\bf r},\tau_1) E_{\beta}({\bf r},\tau_2)
  389         \,d\tau_1\,d\tau_2}_{\equiv I_1}\cr&\qquad
  390       +\varepsilon_0
  391         \underbrace{\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  392           A^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
  393           E_{\alpha}({\bf r},\tau_1) E_{\beta}({\bf r},\tau_2)
  394         \,d\tau_1\,d\tau_2}_{\equiv I_2}\cr
  395       &=\varepsilon_0
  396         \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  397           S^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)
  398           E_{\alpha}({\bf r},\tau_1) E_{\beta}({\bf r},\tau_2)
  399         \,d\tau_1\,d\tau_2\cr&\qquad
  400       -\varepsilon_0
  401         \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  402           A^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)
  403           E_{\alpha}({\bf r},\tau_1) E_{\beta}({\bf r},\tau_2)
  404         \,d\tau_1\,d\tau_2\cr
  405       &=\varepsilon_0
  406         \underbrace{\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  407           S^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)
  408           E_{\beta}({\bf r},\tau_2) E_{\alpha}({\bf r},\tau_1)
  409         \,d\tau_1\,d\tau_2}_{=I_1}\cr&\qquad
  410       -\varepsilon_0
  411         \underbrace{\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  412           A^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1)
  413           E_{\beta}({\bf r},\tau_2) E_{\alpha}({\bf r},\tau_1)
  414         \,d\tau_1\,d\tau_2}_{=I_2}\cr
  415     }
  416   $$
  417   i.~e.~the symmetric part satisfy the trivial identity $I_1=I_1$,
  418   while the antisymmetric part satisfy $I_2=-I_2$, i.~e.~the antisymmetric
  419   part does not contribute to the polarization density, and that we
  420   may set the antiymmetric part to zero, without imposing any
  421   constraint on the validity of the theory.
  422   
  423   The time response $T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)$
  424   is then unique and now chosen to be symmetric,
  425   $$
  426     T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
  427       =T^{(2)}_{\mu\beta\alpha}(t;\tau_2,\tau_1).
  428   $$
  429   This procedure of symmetrization may seem like an all theoretical
  430   contruction, somewhat out of focus of what lies ahead, but in fact
  431   it turns out to be an extremely useful property that we later
  432   on will exploit extensively.
  433   The previously described symmetric property will later on, for
  434   susceptibility tensors in the frequency domain, be denoted as
  435   the {\sl intrinsic permutation symmetry}, a general property
  436   that will hold irregardless of whether the nonlinear interaction
  437   under analysis is highly resonant or far from resonance.
  438   
  439   By again applying the arguments of time invariance, as previously
  440   for the linear response function, we find that
  441   $$
  442     T^{(2)}_{\mu\alpha\beta}(t+t_0;\tau_1,\tau_2)
  443       =T^{(2)}_{\mu\alpha\beta}(t;\tau_1-t_0,\tau_2-t_0)
  444   $$
  445   for all $t$, $\tau_1$, and $\tau_2$.
  446   Hence, by setting $t=0$ and then replacing th arbitrary time $t_0$
  447   by $t$, again as previously done for the linear case, one finds
  448   that $T^{(2)}(t;\tau_1,\tau_2)$ depends only on the two time
  449   differences $t-\tau_1$ and $t-\tau_2$.
  450   To make this fact explicit, we may hence write the response function as
  451   $$
  452     T^{(2)}_{\mu\alpha\beta}(t;\tau_1,\tau_2)
  453       =R^{(2)}_{\mu\alpha\beta}(t-\tau_1,t-\tau_2),
  454   $$
  455   giving the canonical form of the quadratic polarization density as
  456   $$
  457     \eqalign{
  458       P^{(2)}_{\mu}({\bf r},t)
  459       &=\varepsilon_0
  460         \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  461           R^{(2)}_{\mu\alpha\beta}(t-\tau_1,t-\tau_2)
  462           E_{\alpha}({\bf r},\tau_1) E_{\beta}({\bf r},\tau_2)
  463         \,d\tau_1\,d\tau_2\cr
  464       &=\varepsilon_0
  465         \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  466           R^{(2)}_{\mu\alpha\beta}(\tau'_1,\tau'_2)
  467           E_{\alpha}({\bf r},t-\tau'_1) E_{\beta}({\bf r},t-\tau'_2)
  468         \,d\tau'_1\,d\tau'_2.\cr
  469     }\eqno{(10)}
  470   $$
  471   The tensor $R^{(2)}_{\mu\alpha\beta}(\tau_1,\tau_2)$ is called
  472   the quadratic electric polarization response of the medium,
  473   and in similar with the linear response function, arguments
  474   of causality require the response function to be zero whenever
  475   $\tau_1$ and/or $\tau_2$ is negative. Similarly, the reality
  476   condition on $E_{\alpha}({\bf r},t)$ and $P_{\alpha}({\bf r},t)$
  477   requires that $R^{(2)}_{\mu\alpha\beta}(\tau_1,\tau_2)$ is a
  478   real-valued function.
  479   
  480   \section{Higher order polarization response functions}
  481   The $n$th order polarization density can in similar to the linear ($n=1$
  482   and quadratic ($n=2$) ones be written as
  483   $$
  484     \eqalign{
  485       P^{(n)}_{\mu}({\bf r},t)
  486       &=\varepsilon_0
  487         \int^{\infty}_{-\infty}\cdots\int^{\infty}_{-\infty}
  488           R^{(n)}_{\mu\alpha_1\cdots\alpha_n}
  489           (t-\tau_1,\ldots,t-\tau_n)
  490           E_{\alpha_1}({\bf r},\tau_1)\cdots E_{\alpha_n}({\bf r},\tau_n)
  491         \,d\tau_1\,\cdots\,d\tau_n\cr
  492       &=\varepsilon_0
  493         \int^{\infty}_{-\infty}\cdots\int^{\infty}_{-\infty}
  494           R^{(n)}_{\mu\alpha_1\cdots\alpha_n}
  495           (\tau'_1,\ldots,\tau'_n)
  496           E_{\alpha_1}({\bf r},t-\tau'_1)\cdots E_{\alpha_n}({\bf r},t-\tau'_n)
  497         \,d\tau'_1\,\cdots\,d\tau'_n,\cr
  498     }\eqno{(11)}
  499   $$
  500   where the $n$th order response function is a tensor of rank $n+1$,
  501   and a real-valued function of the $n$ parameters $\tau_1,\ldots,\tau_n$,
  502   vanishing whenever any $\tau_k<0$, $k=1,2,\ldots,n$, and with the
  503   {\sl intrinsic permutation symmetry} that it is left invariant under
  504   any of the $n!$ pairwise permutations
  505   of $(\alpha_1,\tau_1),\ldots,(\alpha_n,\tau_n)$.
  506   \bye
  507   

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