Contents of file 'errata.tex':
1 % File: nlopt/errata/errata.tex [pure TeX code]
2 % Last change: March 16, 2003
3 %
4 % Summary of the errors I have found in the book ``The elements of nonlinear
5 % optics'', by P. N. Butcher and D. Cotter (Cambridge University Press, 1991),
6 % being the course litterature in the course ``Nonlinear optics'', held
7 % January-March 2003 at the Royal Institute of Technology, Stockholm, Sweden.
8 %
9 % Copyright (C) 1996-2003, Fredrik Jonsson
10 %
11 % Revision history of this document:
12 %
13 % [2002-11-17] Added the note regarding page 66, line 11.
14 % [2002-12-06] Added the note regarding page 72, last line.
15 % [2003-03-06] Added the note regarding page 240, line 6.
16 % [2003-03-12] Added the note regarding page 241, line 30.
17 % [2003-03-16] Added the note regarding page 159, line 13.
18 % [2003-03-16] Added the note regarding page 160, line 2.
19 %
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31 \def\sech{\mathop{\rm sech}\nolimits}
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35 \def\erritem#1#2#3 {\goodbreak\noindent{\bf p.\hskip 3pt{#1}}%
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37 \centerline{\twelvebf Errors in {\twelvebfsl The Elements of
38 Nonlinear Optics} (1991)}
39 \centerline{\twelvebf by Paul Butcher and David Cotter}\bigskip
40 \centerline{Errata written by Fredrik Jonsson}\medskip
41 \centerline{Updated as of March 16, 2003}\medskip
42 \centerline{
43 Please send additions or corrections to {\tt fredrik.jonsson@proximion.com}}
44 \bigskip
45 {\narrower\noindent
46 Below follows a listing of errors found in P.~N.~Butcher and D.~Cotters
47 book {\sl The Elements of Nonlinear Optics} (Cambridge University Press,
48 Cambridge, 1991). Being a summary of the notes I have made in my personal
49 copy of the book since June 1996, this list should by no means be considered
50 as any kind of ``official'' list of errors, but rather as an attempt to
51 collect the (rather few) misprints in the text. In the list, not only
52 typographical misprints, but also some inconsequent notations -- which do
53 not alter the described theory -- are included.\par}
54 \bigskip
55
56 \erritem{15}{lines 14 and 16}
57 In order not to introduce any ambiguity of the multiple arguments
58 of the symmetric and antisymmetric parts, the arguments $(t;\tau_1,\tau_2)$
59 should be explicitly written in the left-hand sides of the equations.
60 \medskip
61
62 \erritem{15}{line 18}
63 ``$\ldots$ dummy variables $\alpha\tau_1$ and $\beta\tau_2$.''
64 should be replaced by
65 ``$\ldots$ dummy variables $(\alpha,\tau_1)$ and $(\beta,\tau_2)$.'',
66 following the notation as used later in, for example, \S2.3.2
67 and \S4.3.1.
68 \medskip
69
70 \erritem{49}{line 31}
71 $H_1(t)$ should be replaced by $H_{\rm I}(t)$.
72 \medskip
73
74 \erritem{54}{lines 3, 4, and 6}
75 In Eq.~(3.80), the upper limit of integration $t_1$ should be
76 replaced by $\tau_1$.
77 \medskip
78
79 \erritem{54}{lines 8 and 24}
80 In Eqs.~(3.81) and~(3.82), the upper limits of integration $t_1$
81 and $t_{n-1}$ should be replaced by $\tau_1$ and $\tau_{n-1}$,
82 respectively.
83 \medskip
84
85 \erritem{60}{line 24}
86 The sentence ``To achieve this end we $\ldots$'' should be
87 replaced by ``To achieve this we $\ldots$''
88 \medskip
89
90 \erritem{66}{line 11}
91 In the right hand side of Eq.~(4.49), one should in order not to cause
92 confusion with the Einstein convention of summation over repeated indices
93 explicitly state that no summation is implied, and hence the equation
94 should be written as
95 $$H_0 u_i(\Theta)={\Bbb E}_i u_i(\Theta).\quad({\rm no\ sum})\eqno{(4.49)}$$
96 using the common notation as used in tensor calculus.
