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% Lecture No 7 in the course ``Nonlinear optics'', held January-March,
% 2003, at the Royal Institute of Technology, Stockholm, Sweden.
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% Copyright (C) 2002-2003, Fredrik Jonsson
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\lecture{7}
So far, this course has mainly dealt with the dependence of the angular
frequency of the light and molecular interaction strength in the
description of nonlinear optics. In this lecture, we will now end
this development of the description of interaction between light
and matter, in favour of more engineering practical techniques
for describing the theory of an experimental setup in a certain
geometry, and for reducing the number of necessary tensor elements
needed for describing a mediumof a certain crystallographic point-symmetry
group.
\section{Motivation for analysis of susceptibilities in rotated
coordinate systems}
For a given experimental setup, it is often convenient to introduce
some kind of reference coordinate frame, in which one for example express
the wave propagation as a linear motion along some Cartesian coordinate axis.
This laboratory reference frame might be chosen, for example, with the
$z$-axis coinciding with the direction of propagation of the optical wave
at the laser output, in the phase-matched direction of an optical parametric
oscillator (OPO), after some beam aligning mirror, etc.
In some cases, it might be so that this laboratory frame coincide with
the natural coordinate frame\footnote{${}^1$}{The natural coordinate frame
of the crystal is often chosen such that some particular symmetry axis
is chosen as one of the Cartesian axes.} of the nonlinear crystal, in which
case the coordinate indices of the linear as well as nonlinear susceptibility
tensors take the same values as the coordinates of the laboratory frame.
However, we cannot generally assume the coordinate frame of the crystal to
coincide with a conveniently chosen laboratory reference frame, and this
implies that we generally should be prepared to spatially transform the
susceptibility tensors to arbitrarily rotated coordinate frames.
Having formulated these spatial transformation rules, we will also directly
benefit in another aspect of the description of nonlinear optical
interactions, namely the reduction of the susceptibility tensors
to the minimal set of nonzero elements. This is typically performed
by using the knowledge of the so called {\sl crystallographic point
ymmetry group} of the medium, which essentially is a description of the
spatial operations (rotations, inversions etc.) that define the symmetry
operations of the medium.
As a particular example of the applicability of the spatial transformation
rules (which we soon will formulate) is illustrated in Figs.~1 and~2.
In Fig.~1, the procedure for analysis of sum or frequency difference
generation is outlined. Starting from the description of the linear and
nonlinear susceptibility tensors of the medium, as we previously have
derived the relations from a first principle approach in Lectures 1--6,
we obtain the expressions for the electric polarization densities of the
medium as functions of the applied electric fields of the optical wave
inside the nonlinear crystal. These polarization densities are then
inserted into the wave equation, which basically is derived from Maxwell's
equations of motion for the electromagnetic field. In the wave equation,
the polarization densities act as source terms in an otherwise homogeneous
equation for the motion of the electromagnetic field in vacuum.
As the wave equation is solved for the electric field, here taken in complex
notation, we have solved for the general output from the crystal, and we
can then design the experiment in such a way that an optimal efficiency
is obtained.
\vfill\eject
\centerline{\epsfxsize=150mm\epsfbox{../images/nonrotse/nonrotse.1}}
\medskip
\centerline{Figure 1. The setup in which the orientation of the laboratory
and crystal frames coincide.}
\medskip
\noindent
In Fig.~1, this outline is illustrated for the case where the natural
coordinate frame of the crystal happens to coincide with the coordinate
system of the laboratory frame. In this case, all elements of the
susceptibilities taken in the coordinate frame of the crystal (which
naturally is the coordinate frame in which we can obtain tabulated
sets of tensor elements) will coincide with the elements as taken
in the laboratory frame, and the design and interpretation of the
experiment is straightforward.
However, this setup clearly constitutes a rare case, since we have
infinitely many other possibilities of orienting the crystal relative
the laboratory coordinate frame.
