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    1   % File: nlopt/lect1/lect1.tex [pure TeX code]
    2   % Last change: January 7, 2003
    3   %
    4   % Lecture No 1 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{1}
   30   Nonlinear optics is the discipline in physics in which the electric
   31   polarization density of the medium is studied as a nonlinear function
   32   of the electromagnetic field of the light. Being a wide field of
   33   research in electromagnetic wave propagation, nonlinear interaction
   34   between light and matter leads to a wide spectrum of phenomena, such
   35   as optical frequency conversion, optical solitons, phase conjugation,
   36   and Raman scattering. In addition, many of the analytical tools applied
   37   in nonlinear optics are of general character, such as the perturbative
   38   techniques and symmetry considerations, and can equally well be applied
   39   in other disciplines in nonlinear dynamics.
   40   
   41   \section{The contents of this course}
   42   This course is intended as an introduction to the wide field of
   43   phenomena encountered in nonlinear optics.
   44   The course covers:
   45   \smallskip
   46   
   47   \item{$\bullet$}{The theoretical foundation of nonlinear interaction
   48      between light and matter.}
   49   \item{$\bullet$}{Perturbation analysis of nonlinear interaction
   50      between light and matter.}
   51   \item{$\bullet$}{The Bloch equation and its interpretation.}
   52   \item{$\bullet$}{Basics of soliton theory and the inverse
   53      scattering transform.}
   54   \smallskip
   55   
   56   It should be emphasized that the course does not cover state-of-the-art
   57   material constants of nonlinear optical materials, etc.~but rather
   58   focus on the theoretical foundations and ideas of nonlinear optical
   59   interactions between light and matter.
   60   
   61   A central analytical technique in this course is the perurbation
   62   analysis, with its foundation in the analytical mechanics.
   63   This technique will in the course mainly be applied to the
   64   quantum-mechanical description of interaction between light and
   65   matter, but is central in a wide field of cross-disciplinary
   66   physics as well.
   67   In order to give an introduction to the analytical theory of
   68   nonlinear systems, we will therefore start with the analysis
   69   of the nonlinear equations of motion for the mechanical pendulum.
