Contents of file 'lect7/lect7.tex':




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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
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12   % the Euler fraktur font.
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14   \input amssym
15   \font\ninerm=cmr9
16   \font\twelvesc=cmcsc10
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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
36       \else\hfill\fi}
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}
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