Return to previous page

Contents of file 'lect7/lect7.tex':




Deprecated: Function split() is deprecated in /storage/content/45/2011745/jonsson.eu/public_html/php/htmlicise.php on line 46
    1   % File: nlopt/lect7/lect7.tex [pure TeX code]
    2   % Last change: February 17, 2003
    3   %
    4   % Lecture No 7 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{7}
   51   So far, this course has mainly dealt with the dependence of the angular
   52   frequency of the light and molecular interaction strength in the
   53   description of nonlinear optics. In this lecture, we will now end
   54   this development of the description of interaction between light
   55   and matter, in favour of more engineering practical techniques
   56   for describing the theory of an experimental setup in a certain
   57   geometry, and for reducing the number of necessary tensor elements
   58   needed for describing a mediumof a certain crystallographic point-symmetry
   59   group.
   60   
   61   \section{Motivation for analysis of susceptibilities in rotated
   62     coordinate systems}
   63   For a given experimental setup, it is often convenient to introduce
   64   some kind of reference coordinate frame, in which one for example express
   65   the wave propagation as a linear motion along some Cartesian coordinate axis.
   66   This laboratory reference frame might be chosen, for example, with the
   67   $z$-axis coinciding with the direction of propagation of the optical wave
   68   at the laser output, in the phase-matched direction of an optical parametric
   69   oscillator (OPO), after some beam aligning mirror, etc.
   70   
   71   In some cases, it might be so that this laboratory frame coincide with
   72   the natural coordinate frame\footnote{${}^1$}{The natural coordinate frame
   73   of the crystal is often chosen such that some particular symmetry axis
   74   is chosen as one of the Cartesian axes.} of the nonlinear crystal, in which
   75   case the coordinate indices of the linear as well as nonlinear susceptibility
   76   tensors take the same values as the coordinates of the laboratory frame.
   77   However, we cannot generally assume the coordinate frame of the crystal to
   78   coincide with a conveniently chosen laboratory reference frame, and this
   79   implies that we generally should be prepared to spatially transform the
   80   susceptibility tensors to arbitrarily rotated coordinate frames.
   81   
   82   Having formulated these spatial transformation rules, we will also directly
   83   benefit in another aspect of the description of nonlinear optical
   84   interactions, namely the reduction of the susceptibility tensors
   85   to the minimal set of nonzero elements. This is typically performed
   86   by using the knowledge of the so called {\sl crystallographic point
   87   ymmetry group} of the medium, which essentially is a description of the
   88   spatial operations (rotations, inversions etc.) that define the symmetry
   89   operations of the medium.
   90   
   91   As a particular example of the applicability of the spatial transformation
   92   rules (which we soon will formulate) is illustrated in Figs.~1 and~2.
   93   In Fig.~1, the procedure for analysis of sum or frequency difference
   94   generation is outlined. Starting from the description of the linear and
   95   nonlinear susceptibility tensors of the medium, as we previously have
   96   derived the relations from a first principle approach in Lectures 1--6,
   97   we obtain the expressions for the electric polarization densities of the
   98   medium as functions of the applied electric fields of the optical wave
   99   inside the nonlinear crystal. These polarization densities are then
  100   inserted into the wave equation, which basically is derived from Maxwell's
  101   equations of motion for the electromagnetic field. In the wave equation,
  102   the polarization densities act as source terms in an otherwise homogeneous
  103   equation for the motion of the electromagnetic field in vacuum.
  104   
  105   As the wave equation is solved for the electric field, here taken in complex
  106   notation, we have solved for the general output from the crystal, and we
  107   can then design the experiment in such a way that an optimal efficiency
  108   is obtained.
  109   \vfill\eject
  110   
  111   \centerline{\epsfxsize=150mm\epsfbox{../images/nonrotse/nonrotse.1}}
  112   \medskip
  113   \centerline{Figure 1. The setup in which the orientation of the laboratory
  114     and crystal frames coincide.}
  115   \medskip
  116   \noindent
  117   In Fig.~1, this outline is illustrated for the case where the natural
  118   coordinate frame of the crystal happens to coincide with the coordinate
  119   system of the laboratory frame. In this case, all elements of the
  120   susceptibilities taken in the coordinate frame of the crystal (which
  121   naturally is the coordinate frame in which we can obtain tabulated
  122   sets of tensor elements) will coincide with the elements as taken
  123   in the laboratory frame, and the design and interpretation of the
  124   experiment is straightforward.
  125   
  126   However, this setup clearly constitutes a rare case, since we have
  127   infinitely many other possibilities of orienting the crystal relative
  128   the laboratory coordinate frame.