97 \medskip
98
99 \erritem{67}{line 6}
100 ``$\ldots$ express the the unperturbed $\ldots$'' should be
101 replaced by ``$\ldots$ express the unperturbed $\ldots$''
102 \medskip
103
104 \erritem{72}{line 15}
105 ``$(\alpha,\omega_{1)}$'' should be replaced by ``$(\alpha,\omega_1)$''.
106 \medskip
107
108 \erritem{72}{last line}
109 ``$\ldots$ of this type, one of which is an identity.'' should be replaced by
110 ``$\ldots$ of this type, of which one is an identity.''
111 \medskip
112
113 \erritem{86}{line 4}
114 In Eq.~(4.103), ``$\cdots{\bf f}_t(\omega_n\cdot{\bf e}_n\rangle$''
115 should be replaced by ``$\cdots{\bf f}_t(\omega_n)\cdot{\bf e}_n\rangle$''.
116 \medskip
117
118 \erritem{93}{lines 12--13}
119 Strictly speaking, the real part of the susceptibility
120 $\chi^{(1)}(-\omega_{\sigma};\omega)$ is not proportional to the
121 refractive index $n(\omega)$, but rather to $n^2(\omega)-1$.
122 \medskip
123
124 \erritem{97}{lines 15 and 20}
125 Strictly speaking, $\Omega_{fg}$ is the transition angular frequency,
126 and does not have the physical dimension of energy; therefore replace
127 ``$\Omega_{fg}$'' in lines 15 and 20 by ``$\hbar\Omega_{fg}$''.
128 \medskip
129
130 \erritem{106}{line 9}
131 In the first term of Eq.~(4.128), the summation should be performed
132 over index $j$ rather than index $i$, i.~e.~replace $\sum_i{\bf p}_j$
133 by $\sum_j{\bf p}_j$.
134 \medskip
135
136 \erritem{132}{line 8}
137 In Eq.~(5.30), ``$E_i u_i(\Theta)$'' should be replaced by
138 ``${\Bbb E}_i u_i(\Theta)$ (no sum)''.
139 \medskip
140
141 \erritem{132}{line 25}
142 ``$\rho_0(a)=\eta\exp(-E_a/kT)$'' should be replaced by
143 ``$\rho_0(a)=\eta\exp(-{\Bbb E}_a/kT)$''.
144 \medskip
145
146 \erritem{136}{line 19}
147 In Eq.~(5.43), ``$\chi_{ua}(1)(-\omega,\omega)$''
148 should be replaced by ``$\chi_{ua}^{(1)}(-\omega,\omega)$''.
149 \medskip
150
151 \erritem{159}{line 13}
152 In the left-hand side of Eq.~(6.33), the Hamiltonian $H_0$ describing the
153 system is a quantum-mechanical operator, while in the right-hand side, the
154 matrix representation of the corresponding elements
155 $\langle m|H_0|n\rangle=\delta_{mn}{\Bbb E}_{n}$
156 in the energy representation appears. In order to overcome this inconsistency,
157 Eq.~(6.33) should (in analogy with, for example, Eq.~(6.31) for the matrix
158 elements of the density operator) be replaced by either
159 $$
160 \pmatrix{\langle a|H_0|a\rangle&\langle a|H_0|b\rangle\cr
161 \langle b|H_0|a\rangle&\langle b|H_0|b\rangle\cr}
162 =\pmatrix{{\Bbb E}_a&0\cr
163 0&{\Bbb E}_b\cr},
164 $$
165 or
166 $$
167 \pmatrix{[H_0]_{aa}&[H_0]_{ab}\cr
168 [H_0]_{ba}&[H_0]_{bb}\cr}
169 =\pmatrix{{\Bbb E}_a&0\cr
170 0&{\Bbb E}_b\cr}.
