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    1   % File: nlopt/lect8/lect8.tex [pure TeX code]
    2   % Last change: February 24, 2003
    3   %
    4   % Lecture No 8 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   %
   25   % Define a handy macro for the list of symmetry operations
   26   % in Schoenflies notation for point-symmetry groups.
   27   %
   28   \newdimen\citemindent \citemindent=40pt
   29   \newdimen\citemleftskip \citemleftskip=60pt
   30   \def\citem[#1]{\smallbreak\noindent\hbox to 20pt{}%
   31     \hbox to\citemindent{#1\hfill}%
   32     \hangindent\citemleftskip\ignorespaces}
   33   \def\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm
   34     \topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt
   35     \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
   36       \else\hfill\fi}
   37     \def\rightheadline{\tenrm{\it Lecture notes #1}
   38       \hfil{\it Nonlinear Optics 5A5513 (2003)}}
   39     \def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)}
   40       \hfil{\it Lecture notes #1}}
   41     \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps}
   42     \vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}}
   43     \hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1}
   44     \vskip 24pt\noindent}
   45   \def\section #1 {\medskip\goodbreak\noindent{\bf #1}
   46     \par\nobreak\smallskip\noindent}
   47   \def\subsection #1 {\smallskip\goodbreak\noindent{\it #1}
   48     \par\nobreak\smallskip\noindent}
   49   
   50   \lecture{8}
   51   In this lecture, the electric polarisation density of the medium is
   52   finally inserted into Maxwell's equations, and the wave propagation
   53   properties of electromagnetic waves in nonlinear optical media is
   54   for the first time in this course analysed. As an example of wave
   55   propagation in nonlinear optical media, the optical Kerr effect is
   56   analysed for infinite plane continuous waves.
   57   \medskip
   58   
   59   \noindent The outline for this lecture is:
   60   \item{$\bullet$}{Maxwells equations (general electromagnetic wave
   61     propagation)}
   62   \item{$\bullet$}{Time dependent processes (envelopes slowly varying
   63     in space and time)}
   64   \item{$\bullet$}{Time independent processes (envelopes slowly varying
   65     in space but constant in time)}
   66   \item{$\bullet$}{Examples (optical Kerr-effect $\Leftrightarrow$
   67     $\chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega)$)}
   68   \medskip
   69   
   70   \section{Wave propagation in nonlinear media}
   71   \subsection{Maxwell's equations}
   72   The propagation of electromagnetic waves are, from a first principles
   73   approach, governed by the {\sl Maxwell's equations} (here listed in their
   74   real-valued form in SI units),
   75   $$
   76     \eqalign{
   77       \nabla\times {\bf E}({\bf r},t)
   78          &= -{{\partial{\bf B}({\bf r},t)}\over{\partial{t}}},\hskip 50pt
   79          ({\textstyle\rm Faraday's\ law})\cr
   80       \nabla\times{\bf H}({\bf r},t)
   81          &= {\bf J}({\bf r},t)
   82              + {{\partial{\bf D}({\bf r},t)}\over{\partial{t}}},\hskip 18.7pt
   83          ({\textstyle\rm Ampere's\ law})\cr
   84       \nabla\cdot{\bf D}({\bf r},t)&=\rho({\bf r},t),\cr
   85       \nabla\cdot{\bf B}({\bf r},t)&=0,\cr
   86     }
   87   $$
   88   where $\rho({\bf r},t)$ is the density of free charges,
   89   and ${\bf J}({\bf r},t)$ the corresponding current density of free charges.
   90   
   91   \subsection{Constitutive relations}
   92   The constitutive relations are in SI units formulated as
   93   $$
   94     \eqalign{
   95       {\bf D}({\bf r},t)&=\varepsilon_0{\bf E}({\bf r},t)+{\bf P}({\bf r},t),\cr
   96       {\bf B}({\bf r},t)&=\mu_0[{\bf H}({\bf r},t) + {\bf M}({\bf r},t)],\cr
   97     }
   98   $$
   99   where ${\bf P}({\bf r},t)={\bf P}[{\bf E}({\bf r},t),{\bf B}({\bf r},t)]$
  100   is the macroscopic polarization density
  101   (electric dipole moment per unit volume), and
  102   ${\bf M}({\bf r},t)={\bf M}[{\bf E}({\bf r},t),{\bf B}({\bf r},t)]$
  103   the magnetization (magnetic dipole moment per unit volume) of the medium.
  104   
  105   Here ${\bf E}({\bf r},t)$ and ${\bf B}({\bf r},t)$ are considered as
  106   the fundamental macroscopic electric and magnetic field quantities;
  107   ${\bf D}({\bf r},t)$ and ${\bf H}({\bf r},t)$ are the corresponding
  108   derived fields associated with the state of matter,
  109   connected to ${\bf E}({\bf r},t)$ and ${\bf B}({\bf r},t)$
  110   through the electric polarization density ${\bf P}({\bf r},t)$ and
  111   magnetization (magnetic polarization density) ${\bf M}({\bf r},t)$
  112   through the basic constitutive relations.
  113   In fact, the constitutive equations above form the very
  114   definitions\footnote{${}^1$}{J.~D.~Jackson, {\sl Classical Electrodynamics},
  115   2nd ed.~(Wiley, New York, 1975); J.~A.~Stratton, {\sl Electromagnetic Theory}
  116   (Mc\-Graw-Hill, New York, 1941).} of the electric polarization density and
  117   magnetization.