  129   Sometimes the experiment is {\sl designed} with the crystal and laboratory
  130   frames coinciding, in order to simplify the interpretation of an experiment,
  131   and sometimes it is instead {\sl necessary} to rotate the crytal, in order
  132   to achieve phase-matching of nonlinear process, as is the case in for example
  133   most schemes for second-order optical parametric amplification.
  134   
  135   If now the crystal frame is rotated with respect to the laboratory frame,
  136   as shown in Fig.~2, we should make up our mind in which system we would
  137   like the wave propagation to be analyzed. In some cases, it might be so that
  138   the output of the experimental setup is most easily interpreted in the
  139   coordinate frame of the crystal, but in most cases, we have a fixed laboratory
  140   frame (fixed by the orientation of the laser, positions of mirrors, etc.)
  141   in which we would like to express the wave propagation and interaction
  142   between light and matter.
  143   \vfill\eject
  144   
  145   \centerline{\epsfxsize=150mm\epsfbox{../images/rotsetup/rotsetup.1}}
  146   \medskip
  147   \centerline{Figure 2. The setup in which the crystal frame is rotated
  148     relative the laboratory frame.}
  149   \medskip
  150   \noindent
  151   In Fig.~2, we would, in order to express the nonlinear process in the
  152   laboratory frame, like to obtain the naturally appearing susceptibilities
  153   $\chi^{(2)}_{xyz}$, $\chi^{(2)}_{xxx}$, etc., in the laboratory frame
  154   instead, as $\chi^{(2)}_{x'y'z'}$, $\chi^{(2)}_{x'x'x'}$, etc.
  155   
  156   Just to summarize, why are then the transformation rules and spatial
  157   symmetries of the meduim so important?
  158   \smallskip
  159   \item{$\bullet$}{Hard to make physical conclusions about generated
  160     optical fields unless orientation of the laboratory and crystal
  161     frames coincide.}
  162   \item{$\bullet$}{Spatial symmetries often significantly simplifies the
  163     wave propagation problem (by choosing a suitable polarization state
  164     and direction of propagation of the light, etc.).}
  165   \item{$\bullet$}{Useful for reducing the number of necessary elements
  166     of the susceptibility tensors (using Neumann's principle).}
  167   \smallskip
  168   
  169   \section{Optical properties in rotated coordinate frames}
  170   Consider two coordinate systems described by Cartesian coordinates
  171   $x_{\alpha}$ and $x'_{\alpha}$, respectively. The coordinate systems
  172   are rotated with respect to each other, and the relation between
  173   the coordinates are described by the $[3\times 3]$ transformation
  174   matrix $R_{ab}$ as
  175   $$
  176     {\bf x}'={\bf R}{\bf x}\qquad\Leftrightarrow
  177       \qquad x'_{\alpha}=R_{\alpha\beta}x_{\beta},
  178     \eqno{[{\rm B.\,\&\,C.}\ (5.40)]}
  179   $$
  180   where ${\bf x}=(x,y,z)^{\rm T}$ and ${\bf x}'=(x',y',z')^{\rm T}$
  181   are column vectors.
  182   The inverse transformation between the coordinate systems is similarly
  183   given as
  184   $$
  185     {\bf x}={\bf R}^{-1}{\bf x}'\qquad\Leftrightarrow
  186       \qquad x_{\beta}=R_{\alpha\beta}x'_{\alpha}.
  187     \eqno{[{\rm B.\,\&\,C.}\ (5.41)]}
  188   $$
  189   
  190   \centerline{\epsfxsize=65mm\epsfbox{../images/rotframe/rotframe.1}}
  191   \medskip
  192   \noindent{Figure 3. Illustration of proper rotation of the crystal
  193     frame $(x,y,z)$ relative to the laboratory reference frame $(x',y',z')$,
  194     by means of $x'_{\alpha} = R_{\alpha\beta} x_{\beta}$
  195     with $\det{({\bf R})}=1$.}
  196   \medskip
  197   \noindent
  198   
  199   \centerline{\epsfxsize=68mm\epsfbox{../images/rotfig/rotfig.1}}
  200   \medskip
  201   \noindent{Figure 4. The coordinate transformations
  202     (a) ${\bf x}=(x,y,z)\mapsto{\bf x}'=(-x,-y,z)$, constituting
  203     a proper rotation around the $z$-axis, and
  204     (b) the space inversion ${\bf x}\mapsto{\bf x}'=-{\bf x}$,
  205     an improper rotation corresponding to, for example,
  206     a rotation around the $z$-axis followed by an inversion
  207     in the $xy$-plane.}
  208   \medskip
  209   \noindent
  210   
  211   We should notice that there are two types of rotations that are
  212   encountered as transformations:
  213   \smallskip
  214   \item{$\bullet$}{Proper rotations, for which $\det({\bf R})=1$.