171 $$
172 \medskip
173
174 \erritem{160}{line 2}
175 In the left-hand side of Eq.~(6.34), the Hamiltonian $H_{\rm I}(t)$ is a
176 quantum-mechanical operator, while in the right-hand side, the matrix
177 representation of its scalar elements $\langle a|H_{\rm I}(t)|b\rangle$
178 appears. (The same inconsistency appear in Eq.~(6.33).)
179 In order to overcome this inconsistency, Eq.~(6.34) should (in analogy
180 with, for example, Eq.~(6.31) for the matrix elements of the density
181 operator) be replaced by either
182 $$
183 \pmatrix{\langle a|H_{\rm I}(t)|a\rangle&\langle a|H_{\rm I}(t)|b\rangle\cr
184 \langle b|H_{\rm I}(t)|a\rangle&\langle b|H_{\rm I}(t)|b\rangle\cr}
185 =\pmatrix{\delta{\Bbb E}_a&-e{\bf r}_{ab}\cdot{\bf E}(t)\cr
186 -e{\bf r}_{ba}\cdot{\bf E}(t)&\delta{\Bbb E}_b\cr},
187 $$
188 or
189 $$
190 \pmatrix{[H_{\rm I}(t)]_{aa}&[H_{\rm I}(t)]_{ab}\cr
191 [H_{\rm I}(t)]_{ba}&[H_{\rm I}(t)]_{bb}\cr}
192 =\pmatrix{\delta{\Bbb E}_a&-e{\bf r}_{ab}\cdot{\bf E}(t)\cr
193 -e{\bf r}_{ba}\cdot{\bf E}(t)&\delta{\Bbb E}_b\cr}.
194 $$
195 \medskip
196
197 \erritem{164}{line 15}
198 In Eq.~(6.49), ``$i\hbar(1-\rho_{bb})/T_b$'' should be replaced by
199 ``$i\hbar(1-\rho_{aa})/T_b$''.\footnote{$\dagger$}{Cf.~M.~D.~Levenson
200 and S.~S.~Kano, {\sl Introduction to Nonlinear Laser Spectroscopy}
201 (Academic Press, New York, 1988), p.~33, Eqs.~(2.3.1)--(2.3.2).}
202 \medskip
203
204 \erritem{203}{lines 32--33}
205 ``Fig.~4.3'' should be replaced by ``Fig.~4.4(a)''.
206 \medskip
207
208 \erritem{215}{line 11}
209 In Eq.~(7.14),
210 $$
211 \ldots=\mu_0\int^{\infty}_{-\infty}\,d\omega'(\omega+\omega')^2\cdots
212 $$
213 should be replaced by
214 $$
215 \ldots={{1}\over{c^2}}
216 \int^{\infty}_{-\infty}\,d\omega'(\omega+\omega')^2\cdots
217 $$
218 \medskip
219
220 \erritem{220}{section 7.2.1}
221 In the example of second harmonic generation, the wave equation~(7.26) is given
222 without any explanation of which point symmetry classes of media it applies to,
223 and hence it is from the text virtually impossible to relate the effective
224 nonlinear parameters to the elements of
225 $\chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega,\omega)$.
226 \medskip
227
228 \erritem{234}{line 31}
229 In Eq.~(7.45), ``$\ldots=iq^*\hat{E}^*_3>$'' should be replaced by
230 ``$\ldots=iq^*\hat{E}^*_3$''.
231 \medskip
232
233 \erritem{240}{line 6}
234 In the first line of Eq.~(7.55), there is an ambiguity of the denominator,
235 as well as an erroneous dispersion term, and the equation
236 $$u=\tau\sqrt{n_2\omega/c|d^2 k/d\omega^2|^2}\widehat{E}$$
237 should be replaced by
238 $$u=\tau\sqrt{n_2\omega/(c|d^2 k/d\omega^2|)}\widehat{E}.$$
239 (The other lines of Eq.~(7.55) are correct.)