  118   
  119   \section{Two frequent assumptions in nonlinear optics}
  120   \item{$\bullet$}{No free charges present,
  121     $$\rho({\bf r},t)=0,\qquad{\bf J}({\bf r},t)={\bf 0}.$$
  122     (Any relaxation processes etc.~are included in
  123     imaginary parts of the terms of the electric susceptibility.)}
  124   \item{$\bullet$}{No magnetization of the medium,
  125     $${\bf M}({\bf r},t)={\bf 0}.$$}
  126   
  127   \section{The wave equation}
  128   By taking the cross product with the nabla operator and Faraday's law,
  129   one obtains
  130   $$
  131     \eqalign{
  132       \nabla\times\nabla\times{\bf E}({\bf r},t)
  133       &=-{{\partial}\over{\partial{t}}}\nabla\times{\bf B}({\bf r},t)\cr
  134       &=-\mu_0{{\partial}\over{\partial{t}}}\nabla\times{\bf H}({\bf r},t)\cr
  135       &=-\mu_0{{\partial}\over{\partial{t}}}
  136             {{\partial{\bf D}({\bf r},t)}\over{\partial t}}\cr
  137       &=-\mu_0\Big(
  138             \varepsilon_0{{\partial^2{\bf E}({\bf r},t)}\over{\partial t^2}}
  139             +{{\partial^2{\bf P}({\bf r},t)}\over{\partial t^2}}\Big).\cr
  140     }
  141   $$
  142   Since now $\mu_0\varepsilon_0=1/c^2$ in SI units, with $c$ being the speed
  143   of light in vacuum, one hence obtains the basic wave equation, taken in
  144   time domain, as
  145   $$
  146     \nabla\times\nabla\times{\bf E}({\bf r},t)
  147       +{{1}\over{c^2}}{{\partial^2{\bf E}({\bf r},t)}\over{\partial t^2}}
  148       =-\mu_0{{\partial^2{\bf P}({\bf r},t)}\over{\partial t^2}},\eqno{(1)}
  149   $$
  150   where, as in the previous lectures of this course, the polarization density
  151   can be written in terms of the perturbation series as
  152   $$
  153     \eqalign{
  154       {\bf P}({\bf r},t)&=\sum^{\infty}_{k=1}{\bf P}^{(k)}({\bf r},t)
  155          =\underbrace{
  156             \varepsilon_0\int^{\infty}_{-\infty}
  157             \chi^{(1)}_{\mu\alpha}(-\omega;\omega)
  158             E_{\alpha}({\bf r},\omega)\exp(-i\omega t)\,d\omega
  159           }_{={\bf P}^{(1)}({\bf r},t)}
  160          +\underbrace{
  161             \sum^{\infty}_{k=2}{\bf P}^{(k)}({\bf r},t)
  162           }_{={\bf P}^{({\rm NL})}({\bf r},t)}\cr
  163     }
  164   $$
  165   In the left hand side of Eq.~(1), we find the part of the homogeneous wave
  166   equation for propagation of electromagnetic waves in vacuum, while the right
  167   hand side described the modifications to the vacuum propagation due to the
  168   interaction between light and matter. In this respect, it is now clear that
  169   the electric polarisation effectively acts as a source term in the
  170   mathematical description of electromagnetic wave propagation, making the
  171   otherwise homogeneous vacuum problem an inhomogeneous problem (though with
  172   known source terms).
  173   
  174   It should be noticed that whenever the polarization density is calculated
  175   from the Bloch equations (formulated later on, in lecture 10 of this course),
  176   instead of by means of a perturbation series as above, the Maxwell
  177   equations and the wave equation~(1) above are denoted {\sl Maxwell-Bloch
  178   equations}.
  179   In some sense, we can therefore see the choice of method for the calculation
  180   of the polarization density as a switch point not only for using the
  181   susceptibility formalism or not for the description of interaction between
  182   light and matter, but also for the form of the wave propagation problem in
  183   nonlinear media, which mathematically significantly differ between the
  184   ``pure'' Maxwell's equations with susceptibilities and the Maxwell-Bloch
  185   equations.
  186   
  187   \section{The wave equation in frequency domain (optional)}
  188   Frequently in this course, we have rather been studying the electric
  189   fields and polarisation densities in frequency domain, since many
  190   phenomena in optics are properly and conveniently described as static
  191   (in which case the frequency dependence is simply reduced to the interaction
  192   between discrete frequencies in the spectrum).