Sometimes the experiment is {\sl designed} with the crystal and laboratory
frames coinciding, in order to simplify the interpretation of an experiment,
and sometimes it is instead {\sl necessary} to rotate the crytal, in order
to achieve phase-matching of nonlinear process, as is the case in for example
most schemes for second-order optical parametric amplification.
If now the crystal frame is rotated with respect to the laboratory frame,
as shown in Fig.~2, we should make up our mind in which system we would
like the wave propagation to be analyzed. In some cases, it might be so that
the output of the experimental setup is most easily interpreted in the
coordinate frame of the crystal, but in most cases, we have a fixed laboratory
frame (fixed by the orientation of the laser, positions of mirrors, etc.)
in which we would like to express the wave propagation and interaction
between light and matter.
\vfill\eject
\centerline{\epsfxsize=150mm\epsfbox{../images/rotsetup/rotsetup.1}}
\medskip
\centerline{Figure 2. The setup in which the crystal frame is rotated
relative the laboratory frame.}
\medskip
\noindent
In Fig.~2, we would, in order to express the nonlinear process in the
laboratory frame, like to obtain the naturally appearing susceptibilities
$\chi^{(2)}_{xyz}$, $\chi^{(2)}_{xxx}$, etc., in the laboratory frame
instead, as $\chi^{(2)}_{x'y'z'}$, $\chi^{(2)}_{x'x'x'}$, etc.
Just to summarize, why are then the transformation rules and spatial
symmetries of the meduim so important?
\smallskip
\item{$\bullet$}{Hard to make physical conclusions about generated
optical fields unless orientation of the laboratory and crystal
frames coincide.}
\item{$\bullet$}{Spatial symmetries often significantly simplifies the
wave propagation problem (by choosing a suitable polarization state
and direction of propagation of the light, etc.).}
\item{$\bullet$}{Useful for reducing the number of necessary elements
of the susceptibility tensors (using Neumann's principle).}
\smallskip
\section{Optical properties in rotated coordinate frames}
Consider two coordinate systems described by Cartesian coordinates
$x_{\alpha}$ and $x'_{\alpha}$, respectively. The coordinate systems
are rotated with respect to each other, and the relation between
the coordinates are described by the $[3\times 3]$ transformation
matrix $R_{ab}$ as
$$
{\bf x}'={\bf R}{\bf x}\qquad\Leftrightarrow
\qquad x'_{\alpha}=R_{\alpha\beta}x_{\beta},
\eqno{[{\rm B.\,\&\,C.}\ (5.40)]}
$$
where ${\bf x}=(x,y,z)^{\rm T}$ and ${\bf x}'=(x',y',z')^{\rm T}$
are column vectors.
The inverse transformation between the coordinate systems is similarly
given as
$$
{\bf x}={\bf R}^{-1}{\bf x}'\qquad\Leftrightarrow
\qquad x_{\beta}=R_{\alpha\beta}x'_{\alpha}.
\eqno{[{\rm B.\,\&\,C.}\ (5.41)]}
$$
\centerline{\epsfxsize=65mm\epsfbox{../images/rotframe/rotframe.1}}
\medskip
\noindent{Figure 3. Illustration of proper rotation of the crystal
frame $(x,y,z)$ relative to the laboratory reference frame $(x',y',z')$,
by means of $x'_{\alpha} = R_{\alpha\beta} x_{\beta}$
with $\det{({\bf R})}=1$.}
\medskip
\noindent
\centerline{\epsfxsize=68mm\epsfbox{../images/rotfig/rotfig.1}}
\medskip
\noindent{Figure 4. The coordinate transformations
(a) ${\bf x}=(x,y,z)\mapsto{\bf x}'=(-x,-y,z)$, constituting
a proper rotation around the $z$-axis, and
(b) the space inversion ${\bf x}\mapsto{\bf x}'=-{\bf x}$,
an improper rotation corresponding to, for example,
a rotation around the $z$-axis followed by an inversion
in the $xy$-plane.}
\medskip
\noindent
We should notice that there are two types of rotations that are
encountered as transformations:
\smallskip
\item{$\bullet$}{Proper rotations, for which $\det({\bf R})=1$.