   70   
   71   \section{Examples of applications of nonlinear optics}
   72   Some important applications in nonlinear optics:
   73   \medskip
   74   \item{$\bullet$}{Optical parametric amplification (OPA) and
   75      oscillation (OPO), $\hbar\omega_{\rm p}\to\hbar\omega_{\rm s}
   76      +\hbar\omega_{\rm i}$.}
   77   \item{$\bullet$}{Second harmonic generation (SHG),
   78      $\hbar\omega+\hbar\omega\to\hbar(2\omega)$.}
   79   \item{$\bullet$}{Third harmonic generation (THG),
   80      $\hbar\omega+\hbar\omega+\hbar\omega\to\hbar(3\omega)$.}
   81   \item{$\bullet$}{Pockels effect, or the linear electro-optical
   82      effect (applications for optical switching).}
   83   \item{$\bullet$}{Optical bistability (optical logics).}
   84   \item{$\bullet$}{Optical solitons (ultra long-haul communication).}
   85   \vfill\eject
   86   
   87   \section{A brief history of nonlinear optics}
   88   Some important advances in nonlinear optics:
   89   \medskip
   90   \item{$\bullet$}{Townes et al. (1960), invention of the
   91      laser.\footnote{${}^1$}{Charles H.~Townes was in 1964 awarded
   92      with the Nobel Prize for the invention of the ammonia laser.}
   93   }
   94   \item{$\bullet$}{Franken et al. (1961), First observation ever of
   95      nonlinear optical effects, second harmonic generation
   96      (SHG).\footnote{${}^2$}{Franken et al. detected
   97      ulvtraviolet light ($\lambda=347.1$ nm) at twice the frequency of a
   98      ruby laser beam ($\lambda=694.2$ nm) when this beam traversed a
   99      quartz crystal; P.~A.~Franken, A.~E.~Hill, C.~W. Peters, G.~Weinreich,
  100      Phys.~Rev.~Lett. {\bf 7}, 118 (1961).
  101      Second harmonic generation is also the first nonlinear effect ever
  102      observed where a coherent input generates a coherent output.}
  103   }
  104   \item{$\bullet$}{Terhune et al. (1962), First observation of
  105      third harmonic generation (THG).\footnote{${}^3$}{In their experiment,
  106      Terhune et al. detected only about a thousand THG photons per pulse,
  107      at $\lambda=231.3$ nm, corresponding to a conversion of one photon
  108      out of about $10^{15}$ photons at the fundamental wavelength at
  109      $\lambda=693.9$ nm; R.~W. Terhune, P.~D. Maker, and
  110      C.~M. Savage, Phys. Rev. Lett. {\bf 8}, 404 (1962).}
  111   }
  112   \item{$\bullet$}{E.~J.~Woodbury and W.~K.~Ng (1962),
  113      first demonstration of stimulated Raman
  114      scattering.\footnote{${}^4$}{E.~J.~Woodbury and W.~K.~Ng,
  115      Proc.~IRE {\bf 50}, 2347 (1962).}
  116   }
  117   \item{$\bullet$}{Armstrong et al. (1962), formulation of general
  118      permutation symmetry relations in nonlinear
  119      optics.\footnote{${}^5$}{The general permutation symmetry
  120      relations of higher-order susceptibilities were published
  121      by J.~A.~Armstrong, N.~Bloembergen, J.~Ducuing, and
  122      P.~S. Pershan, Phys.~Rev.~{\bf 127}, 1918 (1962).}
  123   }
  124   \item{$\bullet$}{A.~Hasegawa and F.~Tappert (1973),
  125      first theoretical prediction of soliton generation in optical
  126      fibers.\footnote{${}^6$}{A.~Hasegawa and F.~Tappert,
  127      ``Transmission of stationary nonliner optica pulses in dispersive
  128      optical fibers: I, Anomalous dispersion; II Normal dispersion'',
  129      Appl. Phys. Lett. {\bf 23}, 142--144 and 171--172
  130      (August 1 and 15, 1973).}
  131   }
  132   \item{$\bullet$}{H.~M.~Gibbs et al. (1976), first demonstration
  133      and explaination of optical
  134      bistability.\footnote{${}^7$}{H.~M. Gibbs, S.~M. McCall,
  135      and T.~N.~C. Venkatesan, Phys. Rev. Lett. {\bf 36}, 1135 (1976).}
  136   }
  137   \item{$\bullet$}{L.~F.~Mollenauer et al.~(1980),
  138      first confirmation of soliton generation in optical
  139      fibers.\footnote{${}^8$}{L.~F.~Mollenauer, R.~H. Stolen,
  140      and J.~P. Gordon, ``Experimental observation of picosecond
  141      pulse narrowing and solitons in optical fibers'',
  142      Phys. Rev. Lett. {\bf 45}, 1095--1098 (September 29, 1980);
  143      the first reported observation of solitons was though made in 1834 by
  144      John Scott Russell, a Scottish scientist and later famous
  145      Victorian engineer and shipbuilder, while studying water waves
  146      in the Glasgow-Edinburgh channel.}
  147   }
  148   \smallskip
  149   Recently, many advances in nonlinear optics has been made,
  150   with a lot of efforts with fields of, for example, Bose-Einstein
  151   condensation and laser cooling; these fields are, however,
  152   a bit out of focus from the subjects of this course, which
  153   can be said to be an introduction to the 1960s and 1970s
  154   advances in nonlinear optics.
  155   It should also be emphasized that many of the effects observed in
  156   nonlinear optics, such as the Raman scattering, were observed
  157   much earlier in the microwave range.
  158   
  159   \section{Outline for calculations of polarization densities}
  160   \subsection{Metals and plasmas}
  161   From an all-classical point-of-view, the calculation of the
  162   electric polarization density of metals and plasmas, containing
  163   a free electron gas, can be performed using the model of free
  164   charges acting under the Lorenz force of an electromagnetic field,
  165   $$m_{\rm e}{{d^2{\bf r}_{\rm e}}\over{dt^2}}
  166   =-e{\bf E}(t)-e{{d{\bf r}_{\rm e}}\over{dt}}\times{\bf B}(t),$$
  167   where ${\bf E}$ and ${\bf B}$ are all-classical electric and magnetic
  168   fields of the electromagnetic field of the light.
  169   In forming the equation for the motion of the electron, the origin was
  170   chosen to coincide with the center of the nucleus.
  171   
  172   \subsection{Dielectrics}
  173   A very useful model used by Drude and Lorentz\footnote{${}^9$}{R.~Becker,
  174   {\it Elektronen Theorie}, (Teubner, Leipzig, 1933).} to calculate
  175   the linear electric polarization of the medium describes the
  176   electrons as harmonically bound particles.
  177   
  178   For dielectrics in the nonlinear optical regime, as being the focus
  179   of our attention in this course, the calculation of the electric
  180   polarization density is instead performed using a nonlinear spring
  181   model of the bound charges, here quoted for one-dimensional motion as
  182   $$m_{\rm e}{{d^2{x}_{\rm e}}\over{dt^2}}
  183   +\Gamma_{\rm e}{{d{x}_{\rm e}}\over{dt}}+\alpha^{(1)} x_{\rm e}
  184   +\alpha^{(2)} x^2_{\rm e}+\alpha^{(3)} x^3_{\rm e}+\ldots
  185   =-eE_x(t).$$
  186   As in the previous case of metals and plasmas, in forming the equation
  187   for the motion of the electron, the origin was also here chosen to
  188   coincide with the center of the nucleus.