  215     (Righthanded systems keep being righthanded, and lefthanded systems
  216     keep being lefthanded.)}
  217   \item{$\bullet$}{Improper rotations, for which $\det({\bf R})=-1$.
  218     (Righthanded systems are transformed into lefthanded systems, and
  219     vice versa.)}
  220   \smallskip
  221   
  222   The electric field ${\bf E}({\bf r},t)$ and electric polarization
  223   density ${\bf P}({\bf r},t)$ are both polar quantities that transform
  224   in the same way as regular Cartesian coordinates, and hence we have
  225   descriptions of these quantities in coordinate systems $(x,y,z)$
  226   and $(x',y',z')$ related to each other as
  227   $$
  228     E'_{\mu}({\bf r},t)=R_{\mu u}E_u({\bf r},t)
  229     \qquad\Leftrightarrow\qquad
  230     E_u({\bf r},t)=R_{\mu u}E'_{\mu}({\bf r},t),
  231   $$
  232   and
  233   $$
  234     P'_{\mu}({\bf r},t)=R_{\mu u}P_u({\bf r},t)
  235     \qquad\Leftrightarrow\qquad
  236     P_u({\bf r},t)=R_{\mu u}P'_{\mu}({\bf r},t),
  237   $$
  238   respectively.
  239   Using these transformation rules, we will now derive the form of the
  240   susceptibilities in rotated coordinate frames.
  241   \medskip
  242   
  243   \subsection{First order polarization density in rotated coordinate frames}
  244   From the transformation rule for the electric polarization density above,
  245   using the standard form as we previously have expressed the electric
  246   field dependence, we have for the first order polarization density
  247   in the primed coordinate system
  248   $$
  249     \eqalign{
  250       P^{(1)}_{\mu}{}'({\bf r},t)
  251         &=R_{\mu u}P^{(1)}_u({\bf r},t)\cr
  252         &=R_{\mu u}\varepsilon_0\int^{\infty}_{-\infty}
  253           \chi^{(1)}_{ua}(-\omega;\omega)
  254           E_a(\omega)\exp(-i\omega t)\,d\omega\cr
  255         &=R_{\mu u}\varepsilon_0\int^{\infty}_{-\infty}
  256           \chi^{(1)}_{ua}(-\omega;\omega)
  257           R_{\alpha a}E'_{\alpha}(\omega)\exp(-i\omega t)\,d\omega\cr
  258         &=\varepsilon_0\int^{\infty}_{-\infty}
  259           \chi^{(1)}_{\mu\alpha}{}'(-\omega;\omega)
  260           E'_{\alpha}(\omega)\exp(-i\omega t)\,d\omega\cr
  261     }
  262   $$
  263   where
  264   $$
  265     \chi^{(1)}_{\mu\alpha}{}'(-\omega;\omega)
  266       =R_{\mu u}R_{\alpha a}\chi^{(1)}_{ua}(-\omega;\omega)
  267     \eqno{[{\rm B.\,\&\,C.}\ (5.45)]}
  268   $$
  269   is the linear electric susceptibility taken in the primed coordinate system.
  270   \medskip
  271   
  272   \subsection{Second order polarization density in rotated coordinate frames}
  273   Similarly, we have the second order polarization density
  274   in the primed coordinate system as
  275   $$
  276     \eqalign{
  277       P^{(2)}_{\mu}{}'({\bf r},t)
  278         &=R_{\mu u}P^{(2)}_u({\bf r},t)\cr
  279         &=R_{\mu u}\varepsilon_0\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  280           \chi^{(2)}_{uab}(-\omega_{\sigma};\omega_1,\omega_2)
  281           E_a(\omega_1) E_b(\omega_2)
  282           \exp[-i(\omega_1+\omega_1)t]\,d\omega_2\,d\omega_1\cr
  283         &=R_{\mu u}\varepsilon_0\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  284           \chi^{(2)}_{uab}(-\omega_{\sigma};\omega_1,\omega_2)
  285           R_{\alpha a} E'_{\alpha}(\omega_1)
  286           R_{\beta b} E'_{\beta}(\omega_2)
  287           \exp[-i(\omega_1+\omega_1)t]\,d\omega_2\,d\omega_1\cr
  288         &=\varepsilon_0\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
  289           \chi^{(2)}_{\mu\alpha\beta}{}'(-\omega_{\sigma};\omega_1,\omega_2)
  290           E'_{\alpha}(\omega_1) E'_{\beta}(\omega_2)
  291           \exp[-i(\omega_1+\omega_1)t]\,d\omega_2\,d\omega_1\cr
  292     }
  293   $$
  294   where
  295   $$
  296     \chi^{(2)}_{\mu\alpha\beta}{}'(-\omega_{\sigma};\omega_1,\omega_2)
  297       =R_{\mu u}R_{\alpha a}R_{\beta b}
  298        \chi^{(2)}_{uab}(-\omega_{\sigma};\omega_1,\omega_2)
  299     \eqno{[{\rm B.\,\&\,C.}\ (5.46)]}
  300   $$
  301   is the second order electric susceptibility taken in the primed
  302   coordinate system.