(Righthanded systems keep being righthanded, and lefthanded systems
keep being lefthanded.)}
\item{$\bullet$}{Improper rotations, for which $\det({\bf R})=-1$.
(Righthanded systems are transformed into lefthanded systems, and
vice versa.)}
\smallskip
The electric field ${\bf E}({\bf r},t)$ and electric polarization
density ${\bf P}({\bf r},t)$ are both polar quantities that transform
in the same way as regular Cartesian coordinates, and hence we have
descriptions of these quantities in coordinate systems $(x,y,z)$
and $(x',y',z')$ related to each other as
$$
E'_{\mu}({\bf r},t)=R_{\mu u}E_u({\bf r},t)
\qquad\Leftrightarrow\qquad
E_u({\bf r},t)=R_{\mu u}E'_{\mu}({\bf r},t),
$$
and
$$
P'_{\mu}({\bf r},t)=R_{\mu u}P_u({\bf r},t)
\qquad\Leftrightarrow\qquad
P_u({\bf r},t)=R_{\mu u}P'_{\mu}({\bf r},t),
$$
respectively.
Using these transformation rules, we will now derive the form of the
susceptibilities in rotated coordinate frames.
\medskip
\subsection{First order polarization density in rotated coordinate frames}
From the transformation rule for the electric polarization density above,
using the standard form as we previously have expressed the electric
field dependence, we have for the first order polarization density
in the primed coordinate system
$$
\eqalign{
P^{(1)}_{\mu}{}'({\bf r},t)
&=R_{\mu u}P^{(1)}_u({\bf r},t)\cr
&=R_{\mu u}\varepsilon_0\int^{\infty}_{-\infty}
\chi^{(1)}_{ua}(-\omega;\omega)
E_a(\omega)\exp(-i\omega t)\,d\omega\cr
&=R_{\mu u}\varepsilon_0\int^{\infty}_{-\infty}
\chi^{(1)}_{ua}(-\omega;\omega)
R_{\alpha a}E'_{\alpha}(\omega)\exp(-i\omega t)\,d\omega\cr
&=\varepsilon_0\int^{\infty}_{-\infty}
\chi^{(1)}_{\mu\alpha}{}'(-\omega;\omega)
E'_{\alpha}(\omega)\exp(-i\omega t)\,d\omega\cr
}
$$
where
$$
\chi^{(1)}_{\mu\alpha}{}'(-\omega;\omega)
=R_{\mu u}R_{\alpha a}\chi^{(1)}_{ua}(-\omega;\omega)
\eqno{[{\rm B.\,\&\,C.}\ (5.45)]}
$$
is the linear electric susceptibility taken in the primed coordinate system.
\medskip
\subsection{Second order polarization density in rotated coordinate frames}
Similarly, we have the second order polarization density
in the primed coordinate system as
$$
\eqalign{
P^{(2)}_{\mu}{}'({\bf r},t)
&=R_{\mu u}P^{(2)}_u({\bf r},t)\cr
&=R_{\mu u}\varepsilon_0\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
\chi^{(2)}_{uab}(-\omega_{\sigma};\omega_1,\omega_2)
E_a(\omega_1) E_b(\omega_2)
\exp[-i(\omega_1+\omega_1)t]\,d\omega_2\,d\omega_1\cr
&=R_{\mu u}\varepsilon_0\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
\chi^{(2)}_{uab}(-\omega_{\sigma};\omega_1,\omega_2)
R_{\alpha a} E'_{\alpha}(\omega_1)
R_{\beta b} E'_{\beta}(\omega_2)
\exp[-i(\omega_1+\omega_1)t]\,d\omega_2\,d\omega_1\cr
&=\varepsilon_0\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}
\chi^{(2)}_{\mu\alpha\beta}{}'(-\omega_{\sigma};\omega_1,\omega_2)
E'_{\alpha}(\omega_1) E'_{\beta}(\omega_2)
\exp[-i(\omega_1+\omega_1)t]\,d\omega_2\,d\omega_1\cr
}
$$
where
$$
\chi^{(2)}_{\mu\alpha\beta}{}'(-\omega_{\sigma};\omega_1,\omega_2)
=R_{\mu u}R_{\alpha a}R_{\beta b}
\chi^{(2)}_{uab}(-\omega_{\sigma};\omega_1,\omega_2)
\eqno{[{\rm B.\,\&\,C.}\ (5.46)]}
$$
is the second order electric susceptibility taken in the primed
coordinate system.