  189   
  190   This classical mechanical model will later in this lecture be applied
  191   to the derivation of the second-order nonlinear polarzation density
  192   of the medium.
  193   
  194   \section{\bf Introduction to nonlinear dynamical systems}
  195   In this section we will, as a preamble to later analysis of
  196   quantum-mechanical systems, apply perturbation analysis to a
  197   simple mechanical system.
  198   Among the simplest nonlinear dynamical systems is the pendulum,
  199   for which the total mechanical energy of the system, considering the
  200   point of suspension as defining the level of zero potential energy,
  201   is given as the sum of the kinetic and potential energy as
  202   $$E=T+V={1\over2m}|{\bf p}|^2 - mgl(\cos\vartheta-1),\eqno{(1)}$$
  203   where $m$ is the mass, $g$ the gravitation constant, $l$ the length,
  204   and $\vartheta$ the angle of deflection of the pendulum, and where ${\bf p}$
  205   is the momentum of the point mass.
  206   \medskip
  207   \centerline{\epsfxsize=50mm\epsfbox{../images/pendulum/pendulum.1}}
  208   \centerline{Figure 1. The mechanical pendulum.}
  209   \medskip
  210   \noindent
  211   From the total mechanical energy~(1) of the system, the equations of
  212   motion for the point mass is hence given by Lagranges
  213   equations,\footnote{${}^{10}$}{Herbert Goldstein, {\it Classical Mechanics},
  214   2nd ed.~(Addison-Wesley, Massachusetts, 1980).}
  215   $$
  216     {{d}\over{dt}}\bigg({{\partial L}\over{\partial p_j}}\bigg)
  217       -{{\partial L}\over{\partial q_j}}=0,\qquad j=x,y,z,\eqno{(2)}
  218   $$
  219   where $q_j$ are the generalized coordinates, $p_j=\dot{q}_j$ are the
  220   components of the generalized momentum, and $L=T-V$ is the Lagrangian of
  221   the mechanical system. In spherical coordinates $(\rho,\varphi,\vartheta)$,
  222   the momentum for the mass is given as its mass times the velocity,
  223   $$
  224       {\bf p}=ml(\dot{\vartheta}{\bf e}_{\vartheta}
  225                +\dot{\varphi}\sin\vartheta{\bf e}_{\varphi})
  226   $$
  227   and the Lagrangian for the pendulum is hence given as
  228   $$
  229     \eqalign{
  230       L&={{1}\over{2m}}(p^2_{\varphi}+p^2_{\varphi})+mgl(\cos\vartheta-1)\cr
  231        &={{ml^2}\over{2}}(\dot{\vartheta}^2
  232            +\dot{\varphi}^2\sin^2\vartheta)
  233            +mgl(\cos\vartheta-1).\cr
  234     }
  235   $$
  236   As the Lagrangian for the pendulum is inserted into Eq.~(2),
  237   the resulting equations of motion are for $q_j=\varphi$ and $\vartheta$
  238   obtained as
  239   $$
  240     {{d^2\varphi}\over{dt^2}}
  241       +\bigg({{d\varphi}\over{dt}}\bigg)^2\sin\varphi\cos\varphi=0,
  242     \eqno{(3{\rm a})}
  243   $$
  244   and
  245   $$
  246     {{d^2\vartheta}\over{dt^2}}+(g/l)\sin\vartheta=0,
  247     \eqno{(3{\rm b})}
  248   $$
  249   respectively, where we may notice that the motion in ${\bf e}_{\varphi}$ and
  250   ${\bf e}_{\vartheta}$ directions are decoupled.
  251   We may also notice that the equation of motion for $\varphi$ is
  252   independent of any of the physical parameters involved in te Lagrangian,
  253   and the evolution of $\varphi(t)$ in time is entirely determined by
  254   the initial conditions at some time $t=t_0$.
  255   
  256   In the following discussion, the focus will be on the properties of the
  257   motion of $\vartheta(t)$.