  303   
  304   \subsection{Higher order polarization densities in rotated coordinate frames}
  305   In a manner completely analogous to the second order susceptibility,
  306   the transformation rule between the primed and unprimed coordinate
  307   systems can be obtained for the $n$th order elements of the electric
  308   susceptibility tensor as
  309   $$
  310     \chi^{(n)}_{\mu\alpha_1\cdots\alpha_n}{}'
  311     (-\omega_{\sigma};\omega_1,\ldots,\omega_n)
  312       =R_{\mu u}R_{\alpha_1 a_1}\cdots R_{\alpha_n a_n}
  313        \chi^{(n)}_{u a_1\cdots a_n}(-\omega_{\sigma};\omega_1,\ldots,\omega_n).
  314     \eqno{[{\rm B.\,\&\,C.}\ (5.47)]}
  315   $$
  316   
  317   \section{Crystallographic point symmetry groups}
  318   Typically, a particular point symmetry group of the medium can be
  319   described by the {\sl generating matrices} that describe the minimal
  320   set of transformation matrices (describing a set of symmetry operations)
  321   that will be necessary for the reduction of the constitutive tensors.
  322   Two systems are widely used for the description of point symmetry
  323   groups:\footnote{${}^2$}{C.~f.~Table 2 of the handed out Hartmann's
  324   {\sl An Introduction to Crystal Physics}.}
  325   \smallskip
  326   \item{$\bullet$}{The International system, e.~g.~$\bar{4}3m$, $m3m$,
  327     $422$, etc.}
  328   \item{$\bullet$}{The Sch\"{o}nflies system, e.~g.~$T_d$, $O_h$,
  329     $D_4$, etc.}
  330   \smallskip
  331   The crystallographic point symmetry groups may contain any of
  332   the following symmetry operations:
  333   
  334   {\it 1. Rotations through integral multiples of $2\pi/n$ about some axis.}
  335   The axis is called the $n$-fold rotation axis. It is in solid state
  336   physics shown [1--3] that a Bravais lattice can contain only 2-, 3-, 4-,
  337   or 6-fold axes, and since the crystallographic point symmetry groups
  338   are contained in the Bravais lattice point groups, they too can only
  339   have these axes.
  340   
  341   {\it 2. Rotation-reflections.}
  342   Even when a rotation through $2\pi/n$ is not a symmetry element,
  343   sometimes such a rotation followed by a reflection in a plane
  344   perpendicular to the axis may be a symmetry operation.
  345   The axis is then called an $n$-fold rotation-reflection axis.
  346   For example, the groups $S_6$ and $S_4$ have 6- and 4-fold
  347   rotation-reflection axes.
  348   
  349   {\it 3. Rotation-inversions.}
  350   Similarly, sometimes a rotation through $2\pi/n$ followed by an
  351   inversion in a point lying on the rotation axis is a symmetry
  352   element, even though such a rotation by itself is not.
  353   The axis is then called an $n$-fold rotation-inversion axis.
  354   However, the axis in $S_6$ is only a 3-fold rotation-inversion
  355   axis.
  356   
  357   {\it 4. Reflections.}
  358   A reflection takes every point into its mirror image in a plane,
  359   known as a mirror plane.
  360   
  361   {\it 5. Inversions.}
  362   An inversion has a single fixed point. If that point is taken as
  363   the origin, then every other point ${\bf r}$ is taken into $-{\bf r}$.
  364   
  365   \section{Sch\"onflies notation for the non-cubic
  366      crystallographic point groups}
  367   The twenty-seven {\sl non-cubic} crystallographic point symmetry groups
  368   may contain any of the following symmetry operations, here given
  369   in Sch\"onflies notation\footnote{${}^3$}{In Sch\"onflies notation,
  370   $C$ stands for ``cyclic'', $D$ for ``dihedral'', and $S$ for ``spiegel''.