  193   By using the Fourier integral identity\footnote{${}^2$}{From the inverse
  194   Fourier integral identity, it follows that the Fourier transform of a
  195   derivative of a function $f(t)$ is
  196   $$\fourier[f'(t)](\omega)=-i\omega\fourier[f(t)](\omega)\qquad
  197   \Rightarrow\qquad\fourier[f''(t)](\omega)=-\omega^2\fourier[f(t)](\omega).$$}
  198   $$
  199     E_{\alpha}(t)=\int^{\infty}_{-\infty}E_{\alpha}(\omega)
  200       \exp(-i\omega t)\,d\omega=\fourier^{-1}[E_{\alpha}](t),
  201   $$
  202   with inverse relation
  203   $$
  204     E_{\alpha}(\omega)={{1}\over{2\pi}}\int^{\infty}_{-\infty}E_{\alpha}(\tau)
  205       \exp(i\omega\tau)\,d\tau=\fourier[E_{\alpha}](\omega),
  206   $$
  207   we obtain the wave equation~(1) as
  208   $$
  209     \nabla\times\nabla\times{\bf E}({\bf r},\omega)
  210       -{{\omega^2}\over{c^2}}{\bf E}({\bf r},\omega)
  211       =\mu_0\omega^2{\bf P}({\bf r},\omega).
  212   $$
  213   
  214   \section{Quasimonochromatic light - Time dependent problems}
  215   By inserting the perturbation series for the electric polarisation density
  216   into the general wave equation~(1), which apply to arbitrary electric field
  217   distributions and field intensities of the light, one obtains the
  218   equation
  219   $$
  220     \nabla\times\nabla\times{\bf E}({\bf r},t)
  221       +\underbrace{{{1}\over{c^2}}{{\partial^2}\over{\partial t^2}}
  222          \int^{\infty}_{-\infty}{\bf e}_{\mu}\varepsilon_{\mu\alpha}(\omega)
  223          E_{\alpha}({\bf r},\omega)\exp(-i\omega t)\,d\omega
  224        }_{({\rm denote\ this\ integral\ as\ }I{\rm\ for\ later\ use})}
  225       =-\mu_0{{\partial^2{\bf P}^{({\rm NL})}({\bf r},t)}\over{\partial t^2}},
  226       \eqno{(2)}
  227   $$
  228   where
  229   $$
  230     \varepsilon_{\mu\alpha}(\omega)=\delta_{\mu\alpha}
  231       +\chi^{(1)}_{\mu\alpha}(-\omega;\omega)
  232   $$
  233   is a parameter commonly denoted as the
  234   {\sl relative electrical permittivity}.\footnote{${}^3$}{Notice that
  235   for isotropic media, $\chi^{(1)}_{\mu\alpha}(-\omega;\omega)
  236   =\chi^{(1)}_{xx}(-\omega;\omega)\delta_{\mu\alpha}$, which leads to
  237   the simplified form
  238   $$
  239     {\bf e}_{\mu}\varepsilon_{\mu\alpha}(\omega)E_{\alpha}({\bf r},\omega)
  240     =\varepsilon(\omega){\bf E}({\bf r},\omega).
  241   $$
  242   We will here, however, continue with the general form, in order not
  243   to loose generality in discussion that is to follow.}
  244   This wave equation is identical to Eq.~(7.14) in Butcher and Cotter's
  245   book. (Notice though the printing error in Butcher and Cotter's Eq.~(7.14),
  246   where the first $\mu_0$ should be replaced by $1/c^2$.)
  247   
  248   The second term of the left hand side of Eq.~(2) gives all first order
  249   optical contributions to the wave propagation, as well as all linear
  250   optical dispersion effects.
  251   This terms deserves some extra attention, and we will now proceed with
  252   deriving the effect of the frequency dependence of the relative permittivity
  253   upon the wave equation.
  254   First of all, we notice that since
  255   $E_{\alpha}({\bf r},-\omega)=E^*_{\alpha}({\bf r},\omega)$, which simply
  256   is a consequence of the choice of complex Fourier transform of a real
  257   valued field, the reality condition of Eq.~(2) requires that
  258   $$
  259     \varepsilon_{\mu\alpha}(-\omega)=\varepsilon^*_{\mu\alpha}(\omega).
  260   $$
  261   It should be emphasized that this property of the relative electrical
  262   permittivity merely is a convenient mathematical construction, since we
  263   in regular physical terms only consider positive angular frequencies as
  264   argument for the refractive index, etc.
  265   
  266   For quasimonochromatic light, the electric field and polarisation density
  267   are taken as
  268   $$
  269     \eqalign{
  270       {\bf E}({\bf r},t)&=\sum_{\omega_{\sigma}\ge 0}
  271         \Re[{\bf E}_{\omega_{\sigma}}({\bf r},t)\exp(-i\omega_{\sigma} t)],\cr
  272       {\bf P}({\bf r},t)&=\sum_{\omega_{\sigma}\ge 0}
  273         \Re[{\bf P}_{\omega_{\sigma}}({\bf r},t)\exp(-i\omega_{\sigma} t)],\cr
  274     }
  275   $$
  276   where ${\bf E}_{\omega_{\sigma}}({\bf r},t)$ and
  277   ${\bf P}_{\omega_{\sigma}}({\bf r},t)$ are slowly varying envelopes of
  278   the fields. In the frequency domain, the quasimonochromatic fields are
  279   expressed as
  280   $$
  281     \eqalign{
  282       {\bf E}({\bf r},\omega)&={{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}
  283         [{\bf E}_{\omega_{\sigma}}({\bf r},\omega-\omega_{\sigma})
  284           +{\bf E}^*_{\omega_{\sigma}}({\bf r},-\omega-\omega_{\sigma})],\cr
  285       {\bf P}({\bf r},\omega)&={{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}
  286         [{\bf P}_{\omega_{\sigma}}({\bf r},\omega-\omega_{\sigma})
  287           +{\bf P}^*_{\omega_{\sigma}}({\bf r},-\omega-\omega_{\sigma})],\cr
  288     }
  289   $$
  290   where the envelopes have some limited extent around the carrier
  291   frequencies at $\pm\omega_{\sigma}$. Notice that the fields taken in the
  292   frequency domain are expressed entirely in terms of their respective
  293   temporal envelope, that is to say, without the exponential functions
  294   that appear in their counterparts in time domain.