  258   The equation of motion for the $\vartheta$ coordinate is here
  259   described by the so-called Sine-Gordon equation.\footnote{${}^{11}$}{The
  260   term ``Sine-Gordon equation'' has its origin as an allegory over the
  261   similarity between the time-dependent (Sine-Gordon) equation
  262   $${{\partial^2\varphi}\over{\partial z^2}}-{{1}\over{c^2}}
  263   {{\partial^2\varphi}\over{\partial t^2}}=\mu^2_0\sin\varphi,$$
  264   appearing in, for example, relativistic field theories, as compared
  265   to the time-dependent Klein-Gordon equation, which takes the form
  266   $${{\partial^2\varphi}\over{\partial z^2}}-{{1}\over{c^2}}
  267   {{\partial^2\varphi}\over{\partial t^2}}=\mu^2_0\varphi.$$
  268   The Sine-Gordon equation is sometimes also called ``pendulum equation''
  269   in the terminology of classical mechanics.}
  270   This nonlinear differential equation is hard\footnote{$\dagger$}{Impossible?}
  271   to solve analytically, but if the nonlinear term is expanded as a
  272   Taylor series around $\vartheta=0$,
  273   $$
  274     {{d^2\vartheta}\over{dt^2}}+(g/l)
  275       (\vartheta-{{\vartheta^3}\over{3!}}+{{\vartheta^5}\over{5!}}+\ldots)=0,
  276   $$
  277   Before proceeding further with the properties of the solutions to
  278   the approximative Sine-Gordon, including various orders of nonlinearities,
  279   the general properties will now be illustrated.
  280   In order to illustrate the behaviour of the Sine-Gordon equation,
  281   we may normalize it by using the normalized time $\tau=(g/l)^{1/2}t$,
  282   giving the Sine-Gordon equation in the normalized form
  283   $${{d^2\vartheta}\over{d\tau^2}}+\sin\vartheta=0.\eqno{(4)}$$
  284   The numerical solutions to the normalized Sine-Gordon
  285   equation are in Fig.~2 shown for initial conditions (a) $y(0)=0.1$,
  286   (b) $y(0)=2.1$, and (c) $y(0)=3.1$, all cases with $y'(0)=0$.
  287   \medskip
  288   \centerline{\epsfxsize=45mm\epsfbox{sg010.eps}
  289   \hskip 7.5mm\epsfxsize=45mm\epsfbox{sg210.eps}
  290   \hskip 7.5mm\epsfxsize=45mm\epsfbox{sg310.eps}}
  291   \centerline{Figure 2. Numerical solutions to the normalized Sine-Gordon
  292     equation.}
  293   \medskip
  294   \noindent
  295   First of all, we may consider the linear case, for which the approximation
  296   $\sin\vartheta\approx\vartheta$ holds. For this case, the Sine-Gordon
  297   equation~(4) hence reduces to the one-dimensional linear wave-equation,
  298   with solutions $\vartheta=A\sin((g/l)^{1/2}(t-t_0))$.
  299   As seen in the frequency domain, this solution gives a delta peak
  300   at $\omega=(g/l)^{1/2}$ in the power spectrum $|\tilde{\vartheta}(\omega)|^2$,
  301   with no other frequency components present.
  302   However, if we include the nonlinearities, the previous sine-wave
  303   solution will tend to flatten at the peaks, as well as increase in
  304   period, and this changes the power spectrum to be broadened as well
  305   as flattened out.
  306   In other words, the solution to the Sine-Gordon give rise to a wide
  307   spectrum of frequencies, as compared to the delta peaks of the
  308   solutions to the linearized, approximative Sine-Gordon equation.
  309   
  310   From the numerical solutions, we may draw the conclusion that whenever
  311   higher order nonlinear restoring forces come into play, even such a
  312   simple mechanical system as the pendulum will carry frequency components
  313   at a set of frequencies differing from the single frequency given
  314   by the linearized model of motion.
  315   
  316   More generally, hiding the fact that for this particular case the
  317   restoring force is a simple sine function, the equation of motion
  318   for the pendulum can be written as
  319   $${{d^2\vartheta}\over{dt^2}}+a^{(0)}+a^{(1)}\vartheta+a^{(2)}\vartheta^2
  320   +a^{(3)}\vartheta^3+\ldots=0.\eqno{(5)}$$
  321   This equation of motion may be compared with the nonlinear wave
  322   equation for the electromagnetic field of a travelling optical
  323   wave of angular frequency $\omega$, of the form
  324   $$
  325     {{\partial^2 E}\over{\partial z^2}}
  326      +{{\omega^2}\over{c^2}}E
  327      +{{\omega^2}\over{c^2}}(\chi^{(1)}E+\chi^{(2)}E^2+\chi^{(3)}E^3+\ldots)=0,
  328   $$
  329   which clearly shows the similarity between the nonlinear wave propagation
  330   and the motion of the nonlinear pendulum.
  331   
  332   Having solved the particular problem of the nonlinear pendulum,
  333   we may ask ourselves if the equations of motion may be altered
  334   in some way in order to give insight in other areas of nonlinear
  335   physics as well.