  371   The subscripts h, v, and d stand for ``horizontal'', ``vertical'',
  372   and ``diagonal'', respectively, and refer to the placement of the
  373   placement of the mirror planes with respect to the $n$-fold axis,
  374   always considered to be vertical. (The ``diagonal'' planes in
  375   $D_{n{\rm d}}$ are vertical and bisect the angles between the
  376   2-fold axes)}:
  377   
  378   \citem[$C_n$]{These groups contain only an $n$-fold rotation axis.}
  379   \smallskip
  380   
  381   \citem[$C_{n{\rm v}}$]{In addition to the  $n$-fold rotation
  382     axis, these groups have a mirror plane that contains the axis
  383     of rotation, plus as many additional mirror planes as the
  384     existence of the $n$-fold axis requires.}
  385   \smallskip
  386   
  387   \citem[$C_{n{\rm h}}$]{These groups contain in addition to the
  388     $n$-fold rotation axis a single mirror plane that is perpendicular
  389     to the axis.}
  390   \smallskip
  391   
  392   \citem[$S_n$]{These groups contain only an $n$-fold rotation-reflection
  393     axis.}
  394   \smallskip
  395   
  396   \citem[$D_n$]{In addition to the  $n$-fold rotation axis,
  397     these groups contain a 2-fold axis perpendicular to the
  398     $n$-fold rotation axis, plus as many additional 2-fold
  399     axes as are required by the existence of the $n$-fold axis.}
  400   \smallskip
  401   
  402   \citem[$D_{n{\rm h}}$]{These (the most symmetric groups) contain
  403     all the elements of $D_n$ plus a mirror plane perpendicular
  404     to the $n$-fold axis.}
  405   \smallskip
  406   
  407   \citem[$D_{n{\rm d}}$]{These contain the elements of $D_n$ plus
  408     mirror planes containing the $n$-fold axis, which bisect the
  409     angles between the 2-fold axes.}
  410   \smallskip
  411   
  412   \section{Neumann's principle}
  413   Neumann's principle simply states that {\sl any type of symmetry which
  414   is exhibited by the point symmetry group of the medium is also possessed
  415   by every physical property of the medium}.
  416   
  417   In other words, we can reformulate this for the optical properties
  418   as: {\sl the susceptibility tensors of the medium must be left invariant
  419   under any transformation that also is a point symmetry operation of
  420   the medium}, or
  421   $$
  422     \chi^{(n)'}_{\mu\alpha_1\cdots\alpha_n}
  423       (-\omega_{\sigma};\omega_1,\ldots,\omega_n)
  424     =\chi^{(n)}_{\mu\alpha_1\cdots\alpha_n}
  425       (-\omega_{\sigma};\omega_1,\ldots,\omega_n),
  426   $$
  427   where the tensor elements in the primed coordinate system are transformed
  428   according to
  429   $$
  430     \chi^{(n)'}_{\mu\alpha_1\cdots\alpha_n}
  431       (-\omega_{\sigma};\omega_1,\ldots,\omega_n)
  432     =R_{\mu u}R_{\alpha_1 a_1}\cdots R_{\alpha_n a_n}
  433       \chi^{(n)}_{u a_1\cdots a_n}
  434         (-\omega_{\sigma};\omega_1,\ldots,\omega_n),
  435   $$
  436   where the $[3\times 3]$ matrix ${\bf R}$ describes a point symmetry operation
  437   of the system.
  438   
  439   \section{Inversion properties}
  440   If the {\sl coordinate inversion} $R_{\alpha\beta}=-\delta_{\alpha\beta}$,
  441   is a symmetry operation of the medium (i.~e.~if the medium possess so-called
  442   {\sl inversion symmetry}), then it turns out that
  443   $$
  444     \chi^{(n)}_{\mu\alpha_1\cdots\alpha_n}=0
  445   $$
  446   for all {\sl even} numbers $n$. (Question: Is this symmetry operation
  447   a proper or an improper rotation?)
  448   
  449   \section{Euler angles}
  450   As a convenient way of expressing the matrix of proper rotations,
  451   one may use the {\sl Euler angles} of classical
  452   mechanics,\footnote{${}^4$}{C.~f.~Herbert Goldstein, {\sl Classical Mechanics}
  453   (Addison-Wesley, London, 1980).}
  454   $$
  455     {\bf R}(\varphi,\vartheta,\psi)
  456       ={\bf A}(\psi){\bf B}(\vartheta){\bf C}(\varphi),
  457   $$
  458   where
  459   $$
  460     {\bf A}(\psi)=\pmatrix{\cos\psi&\sin\psi&0\cr
  461                     -\sin\psi&\cos\psi&0\cr
  462                      0&0&1\cr},
  463     \ {\bf B}(\vartheta)=\pmatrix{1&0&0\cr
  464                     0&\cos\vartheta&\sin\vartheta\cr
  465                     0&-\sin\vartheta&\cos\vartheta\cr},
  466     \ {\bf C}(\varphi)=\pmatrix{\cos\varphi&\sin\varphi&0\cr
  467                     -\sin\varphi&\cos\varphi&0\cr
  468                      0&0&1\cr}.