\subsection{Higher order polarization densities in rotated coordinate frames}
In a manner completely analogous to the second order susceptibility,
the transformation rule between the primed and unprimed coordinate
systems can be obtained for the $n$th order elements of the electric
susceptibility tensor as
$$
\chi^{(n)}_{\mu\alpha_1\cdots\alpha_n}{}'
(-\omega_{\sigma};\omega_1,\ldots,\omega_n)
=R_{\mu u}R_{\alpha_1 a_1}\cdots R_{\alpha_n a_n}
\chi^{(n)}_{u a_1\cdots a_n}(-\omega_{\sigma};\omega_1,\ldots,\omega_n).
\eqno{[{\rm B.\,\&\,C.}\ (5.47)]}
$$
\section{Crystallographic point symmetry groups}
Typically, a particular point symmetry group of the medium can be
described by the {\sl generating matrices} that describe the minimal
set of transformation matrices (describing a set of symmetry operations)
that will be necessary for the reduction of the constitutive tensors.
Two systems are widely used for the description of point symmetry
groups:\footnote{${}^2$}{C.~f.~Table 2 of the handed out Hartmann's
{\sl An Introduction to Crystal Physics}.}
\smallskip
\item{$\bullet$}{The International system, e.~g.~$\bar{4}3m$, $m3m$,
$422$, etc.}
\item{$\bullet$}{The Sch\"{o}nflies system, e.~g.~$T_d$, $O_h$,
$D_4$, etc.}
\smallskip
The crystallographic point symmetry groups may contain any of
the following symmetry operations:
{\it 1. Rotations through integral multiples of $2\pi/n$ about some axis.}
The axis is called the $n$-fold rotation axis. It is in solid state
physics shown [1--3] that a Bravais lattice can contain only 2-, 3-, 4-,
or 6-fold axes, and since the crystallographic point symmetry groups
are contained in the Bravais lattice point groups, they too can only
have these axes.
{\it 2. Rotation-reflections.}
Even when a rotation through $2\pi/n$ is not a symmetry element,
sometimes such a rotation followed by a reflection in a plane
perpendicular to the axis may be a symmetry operation.
The axis is then called an $n$-fold rotation-reflection axis.
For example, the groups $S_6$ and $S_4$ have 6- and 4-fold
rotation-reflection axes.
{\it 3. Rotation-inversions.}
Similarly, sometimes a rotation through $2\pi/n$ followed by an
inversion in a point lying on the rotation axis is a symmetry
element, even though such a rotation by itself is not.
The axis is then called an $n$-fold rotation-inversion axis.
However, the axis in $S_6$ is only a 3-fold rotation-inversion
axis.
{\it 4. Reflections.}
A reflection takes every point into its mirror image in a plane,
known as a mirror plane.
{\it 5. Inversions.}
An inversion has a single fixed point. If that point is taken as
the origin, then every other point ${\bf r}$ is taken into $-{\bf r}$.
\section{Sch\"onflies notation for the non-cubic
crystallographic point groups}
The twenty-seven {\sl non-cubic} crystallographic point symmetry groups
may contain any of the following symmetry operations, here given
in Sch\"onflies notation\footnote{${}^3$}{In Sch\"onflies notation,
$C$ stands for ``cyclic'', $D$ for ``dihedral'', and $S$ for ``spiegel''.