  336   For example, the series~(5) that define the feedback that tend to
  337   restore the mechanical pendulum to its rest position clearly defines
  338   equations of motion that conserve the total energy of the
  339   mechanical system.
  340   This, however, in generally not true for an arbitrary series
  341   of terms of various power for the restoring force.
  342   As we will later on see, in nonlinear optics we generally have a
  343   complex, though in many cases most predictable, transfer of energy
  344   between modes of different frequencies and directions of propagation.
  345   
  346   \section{The anharmonic oscillator}
  347   Among the simplest models of interaction between light and matter
  348   is the all-classical one-electron oscillator, consisting of a
  349   negatively charged particle (electron) with mass $m_{\rm e}$, mutually
  350   interacting with a positively charged particle (proton)
  351   with mass $m_{\rm p}$, through attractive Coulomb forces.
  352   \medskip
  353   \centerline{\epsfxsize=90mm\epsfbox{../images/spring/udspring.1}}
  354   \smallskip
  355   \centerline{Figure 5. Setup of the one-dimensional undamped spring model.}
  356   \medskip
  357   
  358   In the one-electron oscillator model, several levels of approximations
  359   may be applied to the problem, with increasing algebraic complexity.
  360   At the first level of approximation, the proton is assumed to
  361   be fixed in space, with the electron free to oscillate around
  362   the proton.
  363   Quite generally, at least within the scope of linear optics,
  364   the restoring spring force which confines the electron
  365   can be assumed to be linear with the displacement distance
  366   of the electron from the central position.
  367   Providing the very basic models of the concept of refractive index and
  368   optical dispersion, this model has been applied by numerous authors,
  369   such as Feynman~[R.~P.~Feynman, {\sl Lectures on Physics} (Addison-Wesley,
  370   Massachusetts, 1963)], and Born and Wolf~[M.~Born and E.~Wolf,
  371   {\sl Principles of Optics} (Cambridge University Press, Cambridge, 1980)].
  372   
  373   Moving on to the next level of approximation, the bound proton-electron
  374   pair may be considered as constituting a two-body central force
  375   problem of classical mechanics, in which one may assume a fixed center
  376   of mass of the system, around which the proton as well as the electron
  377   are free to oscillate.
  378   In this level of approximation, by introducing the concept of reduced
  379   mass for the two moving particles, the equations of motion for the
  380   two particles can be reduced to one equation of motion, for the
  381   evolution of the electric dipole moment of the system.
  382   
  383   The third level of approximation which may be identified is when
  384   the center of mass is allowed to oscillate as well, in which case
  385   an equation of motion for the center of mass appears in addition to
  386   the one for the evolution of the electric dipole moment.
  387   
  388   In each of the models, nonlinearities of the restoring central force
  389   field may be introduced as to include nonlinear interactions as well.
  390   It should be emphasized that the spring model, as now will be introduced,
  391   gives an identical form of the set of nonzero elements of the
  392   susceptibility tensors, as compared with those obtained using a
  393   quantum mechanical analysis.
  394   
  395   Throughout this analysis, the wavelength of the electromagnetic
  396   field will be assumed to be sufficiently large in order to neglect
  397   any spatial variations of the fields over the spatial extent of the
  398   oscillator system.
  399   In this model, the central force field is modelled by a mechanical
  400   spring force with spring constant $k_{\rm e}$, as shown schematically
  401   in Fig.~5, and the all-classical Newton's equations
  402   of motion for the electron and nucleus are
  403   $$
  404     \eqalign{
  405       m_{\rm e}{{\partial^2 x_{\rm e}}\over{\partial t^2}}
  406         &=\underbrace{-eE(t)}_{\rm optical}
  407          -\underbrace{k_0(x_{\rm e}-x_{\rm n})
  408                   +k_1(x_{\rm e}-x_{\rm n})^2}_{\rm spring},\cr
  409       m_{\rm n}{{\partial^2 x_{\rm n}}\over{\partial t^2}}
  410         &=\underbrace{+eE(t)}_{\rm optical}
  411          +\underbrace{k_0(x_{\rm e}-x_{\rm n})
  412                   -k_1(x_{\rm e}-x_{\rm n})^2}_{\rm spring},\cr
  413     }
  414   $$
  415   corresponding to a system of two particles connected by a spring with
  416   spring ``constant''
  417   $$
  418     k=-{{\partial F^{({\rm spring})}_{\rm e}}
  419        \over{\partial (x_{\rm e}-x_{\rm n})}}
  420      ={{\partial F^{({\rm spring})}_{\rm n}}
  421        \over{\partial (x_{\rm e}-x_{\rm n})}}
  422      =k_0-2k_1(x_{\rm e}-x_{\rm n}).