  469   $$
  470   
  471   \section{Example of the direct inspection technique applied to
  472     tetragonal media}
  473   Neumann's principle is a highly useful technique, with applications in
  474   a wide range of disciplines in physics. In order to illustrate this,
  475   we will now apply Neumann's principle to a particular problem, namely
  476   the reduction of the number of elements of the second order electric
  477   susceptibility tensor, in a tetragonal medium belonging to point symmetry
  478   group~$422$.
  479   \medskip
  480   \centerline{\epsfxsize=34mm\epsfbox{../images/tetragon/422.1}}
  481   \medskip
  482   \centerline{Figure 5. An object\footnote{${}^5$}{The figure illustrating
  483     the point symmetry group $422$ is taken from N.~W.~Ashcroft and
  484     N.~D.~Mermin, {\sl Solid state physics} (Saunders College Publishing,
  485     Orlando, 1976), page~122.} possessing the symmetries of point symmetry
  486     group $422$.}
  487   \medskip
  488   \noindent
  489   By inspecting Tables~2 and~3 of Hartmann's {\sl An introduction to Crystal
  490   Physics}\footnote{${}^6$}{Ervin Hartmann, {\sl An Introduction
  491   to Crystal Physics} (University of Cardiff Press, International
  492   Union of Crystallography, 1984), ISBN 0-906449-72-3. Notice that
  493   there is a printing error in Table~3, where the twofold rotation
  494   about the $x_3$-axis should be described by a matrix denoted ``$M_2$'',
  495   and not ``$M_1$'' as written in the table.}
  496   one find that the point symmetry group $422$ of tetragonal media is
  497   described by the generating matrices
  498   $$
  499     {\bf M}_4=\pmatrix{1&0&0\cr 0&-1&0\cr 0&0&-1},\qquad
  500     \left[\matrix{{\rm twofold\ rotation}\cr{\rm about\ }x_1{\rm\ axis}}\right]
  501   $$
  502   and
  503   $$
  504     {\bf M}_7=\pmatrix{0&-1&0\cr 1&0&0\cr 0&0&1}.\qquad
  505     \left[\matrix{{\rm fourfold\ rotation}\cr{\rm about\ }x_3{\rm\ axis}}\right]
  506   $$
  507   \medskip
  508   
  509   \subsection{Does the 422 point symmetry group possess inversion symmetry?}
  510   In Fig.~6, the steps involved for transformation of the object into
  511   an inverted coordinate frame are shown.
  512   \medskip
  513   \centerline{
  514     \epsfxsize=40mm\epsfbox{../images/tetragon/422-a.1}
  515     \epsfxsize=40mm\epsfbox{../images/tetragon/422-b.1}
  516     \epsfxsize=40mm\epsfbox{../images/tetragon/422-c.1}
  517   }
  518   \medskip
  519   \centerline{Figure 6. Transformation into an inverted
  520     coordinate system $(x'',y'',z'')=(-x,-y,-z)$.}
  521   \medskip
  522   \noindent
  523   The result of the sequence in Fig.~6 is an object which cannot be reoriented
  524   in such a way that one obtains the same shape as we started with for the
  525   non-inverted coordinate system, and hence the object of point symmetry
  526   group~$422$ does not possess inversion symmetry.