The subscripts h, v, and d stand for ``horizontal'', ``vertical'',
and ``diagonal'', respectively, and refer to the placement of the
placement of the mirror planes with respect to the $n$-fold axis,
always considered to be vertical. (The ``diagonal'' planes in
$D_{n{\rm d}}$ are vertical and bisect the angles between the
2-fold axes)}:
\citem[$C_n$]{These groups contain only an $n$-fold rotation axis.}
\smallskip
\citem[$C_{n{\rm v}}$]{In addition to the $n$-fold rotation
axis, these groups have a mirror plane that contains the axis
of rotation, plus as many additional mirror planes as the
existence of the $n$-fold axis requires.}
\smallskip
\citem[$C_{n{\rm h}}$]{These groups contain in addition to the
$n$-fold rotation axis a single mirror plane that is perpendicular
to the axis.}
\smallskip
\citem[$S_n$]{These groups contain only an $n$-fold rotation-reflection
axis.}
\smallskip
\citem[$D_n$]{In addition to the $n$-fold rotation axis,
these groups contain a 2-fold axis perpendicular to the
$n$-fold rotation axis, plus as many additional 2-fold
axes as are required by the existence of the $n$-fold axis.}
\smallskip
\citem[$D_{n{\rm h}}$]{These (the most symmetric groups) contain
all the elements of $D_n$ plus a mirror plane perpendicular
to the $n$-fold axis.}
\smallskip
\citem[$D_{n{\rm d}}$]{These contain the elements of $D_n$ plus
mirror planes containing the $n$-fold axis, which bisect the
angles between the 2-fold axes.}
\smallskip
\section{Neumann's principle}
Neumann's principle simply states that {\sl any type of symmetry which
is exhibited by the point symmetry group of the medium is also possessed
by every physical property of the medium}.
In other words, we can reformulate this for the optical properties
as: {\sl the susceptibility tensors of the medium must be left invariant
under any transformation that also is a point symmetry operation of
the medium}, or
$$
\chi^{(n)'}_{\mu\alpha_1\cdots\alpha_n}
(-\omega_{\sigma};\omega_1,\ldots,\omega_n)
=\chi^{(n)}_{\mu\alpha_1\cdots\alpha_n}
(-\omega_{\sigma};\omega_1,\ldots,\omega_n),
$$
where the tensor elements in the primed coordinate system are transformed
according to
$$
\chi^{(n)'}_{\mu\alpha_1\cdots\alpha_n}
(-\omega_{\sigma};\omega_1,\ldots,\omega_n)
=R_{\mu u}R_{\alpha_1 a_1}\cdots R_{\alpha_n a_n}
\chi^{(n)}_{u a_1\cdots a_n}
(-\omega_{\sigma};\omega_1,\ldots,\omega_n),
$$
where the $[3\times 3]$ matrix ${\bf R}$ describes a point symmetry operation
of the system.
\section{Inversion properties}
If the {\sl coordinate inversion} $R_{\alpha\beta}=-\delta_{\alpha\beta}$,
is a symmetry operation of the medium (i.~e.~if the medium possess so-called
{\sl inversion symmetry}), then it turns out that
$$
\chi^{(n)}_{\mu\alpha_1\cdots\alpha_n}=0
$$
for all {\sl even} numbers $n$. (Question: Is this symmetry operation
a proper or an improper rotation?)
\section{Euler angles}
As a convenient way of expressing the matrix of proper rotations,
one may use the {\sl Euler angles} of classical
mechanics,\footnote{${}^4$}{C.~f.~Herbert Goldstein, {\sl Classical Mechanics}
(Addison-Wesley, London, 1980).}
$$
{\bf R}(\varphi,\vartheta,\psi)
={\bf A}(\psi){\bf B}(\vartheta){\bf C}(\varphi),
$$
where
$$
{\bf A}(\psi)=\pmatrix{\cos\psi&\sin\psi&0\cr
-\sin\psi&\cos\psi&0\cr
0&0&1\cr},
\ {\bf B}(\vartheta)=\pmatrix{1&0&0\cr
0&\cos\vartheta&\sin\vartheta\cr
0&-\sin\vartheta&\cos\vartheta\cr},
\ {\bf C}(\varphi)=\pmatrix{\cos\varphi&\sin\varphi&0\cr
-\sin\varphi&\cos\varphi&0\cr
0&0&1\cr}.