  295   
  296   For simplicity considering a medium that in the linear optical domain
  297   is isotropic, with the relative electrical permittivity
  298   $$
  299     \varepsilon_{\mu\alpha}(\omega)
  300       =\varepsilon(\omega)\delta_{\mu\alpha}
  301       =n^2_0(\omega)\delta_{\mu\alpha},
  302   $$
  303   where $n_0(\omega)$ is the first order contribution to the refractive
  304   index of the medium, this leads to the middle term of the wave
  305   equation~(1) in the form
  306   $$
  307     \eqalign{
  308      I&\equiv{{1}\over{c^2}}{{\partial^2}\over{\partial t^2}}
  309         \int^{\infty}_{-\infty}{\bf e}_{\mu}\varepsilon_{\mu\alpha}(\omega)
  310         E_{\alpha}({\bf r},\omega)\exp(-i\omega t)\,d\omega\cr
  311       &=-\int^{\infty}_{-\infty}{{\omega^2 n^2(\omega)}\over{c^2}}
  312         \underbrace{
  313         {{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}
  314         [{\bf E}_{\omega_{\sigma}}({\bf r},\omega-\omega_{\sigma})
  315           +{\bf E}^*_{\omega_{\sigma}}({\bf r},-\omega-\omega_{\sigma})]}_{
  316           {\rm quasimonochromatic\ form\ of\ }{\bf E}({\bf r},\omega)}
  317         \exp(-i\omega t)\,d\omega\cr
  318       &=\{{\rm denote\ }\omega^2\varepsilon(\omega)/c^2
  319           \equiv\omega^2 n^2_0(\omega)/c^2\equiv k^2(\omega)\}\cr
  320       &=-{{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}
  321          \int^{\infty}_{-\infty}k^2(\omega)
  322          [{\bf E}_{\omega_{\sigma}}({\bf r},\omega-\omega_{\sigma})
  323            +{\bf E}^*_{\omega_{\sigma}}({\bf r},-\omega-\omega_{\sigma})]
  324         \exp(-i\omega t)\,d\omega.\cr
  325       &=-{{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}
  326          \int^{\infty}_{-\infty}k^2(\omega)
  327          {\bf E}_{\omega_{\sigma}}({\bf r},\omega-\omega_{\sigma})
  328         \exp(-i\omega t)\,d\omega + {\rm c.\,c.}\cr
  329     }
  330   $$
  331   If now the field envelopes decay to zero rapidly enough in the vicinity
  332   of the carrier frequencies (as we would expect for quasimonochromatic light,
  333   with a strong spectral confinement around the carrier frequency of the light),
  334   then we may expect that a good approximation is to make a Taylor expansion
  335   of $k^2(\omega)$, in the neighbourhood of respective carrier frequency of
  336   the light, as
  337   $$
  338     \eqalign{
  339       k^2(\omega)&\approx\Big(k(\omega_{\sigma})
  340         +{{dk}\over{d\omega}}\Big|_{\omega_{\sigma}}(\omega-\omega_{\sigma})
  341         +{{1}\over{2!}}{{d^2 k}\over{d\omega^2}}
  342           \Big|_{\omega_{\sigma}}(\omega-\omega_{\sigma})^2
  343         \Big)^2\cr
  344       &\approx k^2_{\sigma}
  345         +2k_{\sigma}{{dk}\over{d\omega}}
  346           \Big|_{\omega_{\sigma}}(\omega-\omega_{\sigma})
  347         +k_{\sigma}{{d^2 k}\over{d\omega^2}}
  348           \Big|_{\omega_{\sigma}}(\omega-\omega_{\sigma})^2,
  349         \cr
  350     }
  351   $$
  352   where the notation $k_{\sigma}=k(\omega_{\sigma})$ was introduced, and
  353   hence\footnote{${}^4$}{Notice that unless we apply the second approximation
  354   in the Taylor expansion of $k^2(\omega)$, terms containing the {\sl squares}
  355   of the derivatives will appear, which will lead to wave equations that
  356   differ from the ones given by Butcher and Cotter.