  423   $$
  424   By introducing the reduced mass\footnote{${}^{12}$}{Herbert Goldstein,
  425   {\it Classical Mechanics}, 2nd ed.~(Addison-Wesley, Massachusetts, 1980).}
  426   $m_{\rm r} = m_{\rm e} m_{\rm n} / (m_{\rm e} + m_{\rm n})$ of the system,
  427   the equation of motion for the electric dipole moment
  428   $p=-e(x_{\rm e}-x_{\rm n})$ is then obtained as
  429   $$
  430     {{\partial^2 p}\over{\partial t^2}}+{{k_0}\over{m_{\rm r}}}p
  431       +{{k_1}\over{e m_{\rm r}}}p^2
  432       ={{e^2}\over{m_{\rm r}}}E(t).\eqno(6)
  433   $$
  434   This inhomogeneous nonlinear ordinary differential equation for the
  435   electric dipole moment is the primary interest in the discussion
  436   that now is to follow.
  437   
  438   The electric dipole moment of the anharmonic oscillator is now
  439   expressed in terms of a perturbation series as
  440   $$
  441     p(t)=p^{(0)}(t)+\underbrace{p^{(1)}(t)}_{\propto E(t)}
  442                    +\underbrace{p^{(2)}(t)}_{\propto E^2(t)}
  443                    +\underbrace{p^{(3)}(t)}_{\propto E^3(t)}
  444                    +\ldots,
  445   $$
  446   where each term in the series is proportional to the applied electrical
  447   field strength to the power as indicated in the superscript of repective
  448   term, and formulate the system of $n+1$ equations for $p^{(k)}$,
  449   $k=0,1,2,\ldots,n$, that define the time evolution of the electric
  450   dipole. By inserting the perturbation series into Eq.~(6), we hence
  451   have the equation
  452   $$
  453     \eqalign{
  454       {{\partial^2 p^{(0)}}\over{\partial t^2}}&
  455          +{{\partial^2 p^{(1)}}\over{\partial t^2}}
  456          +{{\partial^2 p^{(2)}}\over{\partial t^2}}
  457          +{{\partial^2 p^{(3)}}\over{\partial t^2}}
  458          +\ldots\cr
  459       &+{{k_0}\over{m_{\rm r}}}p^{(0)}
  460          +{{k_0}\over{m_{\rm r}}}p^{(1)}
  461          +{{k_0}\over{m_{\rm r}}}p^{(2)}
  462          +{{k_0}\over{m_{\rm r}}}p^{(3)}
  463          +\ldots\cr
  464       &+{{k_1}\over{e m_{\rm r}}}
  465          (p^{(0)}+p^{(1)}+p^{(2)}+\ldots)(p^{(0)}+p^{(1)}+p^{(2)}+\ldots)
  466       ={{e^2}\over{m_{\rm r}}}E(t).\cr
  467     }
  468   $$
  469   Since this equation is to hold for an arbitrary electric field $E(t)$,
  470   that is to say, at least within the limits of the validity of the
  471   perturbation analysis, each set of terms with equal power dependence
  472   of the electric field must individually satisfy the relation.
  473   By sorting out the various powers and identifying terms in the left
  474   and right hand sides of the equation, we arrive at the system
  475   of equations
  476   $$
  477     \eqalign{
  478       &{{\partial^2 p^{(0)}}\over{\partial t^2}}
  479         +{{k_0}\over{m_{\rm r}}}p^{(0)}
  480         +{{k_1}\over{e m_{\rm r}}}p^{(0)}{}^2=0,\cr
  481       &{{\partial^2 p^{(1)}}\over{\partial t^2}}
  482         +{{k_0}\over{m_{\rm r}}}p^{(1)}
  483         +{{k_1}\over{e m_{\rm r}}}2p^{(0)}p^{(1)}
  484         ={{e^2}\over{m_{\rm r}}}E(t),\cr
  485       &{{\partial^2 p^{(2)}}\over{\partial t^2}}
  486         +{{k_0}\over{m_{\rm r}}}p^{(2)}
  487         +{{k_1}\over{e m_{\rm r}}}(2p^{(0)}p^{(2)}+p^{(1)}{}^2)=0,\cr
  488       &{{\partial^2 p^{(3)}}\over{\partial t^2}}
  489         +{{k_0}\over{m_{\rm r}}}p^{(3)}
  490         +{{k_1}\over{e m_{\rm r}}}(2p^{(0)}p^{(3)}+2p^{(1)}p^{(2)})=0,\cr
  491     }
  492   $$
  493   where we kept terms with powers of the electric field up to and including
  494   order three.