$$
\section{Example of the direct inspection technique applied to
tetragonal media}
Neumann's principle is a highly useful technique, with applications in
a wide range of disciplines in physics. In order to illustrate this,
we will now apply Neumann's principle to a particular problem, namely
the reduction of the number of elements of the second order electric
susceptibility tensor, in a tetragonal medium belonging to point symmetry
group~$422$.
\medskip
\centerline{\epsfxsize=34mm\epsfbox{../images/tetragon/422.1}}
\medskip
\centerline{Figure 5. An object\footnote{${}^5$}{The figure illustrating
the point symmetry group $422$ is taken from N.~W.~Ashcroft and
N.~D.~Mermin, {\sl Solid state physics} (Saunders College Publishing,
Orlando, 1976), page~122.} possessing the symmetries of point symmetry
group $422$.}
\medskip
\noindent
By inspecting Tables~2 and~3 of Hartmann's {\sl An introduction to Crystal
Physics}\footnote{${}^6$}{Ervin Hartmann, {\sl An Introduction
to Crystal Physics} (University of Cardiff Press, International
Union of Crystallography, 1984), ISBN 0-906449-72-3. Notice that
there is a printing error in Table~3, where the twofold rotation
about the $x_3$-axis should be described by a matrix denoted ``$M_2$'',
and not ``$M_1$'' as written in the table.}
one find that the point symmetry group $422$ of tetragonal media is
described by the generating matrices
$$
{\bf M}_4=\pmatrix{1&0&0\cr 0&-1&0\cr 0&0&-1},\qquad
\left[\matrix{{\rm twofold\ rotation}\cr{\rm about\ }x_1{\rm\ axis}}\right]
$$
and
$$
{\bf M}_7=\pmatrix{0&-1&0\cr 1&0&0\cr 0&0&1}.\qquad
\left[\matrix{{\rm fourfold\ rotation}\cr{\rm about\ }x_3{\rm\ axis}}\right]
$$
\medskip
\subsection{Does the 422 point symmetry group possess inversion symmetry?}
In Fig.~6, the steps involved for transformation of the object into
an inverted coordinate frame are shown.
\medskip
\centerline{
\epsfxsize=40mm\epsfbox{../images/tetragon/422-a.1}
\epsfxsize=40mm\epsfbox{../images/tetragon/422-b.1}
\epsfxsize=40mm\epsfbox{../images/tetragon/422-c.1}
}
\medskip
\centerline{Figure 6. Transformation into an inverted
coordinate system $(x'',y'',z'')=(-x,-y,-z)$.}
\medskip
\noindent
The result of the sequence in Fig.~6 is an object which cannot be reoriented
in such a way that one obtains the same shape as we started with for the
non-inverted coordinate system, and hence the object of point symmetry
group~$422$ does not possess inversion symmetry.