  357   In particular, this situation will arise even if one uses the suggested
  358   expansion given by Eq.~(7.23) in Butcher and Cotter's book, which hence
  359   should be taken with some care if one wish to build a strict foundation
  360   for the time-dependent wave equation.}
  361   $$
  362     \eqalign{
  363      I&\approx -{{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}\int^{\infty}_{-\infty}
  364        \Big(k^2_{\sigma}
  365          +2k_{\sigma}{{dk}\over{d\omega}}
  366            \Big|_{\omega_{\sigma}}(\omega-\omega_{\sigma})
  367          +k_{\sigma}{{d^2 k}\over{d\omega^2}}
  368            \Big|_{\omega_{\sigma}}(\omega-\omega_{\sigma})^2
  369        \Big)
  370   \cr&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times
  371         {\bf E}_{\omega_{\sigma}}({\bf r},\omega-\omega_{\sigma})
  372         \exp(-i\omega t)\,d\omega + {\rm c.\,c.}\cr
  373      &=\{{\rm change\ variable\ of\ integration\ }
  374          \omega'=\omega-\omega_{\sigma}\}\cr
  375      &=-{{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}\int^{\infty}_{-\infty}
  376        \Big(k^2_{\sigma}
  377          +2k_{\sigma}{{dk}\over{d\omega}}
  378            \Big|_{\omega_{\sigma}}\omega'
  379          +k_{\sigma}{{d^2 k}\over{d\omega^2}}
  380            \Big|_{\omega_{\sigma}}\omega'^2
  381        \Big)
  382   \cr&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times
  383         {\bf E}_{\omega_{\sigma}}({\bf r},\omega')
  384         \exp(-i\omega't)\,d\omega'\,\exp(-i\omega_{\sigma} t)
  385         + {\rm c.\,c.}\cr
  386      &=\bigg\{{\rm use\ }\int^{\infty}_{-\infty}\omega^n f(\omega)
  387             \exp(-i\omega t)\,d\omega=\fourier^{-1}[\omega^n f(\omega)](t)
  388               =i^n{{d^n f(t)}\over{d t^n}}\bigg\}\cr
  389      &=-{{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}\exp(-i\omega_{\sigma} t)
  390        \Big(k^2_{\sigma}
  391          +i2k_{\sigma}{{dk}\over{d\omega}}
  392            \Big|_{\omega_{\sigma}}{{\partial}\over{\partial t}}
  393          -k_{\sigma}{{d^2 k}\over{d\omega^2}}
  394            \Big|_{\omega_{\sigma}}{{\partial^2}\over{\partial t^2}}
  395        \Big){\bf E}_{\omega_{\sigma}}({\bf r},t)
  396         + {\rm c.\,c.}\cr
  397     }
  398   $$
  399   As this result is inserted back into the wave equation~(2), one obtains
  400   $$
  401     \eqalign{
  402       {{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}&\exp(-i\omega_{\sigma} t)
  403           \Big[\nabla\times\nabla\times{\bf E}_{\omega_{\sigma}}({\bf r},t)
  404   \cr&\qquad\qquad\qquad
  405         -\Big(k^2_{\sigma}
  406            +2ik_{\sigma}{{dk}\over{d\omega}}
  407              \Big|_{\omega_{\sigma}}{{\partial}\over{\partial t}}
  408            -k_{\sigma}{{d^2 k}\over{d\omega^2}}
  409              \Big|_{\omega_{\sigma}}{{\partial^2}\over{\partial t^2}}
  410          \Big){\bf E}_{\omega_{\sigma}}({\bf r},t)\Big]
  411           + {\rm c.\,c.}\cr
  412         &=-\mu_0{{\partial^2{\bf P}^{({\rm NL})}({\bf r},t)}
  413                  \over{\partial t^2}}\cr
  414         &=-\mu_0{{\partial^2}\over{\partial t^2}}
  415            {{1}\over{2}}
  416            \sum_{\omega_{\sigma}\ge 0}
  417            {\bf P}^{({\rm NL})}_{\omega_{\sigma}}({\bf r},t)
  418            \exp(-i\omega_{\sigma} t)+{\rm c.\,c.}\cr
  419         &\approx\mu_0{{1}\over{2}}\sum_{\omega_{\sigma}\ge 0}\omega^2_{\sigma}
  420            {\bf P}^{({\rm NL})}_{\omega_{\sigma}}({\bf r},t)
  421            \exp(-i\omega_{\sigma} t)+{\rm c.\,c.}\cr
  422     }
  423   $$
  424   As we separate out the respective frequency components at
  425   $\omega=\omega_{\sigma}$ of this equation, one obtains the time
  426   dependent wave equation for the {\sl temporal envelope components}
  427   of the electric field as
  428   $$
  429     \eqalign{
  430        \nabla\times\nabla\times{\bf E}_{\omega_{\sigma}}({\bf r},t)
  431         -\Big(k^2_{\sigma}
  432            +i2k_{\sigma}
  433             {{1}\over{v_{\rm g}}}
  434             {{\partial}\over{\partial t}}
  435            -k_{\sigma}{{d^2 k}\over{d\omega^2}}&
  436              \Big|_{\omega_{\sigma}}{{\partial^2}\over{\partial t^2}}
  437          \Big){\bf E}_{\omega_{\sigma}}({\bf r},t)
  438            =\mu_0\omega^2_{\sigma}
  439             {\bf P}^{({\rm NL})}_{\omega_{\sigma}}({\bf r},t),\cr
  440     }
  441     \eqno{(3)}
  442   $$
  443   where
  444   $$
  445     v_{\rm g}=\Big({{dk}\over{d\omega}}\Big|_{\omega_{\sigma}}\Big)^{-1}.
  446   $$
  447   \vfill\eject
  448   
  449   \section{Three practical approximations}
  450   \item{$[1]$}{The infinite plane wave approximation,
  451     $$
  452       {\bf E}_{\omega_{\sigma}}({\bf r},t)
  453         ={\bf E}_{\omega_{\sigma}}(z,t)\bot{\bf e}_z
  454         \qquad\Rightarrow\qquad
  455         \nabla\times\nabla\times\to -{{\partial^2}\over{\partial z^2}}.