  527   \medskip
  528   
  529   \subsection{Step one -- Point symmetry under twofold rotation around
  530     the $x_1$-axis}
  531   Considering the point symmetry imposed by the ${\bf R}={\bf M}_4$ matrix,
  532   we find that (for simplicity omitting the frequency arguments of the
  533   susceptibility tensor) the second order susceptibility in the rotated
  534   coordinate frame is described by the diagonal elements
  535   $$
  536     \eqalign{
  537       \chi^{(2)'}_{111}
  538         &=R_{1\mu}R_{1\alpha}R_{1\beta}
  539           \chi^{(2)}_{\mu\alpha\beta}\cr
  540         &=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
  541           R_{1\mu}R_{1\alpha}R_{1\beta}
  542           \chi^{(2)}_{\mu\alpha\beta}\cr
  543         &=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
  544           \delta_{1\mu}\delta_{1\alpha}\delta_{1\beta}
  545           \chi^{(2)}_{\mu\alpha\beta}=\chi^{(2)}_{111},
  546           \qquad({\rm identity})\cr
  547     }
  548   $$
  549   and
  550   $$
  551     \eqalign{
  552       \chi^{(2)'}_{222}
  553         &=R_{2\mu}R_{2\alpha}R_{2\beta}
  554           \chi^{(2)}_{\mu\alpha\beta}\cr
  555         &=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
  556           R_{2\mu}R_{2\alpha}R_{2\beta}
  557           \chi^{(2)}_{\mu\alpha\beta}\cr
  558         &=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
  559           (-\delta_{2\mu})(-\delta_{2\alpha})(-\delta_{2\beta})
  560           \chi^{(2)}_{\mu\alpha\beta}
  561          =-\chi^{(2)}_{222}\cr
  562         &=\{{\rm Neumann's\ principle}\}
  563          =\chi^{(2)}_{222}=0\cr
  564     }
  565   $$
  566   which, by noticing that the similar form $R_{3\alpha}=-\delta_{3\alpha}$
  567   holds for the $333$-component (i.~e.~the $zzz$-component), also gives
  568   $\chi_{333}=-\chi_{333}=0$.
  569   Further we have for the $231$-component
  570   $$
  571     \eqalign{
  572       \chi^{(2)'}_{231}
  573         &=R_{2\mu}R_{3\alpha}R_{1\beta}
  574           \chi^{(2)}_{\mu\alpha\beta}\cr
  575         &=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
  576           R_{2\mu}R_{3\alpha}R_{1\beta}
  577           \chi^{(2)}_{\mu\alpha\beta}\cr
  578         &=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
  579           (-\delta_{2\mu})(-\delta_{3\alpha})\delta_{1\beta}
  580           \chi^{(2)}_{\mu\alpha\beta}=\chi^{(2)}_{231},
  581           \qquad({\rm identity})\cr
  582     }
  583   $$
  584   etc., and by continuing in this manner for all 27 elements of
  585   $\chi^{(2)'}_{\mu\alpha\beta}$, one finds that the symmetry operation
  586   ${\bf R}={\bf M}_4$ leaves us with the tensor elements listed in Table 1.
  587   $$\vcenter{\halign{
  588     \qquad\quad\hfil#\hfil\quad&    % Justification of first column
  589     \quad\hfil#\hfil\quad\qquad\cr  % Justification of second column
  590     \noalign{{\hrule width 320pt}\vskip 1pt}
  591     \noalign{{\hrule width 320pt}\smallskip}
  592      Zero elements & Identities (no further info)\cr
  593     \noalign{\smallskip{\hrule width 320pt}\smallskip}
  594       $\chi^{(2)}_{112}$, $\chi^{(2)}_{113}$,
  595       $\chi^{(2)}_{121}$, $\chi^{(2)}_{131}$,
  596         & \cr
  597       $\chi^{(2)}_{211}$, $\chi^{(2)}_{222}$,
  598       $\chi^{(2)}_{223}$, $\chi^{(2)}_{232}$,
  599         & (all other 13 elements)\cr
  600       $\chi^{(2)}_{233}$, $\chi^{(2)}_{311}$,
  601       $\chi^{(2)}_{322}$, $\chi^{(2)}_{323}$,
  602         & \cr
  603       $\chi^{(2)}_{332}$, $\chi^{(2)}_{333}$
  604         & \cr
  605     \noalign{\smallskip}
  606     \noalign{{\hrule width 320pt}\vskip 1pt}
  607     \noalign{{\hrule width 320pt}\smallskip}
  608   }}
  609   $$
  610   \centerline{Table 1. Reduced set of tensor elements after the symmetry
  611     operation ${\bf R}={\bf M}_4$.}
  612   \medskip
  613   
  614   \subsection{Step two -- Point symmetry under fourfold rotation around
  615     the $x_3$-axis}
  616   Proceeding with the next point symmetry operation, described by