\medskip
\subsection{Step one -- Point symmetry under twofold rotation around
the $x_1$-axis}
Considering the point symmetry imposed by the ${\bf R}={\bf M}_4$ matrix,
we find that (for simplicity omitting the frequency arguments of the
susceptibility tensor) the second order susceptibility in the rotated
coordinate frame is described by the diagonal elements
$$
\eqalign{
\chi^{(2)'}_{111}
&=R_{1\mu}R_{1\alpha}R_{1\beta}
\chi^{(2)}_{\mu\alpha\beta}\cr
&=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
R_{1\mu}R_{1\alpha}R_{1\beta}
\chi^{(2)}_{\mu\alpha\beta}\cr
&=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
\delta_{1\mu}\delta_{1\alpha}\delta_{1\beta}
\chi^{(2)}_{\mu\alpha\beta}=\chi^{(2)}_{111},
\qquad({\rm identity})\cr
}
$$
and
$$
\eqalign{
\chi^{(2)'}_{222}
&=R_{2\mu}R_{2\alpha}R_{2\beta}
\chi^{(2)}_{\mu\alpha\beta}\cr
&=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
R_{2\mu}R_{2\alpha}R_{2\beta}
\chi^{(2)}_{\mu\alpha\beta}\cr
&=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
(-\delta_{2\mu})(-\delta_{2\alpha})(-\delta_{2\beta})
\chi^{(2)}_{\mu\alpha\beta}
=-\chi^{(2)}_{222}\cr
&=\{{\rm Neumann's\ principle}\}
=\chi^{(2)}_{222}=0\cr
}
$$
which, by noticing that the similar form $R_{3\alpha}=-\delta_{3\alpha}$
holds for the $333$-component (i.~e.~the $zzz$-component), also gives
$\chi_{333}=-\chi_{333}=0$.
Further we have for the $231$-component
$$
\eqalign{
\chi^{(2)'}_{231}
&=R_{2\mu}R_{3\alpha}R_{1\beta}
\chi^{(2)}_{\mu\alpha\beta}\cr
&=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
R_{2\mu}R_{3\alpha}R_{1\beta}
\chi^{(2)}_{\mu\alpha\beta}\cr
&=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
(-\delta_{2\mu})(-\delta_{3\alpha})\delta_{1\beta}
\chi^{(2)}_{\mu\alpha\beta}=\chi^{(2)}_{231},
\qquad({\rm identity})\cr
}
$$
etc., and by continuing in this manner for all 27 elements of
$\chi^{(2)'}_{\mu\alpha\beta}$, one finds that the symmetry operation
${\bf R}={\bf M}_4$ leaves us with the tensor elements listed in Table 1.
$$\vcenter{\halign{
\qquad\quad\hfil#\hfil\quad& % Justification of first column
\quad\hfil#\hfil\quad\qquad\cr % Justification of second column
\noalign{{\hrule width 320pt}\vskip 1pt}
\noalign{{\hrule width 320pt}\smallskip}
Zero elements & Identities (no further info)\cr
\noalign{\smallskip{\hrule width 320pt}\smallskip}
$\chi^{(2)}_{112}$, $\chi^{(2)}_{113}$,
$\chi^{(2)}_{121}$, $\chi^{(2)}_{131}$,
& \cr
$\chi^{(2)}_{211}$, $\chi^{(2)}_{222}$,
$\chi^{(2)}_{223}$, $\chi^{(2)}_{232}$,
& (all other 13 elements)\cr
$\chi^{(2)}_{233}$, $\chi^{(2)}_{311}$,
$\chi^{(2)}_{322}$, $\chi^{(2)}_{323}$,
& \cr
$\chi^{(2)}_{332}$, $\chi^{(2)}_{333}$
& \cr
\noalign{\smallskip}
\noalign{{\hrule width 320pt}\vskip 1pt}
\noalign{{\hrule width 320pt}\smallskip}
}}
$$
\centerline{Table 1. Reduced set of tensor elements after the symmetry
operation ${\bf R}={\bf M}_4$.}
\medskip
\subsection{Step two -- Point symmetry under fourfold rotation around
the $x_3$-axis}
Proceeding with the next point symmetry operation, described by