  456     $$}
  457   \item{$[2]$}{Unidirectional propagation,
  458     $$
  459       \eqalign{
  460       {\bf E}_{\omega_{\sigma}}(z,t)&={\bf A}_{\omega_{\sigma}}(z,t)
  461          \exp(\pm ik_{\sigma}z)\cr
  462       &\Downarrow\cr
  463       \nabla\times\nabla\times{\bf E}_{\omega_{\sigma}}(z,t)&=
  464         -[{{\partial^2{\bf A}_{\omega_{\sigma}}}\over{\partial z^2}}
  465           \pm 2ik_{\sigma}{{\partial{\bf A}_{\omega_{\sigma}}}\over{\partial z}}
  466           -k^2_{\sigma}{\bf A}_{\omega_{\sigma}}
  467         ]\exp(\pm ik_{\sigma}z),\cr
  468       }
  469     $$
  470     for waves propagating in the positive/negative $z$-direction.
  471     In this case, the real-valued electric field hence takes the form
  472     $$
  473       \eqalign{
  474         {\bf E}({\bf r},t)&=\sum_{\omega_{\sigma}\ge 0}
  475           \Re[{\bf E}_{\omega_{\sigma}}({\bf r},t)\exp(-i\omega_{\sigma} t)]\cr
  476         &=\sum_{\omega_{\sigma}\ge 0}
  477           \Re[{\bf A}_{\omega_{\sigma}}(z,t)
  478           \exp(\pm ik_{\sigma}z-i\omega_{\sigma} t)]\cr
  479         &=\sum_{\omega_{\sigma}\ge 0}|{\bf A}_{\omega_{\sigma}}(z,t)|
  480           \Re\{\exp[ik_{\sigma}z\mp i\omega_{\sigma} t+i\phi(z)]\}\cr
  481         &=\sum_{\omega_{\sigma}\ge 0}|{\bf A}_{\omega_{\sigma}}(z,t)|
  482           \cos(k_{\sigma}z\mp \omega_{\sigma} t+\phi(z)),\cr
  483       }
  484     $$
  485     where $\phi(z,t)$ describes the spatially and temporally varying phase
  486     of the complex-valued slowly varying envelope function
  487     ${\bf A}_{\omega_{\sigma}}(z,t)$ of the electric field.
  488   }
  489   \medskip
  490   \item{$[3]$}{The slowly varying envelope approximation,
  491     $$\Big|{{\partial^2{\bf A}_{\omega_{\sigma}}}\over{\partial z^2}}\Big|
  492       \ll \Big|k_{\sigma}
  493        {{\partial{\bf A}_{\omega_{\sigma}}}\over{\partial z}}\Big|.$$}
  494   \medskip
  495   \noindent
  496   These approximations, whenever applicable, further reduce the time dependent
  497   wave equation to
  498   $$
  499     \eqalign{
  500       \Big(
  501         \pm i{{\partial}\over{\partial z}}
  502          +i{{1}\over{v_{\rm g}}}
  503             {{\partial}\over{\partial t}}
  504            -{{1}\over{2}}{{d^2 k}\over{d\omega^2}}&
  505              \Big|_{\omega_{\sigma}}{{\partial^2}\over{\partial t^2}}
  506          \Big){\bf A}_{\omega_{\sigma}}(z,t)
  507            =-{{\mu_0\omega^2_{\sigma}}\over{2k_{\sigma}}}
  508             {\bf P}^{({\rm NL})}_{\omega_{\sigma}}({\bf r},t)
  509             \exp(\mp ik_{\sigma}z).\cr
  510     }
  511     \eqno{(4)}
  512   $$
  513   This form of the wave equation is identical to Butcher and Cotter's
  514   Eq.~(7.24), with the exception that here waves propagating in positive
  515   (upper signs) as well as negative (lower signs) $z$-direction are
  516   considered.
  517   \vfill\eject
  518   
  519   \section{Monochromatic light}
  520   \subsection{Monochromatic optical field}
  521   $$
  522     \eqalign{
  523       {\bf E}({\bf r},t)&=\sum_{\sigma}
  524         \Re[{\bf E}_{\omega_{\sigma}}({\bf r})\exp(-i\omega_{\sigma} t)],
  525       \qquad\omega_{\sigma}\ge 0\cr
  526       {\bf E}({\bf r},\omega)
  527       &={{1}\over{2}}\sum_{\sigma}
  528         [{\bf E}_{\omega_{\sigma}}({\bf r})\delta(\omega-\omega_{\sigma})
  529         +{\bf E}^*_{\omega_{\sigma}}({\bf r})\delta(\omega+\omega_{\sigma})]\cr