  495   At a first glance, this system seem to suggest that only the first
  496   order of the perturbation series depends on the applied electric
  497   field of the light; however, taking a closer look at the system,
  498   one can easily verify that all orders of the dipole moment
  499   is coupled directly to the lower order terms.
  500   The system of equations for $p^{(k)}$ can now be solved for
  501   $k=0,1,2,\ldots$, in that order, to successively provide the
  502   basis of solutions for higher and higher order terms, until reaching
  503   some $k=n$ after which we may safely neglect the reamaining terms,
  504   hence providing an approximate solution.\footnote{${}^{13}$}{It should
  505   though be emphasized that in the limit $n\to\infty$, the described
  506   theory still is an exact description of the motion of the electric
  507   dipole moment within this model of interaction between light and matter.}
  508   
  509   The zeroth order term in the perturbation series is decribed
  510   by a nonlinear ordinary differential equation of order two,
  511   a so-calles {\sl Riccati equation}, which analytically can be
  512   solved exactly, either by directly applying the theory of Jacobian
  513   elliptic integrals of by applying the Riccati
  514   transormation.\footnote{${}^{14}$}{For examples of the application of the
  515   Riccati transformation, see Zwillinger, {\it Handbook of Differential
  516   Equations}, 2nd ed.~(Academic Press, Boston, 1992).}
  517   However, by considering a system starting from rest, at a state of
  518   equilibrium, we can immediately draw the conclusion that $p^{(0)}(t)$
  519   must be identically zero for all times $t$.
  520   This, of course, only holds for this particular model; in many
  521   molecular systems, such in water, a permanent static dipole moment
  522   is present, something that is left out in this particular spring model
  523   of ours. (Not to be confused with the static polarization induced
  524   by the electric field, which by definition of the terms in the
  525   perturbation series is included in higher order terms, depending
  526   on the power of the electric field.)
  527   
  528   The first order term in the perturbation series is the first and only
  529   one with an explicit dependence of the electric field of the light.
  530   Since the zeroth order perturbation term is zero, the differential
  531   equation for the first order term is linear, which simplifies the
  532   calculus.
  533   However, since it is an inhomogeneous differential equation, we must
  534   generally look for a total solution to the equation as a sum of
  535   a homogeneous solution (with zero right hand side) and a particular
  536   solution (with the electric field in the right hand side present).
  537   The homogeneous solution, which will contain two constants of integration
  538   (since we are considering  second-order ordinary differential equation)
  539   will though only give the part of the solution which depend on initial
  540   conditions, that is to say, in this case a harmonic natural oscillation
  541   of the spring system which in the presence of damping terms rapidly
  542   would decrease to zero.
  543   This implies that in order to find steady-state solutions, in which
  544   the oscillation of the dipole moment directly follows the oscillation
  545   of the electric field of the light, we may directly start looking for
  546   the particular solution.
  547   For a time harmonic electric field, here taken as
  548   $$E(t)=E_{\omega}\sin(\omega t),$$
  549   the particular solution for the first order term is after some
  550   straightforward algebra given as\footnote{${}^{15}$}{For the sake
  551   of self consistency, the general solution for the first order
  552   term is given as
  553   $$p^{(1)}=A\cos((k_0/m_{\rm r})^{1/2}t)+B\sin((k_0/m_{\rm r})^{1/2}t)
  554     +{{e^2/m_{\rm r}}\over{k_0/m_{\rm r}-\omega^2}}E_{\omega}\sin(\omega t),$$
  555   where $A$ and $B$ are constants of integration, determined by initial
  556   conditions.}
  557   $$
  558     p^{(1)}={{(e^2/m_{\rm r})}\over{(k_0/m_{\rm r}-\omega^2)}}E_{\omega}
  559               \sin(\omega t),\qquad\omega^2\ne(k_0/m_{\rm r}).
  560   $$
  561   For a material consisting of $N$ dipoles per unit volume, and by
  562   following the conventions for the linear electric susceptibility
  563   in SI units, this corresponds to a first order electric polarization
  564   density of the form
  565   $$
  566     \eqalign{
  567       P^{(1)}(t)
  568         &=P^{(1)}_{\omega}\sin(\omega t)\cr
  569         &=\varepsilon_0\chi^{(1)}(\omega)E_{\omega}\sin(\omega t),\cr
  570     }
  571   $$
  572   with the {\sl first order (linear) electric susceptibility} given as
  573   $$
  574     \chi^{(1)}(\omega)
  575       =\chi^{(1)}(-\omega;\omega)
  576       ={{N}\over{\varepsilon_0}}{{(e^2/m_{\rm r})}\over{(\Omega^2-\omega^2)}},
  577   $$
  578   where the resonance frequency $\Omega^2=k_0/m_{\rm r}$ was introduced.