${\bf R}={\bf M}_7$, one finds for the remaining 13 elements that,
for example, for the $123$-element
$$
\eqalign{
\chi^{(2)'}_{123}
&=R_{1\mu}R_{2\alpha}R_{3\beta}
\chi^{(2)}_{\mu\alpha\beta}\cr
&=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
R_{1\mu}R_{2\alpha}R_{3\beta}
\chi^{(2)}_{\mu\alpha\beta}\cr
&=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
(-\delta_{2\mu})\delta_{1\alpha}\delta_{3\beta}
\chi^{(2)}_{\mu\alpha\beta}=-\chi^{(2)}_{213}\cr
&=\{{\rm Neumann's\ principle}\}
=\chi^{(2)}_{123},\cr
}
$$
and for the $132$-element
$$
\eqalign{
\chi^{(2)'}_{132}
&=R_{1\mu}R_{3\alpha}R_{2\beta}
\chi^{(2)}_{\mu\alpha\beta}\cr
&=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
R_{1\mu}R_{3\alpha}R_{2\beta}
\chi^{(2)}_{\mu\alpha\beta}\cr
&=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
(-\delta_{2\mu})\delta_{3\alpha}\delta_{1\beta}
\chi^{(2)}_{\mu\alpha\beta}=-\chi^{(2)}_{231}\cr
&=\{{\rm Neumann's\ principle}\}
=\chi^{(2)}_{132},\cr
}
$$
while the $111$-element (which previously, by using the ${\bf R}={\bf M}_4$
point symmetry, just gave an identity with no further information) now gives
$$
\eqalign{
\chi^{(2)'}_{111}
&=R_{1\mu}R_{1\alpha}R_{1\beta}
\chi^{(2)}_{\mu\alpha\beta}\cr
&=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
R_{1\mu}R_{1\alpha}R_{1\beta}
\chi^{(2)}_{\mu\alpha\beta}\cr
&=\sum^3_{\mu=1}\sum^3_{\alpha=1}\sum^3_{\beta=1}
(-\delta_{2\mu})(-\delta_{2\alpha})(-\delta_{2\beta})
\chi^{(2)}_{\mu\alpha\beta}
=-\chi^{(2)}_{222}\cr
&=\{{\rm from\ previous\ result\ for\ }\chi^{(2)}_{222}\}
=0\cr
&=\{{\rm Neumann's\ principle}\}
=\chi^{(2)}_{111}.\cr
}
$$
By (again) proceeding for all 27 elements of $\chi^{(2)'}_{\mu\alpha\beta}$,
one finds the set of tensor elements as listed in Table~2. (See also the
tabulated set in Butcher and Cotter's book, Table A3.2, page 299.)
$$\vcenter{\halign{
\qquad\quad\hfil#\hfil\quad& % Justification of first column
\quad\hfil#\hfil\quad\qquad\cr % Justification of second column
\noalign{{\hrule width 340pt}\vskip 1pt}
\noalign{{\hrule width 340pt}\smallskip}
Zero elements & Nonzero elements\cr
\noalign{\smallskip{\hrule width 340pt}\smallskip}
$\chi^{(2)}_{111}$, $\chi^{(2)}_{112}$, $\chi^{(2)}_{113}$,
$\chi^{(2)}_{121}$, $\chi^{(2)}_{122}$,
& $\chi^{(2)}_{123}=-\chi^{(2)}_{213}$,\cr
$\chi^{(2)}_{131}$, $\chi^{(2)}_{133}$, $\chi^{(2)}_{211}$,
$\chi^{(2)}_{212}$, $\chi^{(2)}_{221}$,
& $\chi^{(2)}_{132}=-\chi^{(2)}_{231}$,\cr
$\chi^{(2)}_{222}$, $\chi^{(2)}_{223}$, $\chi^{(2)}_{232}$,
$\chi^{(2)}_{233}$, $\chi^{(2)}_{311}$,
& $\chi^{(2)}_{321}=-\chi^{(2)}_{312}$,\cr
$\chi^{(2)}_{313}$, $\chi^{(2)}_{322}$, $\chi^{(2)}_{323}$,
$\chi^{(2)}_{331}$, $\chi^{(2)}_{332}$, $\chi^{(2)}_{333}$
& (6 nonzero, 3 independent)\cr
\noalign{\smallskip}
\noalign{{\hrule width 340pt}\vskip 1pt}
\noalign{{\hrule width 340pt}\smallskip}
}}
$$
\centerline{Table 2. Reduced set of tensor elements after symmetry
operations ${\bf R}={\bf M}_4$ and ${\bf R}={\bf M}_7$.}
\bye