  617   ${\bf R}={\bf M}_7$, one finds for the remaining 13 elements that,
  618   for example, for the $123$-element
  619   $$
  620     \eqalign{
  621       \chi^{(2)'}_{123}
  622         &=R_{1\mu}R_{2\alpha}R_{3\beta}
  623           \chi^{(2)}_{\mu\alpha\beta}\cr
  624         &=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
  625           R_{1\mu}R_{2\alpha}R_{3\beta}
  626           \chi^{(2)}_{\mu\alpha\beta}\cr
  627         &=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
  628           (-\delta_{2\mu})\delta_{1\alpha}\delta_{3\beta}
  629           \chi^{(2)}_{\mu\alpha\beta}=-\chi^{(2)}_{213}\cr
  630         &=\{{\rm Neumann's\ principle}\}
  631          =\chi^{(2)}_{123},\cr
  632     }
  633   $$
  634   and for the $132$-element
  635   $$
  636     \eqalign{
  637       \chi^{(2)'}_{132}
  638         &=R_{1\mu}R_{3\alpha}R_{2\beta}
  639           \chi^{(2)}_{\mu\alpha\beta}\cr
  640         &=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
  641           R_{1\mu}R_{3\alpha}R_{2\beta}
  642           \chi^{(2)}_{\mu\alpha\beta}\cr
  643         &=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
  644           (-\delta_{2\mu})\delta_{3\alpha}\delta_{1\beta}
  645           \chi^{(2)}_{\mu\alpha\beta}=-\chi^{(2)}_{231}\cr
  646         &=\{{\rm Neumann's\ principle}\}
  647          =\chi^{(2)}_{132},\cr
  648     }
  649   $$
  650   while the $111$-element (which previously, by using the ${\bf R}={\bf M}_4$
  651   point symmetry, just gave an identity with no further information) now gives
  652   $$
  653     \eqalign{
  654       \chi^{(2)'}_{111}
  655         &=R_{1\mu}R_{1\alpha}R_{1\beta}
  656           \chi^{(2)}_{\mu\alpha\beta}\cr
  657         &=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
  658           R_{1\mu}R_{1\alpha}R_{1\beta}
  659           \chi^{(2)}_{\mu\alpha\beta}\cr
  660         &=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
  661           (-\delta_{2\mu})(-\delta_{2\alpha})(-\delta_{2\beta})
  662           \chi^{(2)}_{\mu\alpha\beta}
  663          =-\chi^{(2)}_{222}\cr
  664         &=\{{\rm from\ previous\ result\ for\ }\chi^{(2)}_{222}\}
  665          =0\cr
  666         &=\{{\rm Neumann's\ principle}\}
  667          =\chi^{(2)}_{111}.\cr
  668     }
  669   $$
  670   By (again) proceeding for all 27 elements of $\chi^{(2)'}_{\mu\alpha\beta}$,
  671   one finds the set of tensor elements as listed in Table~2. (See also the
  672   tabulated set in Butcher and Cotter's book, Table A3.2, page 299.)
  673   $$\vcenter{\halign{
  674     \qquad\quad\hfil#\hfil\quad&    % Justification of first column
  675     \quad\hfil#\hfil\quad\qquad\cr  % Justification of second column
  676     \noalign{{\hrule width 340pt}\vskip 1pt}
  677     \noalign{{\hrule width 340pt}\smallskip}
  678      Zero elements & Nonzero elements\cr
  679     \noalign{\smallskip{\hrule width 340pt}\smallskip}
  680       $\chi^{(2)}_{111}$, $\chi^{(2)}_{112}$, $\chi^{(2)}_{113}$,
  681       $\chi^{(2)}_{121}$, $\chi^{(2)}_{122}$,
  682         & $\chi^{(2)}_{123}=-\chi^{(2)}_{213}$,\cr
  683       $\chi^{(2)}_{131}$, $\chi^{(2)}_{133}$, $\chi^{(2)}_{211}$,
  684       $\chi^{(2)}_{212}$, $\chi^{(2)}_{221}$,
  685         & $\chi^{(2)}_{132}=-\chi^{(2)}_{231}$,\cr
  686       $\chi^{(2)}_{222}$, $\chi^{(2)}_{223}$, $\chi^{(2)}_{232}$,
  687       $\chi^{(2)}_{233}$, $\chi^{(2)}_{311}$,
  688         & $\chi^{(2)}_{321}=-\chi^{(2)}_{312}$,\cr
  689       $\chi^{(2)}_{313}$, $\chi^{(2)}_{322}$, $\chi^{(2)}_{323}$,
  690       $\chi^{(2)}_{331}$, $\chi^{(2)}_{332}$, $\chi^{(2)}_{333}$
  691         & (6 nonzero, 3 independent)\cr
  692     \noalign{\smallskip}
  693     \noalign{{\hrule width 340pt}\vskip 1pt}
  694     \noalign{{\hrule width 340pt}\smallskip}
  695   }}
  696   $$
  697   \centerline{Table 2. Reduced set of tensor elements after symmetry
  698     operations ${\bf R}={\bf M}_4$ and ${\bf R}={\bf M}_7$.}
  699   
  700   \bye
  701   

Return to previous page

Generated by ::viewsrc::

Last modified Wednesday 16 Nov 2011