  579   The Lorenzian shape of the frequency dependence is shown in Fig.~6.
  580   \vfill\eject
  581   
  582   \medskip
  583   \centerline{\epsfxsize=90mm\epsfbox{chi1.eps}}
  584   \smallskip
  585   \centerline{Figure 6. Lorenzian shape of the linear susceptibility
  586     $\chi^{(1)}(-\omega;\omega)$.}
  587   \medskip
  588   Continuing with the second order perturbation term, some straightforward
  589   algebra gives that the particular solution for the second order term
  590   of the electric dipole moment becomes
  591   $$
  592     \eqalign{
  593       p^{(2)}(t)&=-{{k_1e^3}\over{2k_0 m^2_{\rm r}}}
  594                 {{1}\over{(\Omega^2-\omega^2)}}E^2_{\omega}
  595                +{{k_1e^3}\over{2m^3_{\rm r}}}
  596                 {{1}\over{(\Omega^2-\omega^2)(\Omega^2-4\omega^2)}}
  597                 E^2_{\omega}\cr
  598                &\qquad\qquad+{{k_1e^3}\over{m^3_{\rm r}}}
  599                 {{1}\over{(\Omega^2-\omega^2)(\Omega^2-4\omega^2)}}
  600                 E^2_{\omega}\sin^2(\omega t).\cr
  601     }
  602   $$
  603   In terms of the polarization density of the medium, still with $N$
  604   dipoles per unit volume and following the conventions in regular
  605   SI units, this can be written as
  606   $$
  607     \eqalign{
  608       P^{(2)}(t)
  609         &=P^{(0)}_{0}+P^{(0)}_{2\omega}\sin(2\omega t)\cr
  610         &=\underbrace{\varepsilon_0\chi^{(2)}(0;\omega,-\omega)
  611            E_{\omega}E_{\omega}}_{\rm DC\ polarization}
  612         +\underbrace{\varepsilon_0\chi^{(2)}(-2\omega;\omega,\omega)
  613            E_{\omega}E_{\omega}\sin(2\omega)}_{\rm second
  614            \ harmonic\ polarization}\cr
  615     }
  616   $$
  617   with the {\sl second order (quadratic) electric susceptibility} given as
  618   $$
  619     \eqalign{
  620       \chi^{(2)}(0;\omega,-\omega)&=
  621         {{N}\over{\varepsilon_0}}{{k_1e^3}\over{2m^3_{\rm r}}}
  622         \bigg[{{1}\over{(\Omega^2-\omega^2)(\Omega^2-4\omega^2)}}
  623               -{{1}\over{\Omega^2(\Omega^2-\omega^2)}}\bigg],\cr
  624       \chi^{(2)}(2\omega;\omega,\omega)&=
  625         {{N}\over{\varepsilon_0}}{{k_1e^3}\over{m^3_{\rm r}}}
  626         {{1}\over{(\Omega^2-\omega^2)(\Omega^2-4\omega^2)}}.\cr
  627     }
  628   $$
  629   From this we may notice that for one-photon resonances, the nonlinearities
  630   are enhanced whenever $\omega\approx\Omega$ or $2\omega\approx\Omega$,
  631   for the induced DC as well as the second harmonic polarization density.
  632   
  633   The explicit frequency dependencies of the susceptibilities
  634   $\chi^{(2)}(-2\omega;\omega,\omega)$ and $\chi^{(2)}(0;\omega,-\omega)$
  635   are shown in Figs.~7 and 8.
  636   \vfill\eject
  637   
  638   \medskip
  639   \centerline{\epsfxsize=90mm\epsfbox{chi2a.eps}}
  640   \smallskip
  641   \centerline{Figure 7. Lorenzian shape of the linear susceptibility
  642     $\chi^{(2)}(-2\omega;\omega,\omega)$ (SHG).}
  643   \bigskip
  644   \centerline{\epsfxsize=90mm\epsfbox{chi2b.eps}}
  645   \smallskip
  646   \centerline{Figure 8. Lorenzian shape of the linear susceptibility
  647     $\chi^{(2)}(0;\omega,-\omega)$ (DC).}
  648   \medskip
  649   
  650   A well known fact in electromagnetic theory is that an electric dipole
  651   that oscillates at a certain angular frequency, say at $2\omega$,
  652   also emits electromagnetic radiation at this frequency.
  653   In particular, this implies that the term described by the
  654   susceptibility $\chi^{(2)}(-2\omega;\omega,\omega)$ will generate
  655   light at twice the angular frequency of the light, hence generating
  656   a second harmonic light wave.
  657   \bye
  658   

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