  530     }
  531   $$
  532   
  533   \subsection{Polarization density induced by monochromatic optical field}
  534   $$
  535     {\bf P}^{(n)}({\bf r},t)=\sum_{\omega_{\sigma}\ge 0}
  536        {\rm Re}[{\bf P}^{(n)}_{\omega_{\sigma}}\exp(-i\omega_{\sigma} t)],
  537     \qquad\omega_{\sigma}=\omega_1+\omega_2+\ldots+\omega_n
  538   $$
  539   (For construction of ${\bf P}^{(n)}_{\omega_{\sigma}}$, see notes on the
  540   Butcher and Cotter convention handed out during the third lecture.)
  541   
  542   \section{Monochromatic light - Time independent problems}
  543   For strictly monochromatic light, as for example the output light of
  544   continuous wave lasers, the temporal field envelopes are constants in time,
  545   and the wave equation~(3) is reduced to
  546   $$
  547     \eqalign{
  548        \nabla\times\nabla\times{\bf E}_{\omega_{\sigma}}({\bf r})
  549         -k^2_{\sigma}{\bf E}_{\omega_{\sigma}}({\bf r})
  550            =\mu_0\omega^2_{\sigma}
  551             {\bf P}^{({\rm NL})}_{\omega_{\sigma}}({\bf r}).\cr
  552     }
  553     \eqno{(5)}
  554   $$
  555   By applying the above listed approximations, one immediately finds
  556   the monochromatic, time independent form of Eq.~(3) in the infinite
  557   plane wave limit and slowly varying approximation as
  558   $$
  559     \eqalign{
  560         {{\partial}\over{\partial z}}{\bf A}_{\omega_{\sigma}}
  561            =\pm i{{\mu_0\omega^2_{\sigma}}\over{2k_{\sigma}}}
  562             {\bf P}^{({\rm NL})}_{\omega_{\sigma}}
  563             \exp(\mp ik_{\sigma}z),\cr
  564     }
  565     \eqno{(6)}
  566   $$
  567   where the upper/lower sign correspond to a wave propagating in the
  568   positive/negative $z$-direction. This equation corresponds to Butcher
  569   and Cotter's Eq.~(7.17).
  570   \vfill\eject
  571   
  572   \section{Example I: Optical Kerr-effect - Time independent case}
  573   In this example, we consider continuous wave
  574   propagation\footnote{${}^5$}{That is to say, a time independent problem
  575   with the temporal envelope of the electrical field being constant in time.}
  576   in optical Kerr-media, using
  577   light polarized in the $x$-direction and propagating along the positive
  578   direction of the $z$-axis,
  579   $$
  580     {\bf E}({\bf r},t)=\Re[{\bf E}_{\omega}(z)\exp(-i\omega t)],
  581     \qquad{\bf E}_{\omega}(z)={\bf A}_{\omega}(z)\exp(ikz)
  582                              ={\bf e}_x A^x_{\omega}(z)\exp(ikz),
  583   $$
  584   where, as previously, $k=\omega n_0/c$.
  585   From material handed out during the third lecture (notes on the Butcher
  586   and Cotter convention), the nonlinear polarization density for $x$-polarized
  587   light is given as ${\bf P}^{({\rm NL})}_{\omega}={\bf P}^{(3)}_{\omega}$, with
  588   $$
  589     \eqalign{
  590       {\bf P}^{(3)}_{\omega}
  591         &=\varepsilon_0(3/4){\bf e}_x\chi^{(3)}_{xxxx}
  592           (-\omega;\omega,\omega,-\omega)
  593           |E^x_{\omega}|^2 E^x_{\omega}\cr
  594         &=\varepsilon_0(3/4)\chi^{(3)}_{xxxx}
  595           |{\bf E}_{\omega}|^2 {\bf E}_{\omega}\cr
  596         &=\varepsilon_0(3/4)\chi^{(3)}_{xxxx}
  597           |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\exp(ikz),\cr
  598     }
  599   $$
  600   and the time independent wave equation for the field envelope
  601   ${\bf A}_{\omega}$, using Eq.~(6), becomes
  602   $$
  603     \eqalign{
  604         {{\partial}\over{\partial z}}{\bf A}_{\omega}
  605            &=i{{\mu_0\omega^2}\over{2k}}
  606              \underbrace{\varepsilon_0(3/4)\chi^{(3)}_{xxxx}
  607              |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\exp(ikz)}_{
  608                ={\bf P}^{({\rm NL})}_{\omega}(z)}\exp(-ikz)\cr
  609            &=i{{3\omega^2}\over{8c^2k}}\chi^{(3)}_{xxxx}
  610              |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\cr
  611            &=\{{\rm since\ }k=\omega n_0(\omega)/c\}\cr
  612            &=i{{3\omega}\over{8cn_0}}
  613              \chi^{(3)}_{xxxx}
  614              |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\cr
  615            &=\{{\rm in\ analogy\ with\ Butcher\ and\ Cotter\ Eq.~(6.63),\ }
  616                n_2=({{3}/{8n_0}})\chi^{(3)}_{xxxx}\}\cr
  617            &=i {{\omega n_2}\over{c}} |{\bf A}_{\omega}|^2 {\bf A}_{\omega},\cr
  618     }
  619   $$
  620   or, equivalently, in its scalar form
  621   $$
  622     {{\partial}\over{\partial z}}A^x_{\omega}
  623       =i {{\omega n_2}\over{c}} |A^x_{\omega}|^2 A^x_{\omega}.
  624   $$
  625   If the medium of interest now is analyzed at an angular frequency far
  626   from any resonance, we may look for solutions to this equation with
  627   $|{\bf A}_{\omega}|$ being constant (for a lossless medium).
