Contents of file 'lect7/lect7.tex':
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.
13 %
14 \input amssym
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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 %
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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
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