  628   For such a case it is straightforward to integrate the final wave
  629   equation to yield the general solution
  630   $$
  631     {\bf A}_{\omega}(z)=
  632       {\bf A}_{\omega}(z_0)\exp[i\omega n_2 |{\bf A}_{\omega}(z_0)|^2 z/c],
  633   $$
  634   or, again equivalently, in the scalar form
  635   $$
  636     A^x_{\omega}(z)=
  637       A^x_{\omega}(z_0)\exp[i\omega n_2 |A^x_{\omega}(z_0)|^2 z/c],
  638   $$
  639   which hence gives the solution for the real-valued electric field
  640   ${\bf E}({\bf r},t)$ as
  641   $$
  642     \eqalign{
  643       {\bf E}({\bf r},t)&=\Re[{\bf E}_{\omega}(z)\exp(-i\omega t)]\cr
  644         &=\Re\{{\bf A}_{\omega}(z)\exp[i(kz-\omega t)]\}\cr
  645         &=\Re\{{\bf A}_{\omega}(z_0)
  646             \exp[i(kz+\omega n_2 |{\bf A}_{\omega}(z_0)|^2 z/c-\omega t)]\}.\cr
  647     }
  648   $$
  649   From this solution, one immediately finds that the wave propagates
  650   with an effective propagation constant
  651   $$
  652     k+\omega n_2 |{\bf A}_{\omega}(z_0)|^2/c
  653     = (\omega/c) (n_0+ n_2 |{\bf A}_{\omega}(z_0)|^2),
  654   $$
  655   that is to say, experiencing the intensity dependent refractive index
  656   $$
  657     n_{\rm eff}=n_0+ n_2 |{\bf A}_{\omega}(z_0)|^2.
  658   $$
  659   \vfill\eject
  660   
  661   \section{Example II: Optical Kerr-effect - Time dependent case}
  662   We now consider a {\sl time dependent} envelope ${\bf E}_{\omega}(z,t)$,
  663   of an optical wave propagating in the same medium and geometry as in the
  664   previous example, for which now
  665   $$
  666     {\bf E}({\bf r},t)=\Re[{\bf E}_{\omega}(z,t)\exp(-i\omega t)],
  667     \qquad{\bf E}_{\omega}(z,t)={\bf A}_{\omega}(z,t)\exp(ikz)
  668                              ={\bf e}_x A^x_{\omega}(z,t)\exp(ikz).
  669   $$
  670   The proper wave equation to apply for this case is the time dependent
  671   wave equation~(4), and since the nonlinear polarization density of
  672   the medium still is given by the optical Kerr-effect, we obtain
  673   $$
  674     \eqalign{
  675       \Big(
  676         i{{\partial}\over{\partial z}}
  677          &+i{{1}\over{v_{\rm g}}}
  678             {{\partial}\over{\partial t}}
  679            -{{1}\over{2}}{{d^2 k}\over{d\omega^2}}
  680              \Big|_{\omega_{\sigma}}{{\partial^2}\over{\partial t^2}}
  681          \Big){\bf A}_{\omega}(z,t)\cr
  682            &=-{{\mu_0\omega^2}\over{2k}}
  683              \underbrace{\varepsilon_0(3/4)\chi^{(3)}_{xxxx}
  684              |{\bf A}_{\omega}(z,t)|^2 {\bf A}_{\omega}(z,t)\exp(ikz)}_{
  685                ={\bf P}^{({\rm NL})}_{\omega}(z,t)}
  686             \exp(-ikz)\cr
  687            &=-{{3\omega^2}\over{8c^2k}}
  688              \chi^{(3)}_{xxxx}
  689              |{\bf A}_{\omega}(z,t)|^2 {\bf A}_{\omega}(z,t)\cr
  690            &=\{{\rm as\ in\ previous\ example}\}\cr
  691            &=-{{\omega n_2}\over{c}}
  692              |{\bf A}_{\omega}(z,t)|^2 {\bf A}_{\omega}(z,t).\cr
  693     }
  694   $$
  695   The resulting wave equation
  696   $$
  697     \eqalign{
  698       \Big(
  699         i{{\partial}\over{\partial z}}
  700          &+i{{1}\over{v_{\rm g}}}
  701             {{\partial}\over{\partial t}}
  702            -{{1}\over{2}}{{d^2 k}\over{d\omega^2}}
  703              \Big|_{\omega_{\sigma}}{{\partial^2}\over{\partial t^2}}
  704          \Big){\bf A}_{\omega}
  705            =-{{\omega n_2}\over{c}}
  706              |{\bf A}_{\omega}|^2 {\bf A}_{\omega}\cr
  707     }
  708   $$
  709   is the starting point for analysis of solitons and solitary waves
  710   in optical Kerr-media.
  711   The obtained equation is the non-normalized form of the
  712   in nonlinear physics (not only nonlinear optics!) often
  713   encountered {\sl nonlinear Schr\"odinger equation}
  714   (or NLSE, as its common acronym yields).
  715   
  716   For a discussion on the transformation that cast the nonlinear Schr\"odinger
  717   equation into the {\sl normalized nonlinear Schr\"odinger equation},
  718   see Butcher and Cotter's book, page~240.
  719   \bye
  720   

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