240 \medskip
241
242 \erritem{241}{line 30}
243 The fundamental bright soliton solution to the nonlinear Schr\"{o}dinger
244 equation should yield ``$u(\zeta,s)=\sech(s)\exp(i\zeta/2)$'', that is to say,
245 {\sl without} any minus sign in the exponential.
246 \medskip
247
248 \erritem{251}{line 6}
249 In Eq.~(8.5), there is parenthesis mismatch in both right- and
250 lefthand sides;
251 $$
252 f_0[({\Bbb E}_n({\bf k})]
253 =\{\exp[{\Bbb E}_n({\bf k})-{\Bbb E}_{\rm F}]/kT+1\}^{-1}
254 $$
255 should be replaced by
256 $$
257 f_0[{\Bbb E}_n({\bf k})]
258 =\{\exp[({\Bbb E}_n({\bf k})-{\Bbb E}_{\rm F})/kT]+1\}^{-1}.
259 $$
260 \medskip
261
262 \erritem{252}{line 6}
263 In Eq.~(8.7), $n_{\rm ph}(\omega_s({\bf q}))$ should be replaced
264 by $n_{\rm ph}[\omega_s({\bf q})]$, in order to follow the
265 functional style of notation as used in, for example, Eq.~(8.5).
266 \medskip
267
268 \erritem{253}{line 20}
269 In Eq.~(8.11), insert a ``$]$'' after ${\Bbb E}_n({\bf k})$.
270 \medskip
271
272 \erritem{298}{Table A3.2}
273 ``$\ldots$ no centre of symmetry $\ldots$'' should be replaced
274 by ``$\ldots$ no centre of inversion $\ldots$''.
275 \medskip
276
277 \erritem{317}{lines 1, 9, and 24}
278 In Appendix 9, there is an inconsistency in the notation for
279 the polarisation density and the electric dipole operator, as
280 compared to the one used in Chapters 3 and 4.
281 While ${\bf P}^{\rm D}$, ${\bf P}^{\rm Q}$, and ${\bf M}^{\rm D}$
282 in Eq.~(A9.1) (and in line 9 on the same page)
283 denote the all-classical electric dipolar, electric quadrupolar
284 and magnetic dipolar polarization densities of the medium, they
285 in Eqs.~(A9.6) and~(A9.7) clearly denote quantum-mechanical operators.
286 In order to overcome this inconsistency in notation, which in
287 addition gives a wrong answer if properly inserted into the
288 perturbation calculus etc., one should chose either of the
289 conventions. By choosing ${\bf P}^{\rm D}$, ${\bf P}^{\rm Q}$,
290 and ${\bf M}^{\rm D}$ to denote the corresponding quantum-mechanical
291 operators, which seem to be the easiest way of correcting this
292 inconsistency, the following corrections to the text should be made:
293
294 {[{\it line 1}]}
295 $$
296 {\bf P}={\bf P}^{\rm D}+{\bf P}^{\rm Q},\qquad
297 {\bf M}={\bf M}^{\rm D},
298 \eqno{({\rm A}9.1)}
299 $$
300 should be replaced by
301 $$
302 {\bf P}=\langle{\bf P}^{\rm D}\rangle+\langle{\bf P}^{\rm Q}\rangle,\qquad
303 {\bf M}=\langle{\bf M}^{\rm D}\rangle,
304 \eqno{({\rm A}9.1)}
305 $$
306
307 {[{\it line 9}]}
308 Remove ``${\bf P}^{\rm D}$'' or replace with
309 ``$\langle{\bf P}^{\rm D}\rangle$''.
310
311 {[{\it line 24}]} Somewhere in Appendix 9, there should be
312 a clarifying statement that the nabla operator appearing in Eq.~(A9.7)
313 only is operating on the all-classical, macroscopic electric field
314 of the light, and hence should be regarded as a classical vector
315 when evaluating the quantum-mechanical trace that is involved in
316 the expectation value of, for example, the corrected form of Eq.~(A9.1).
317 \medskip
318
319 \erritem{318}{line 12}
320 ``${\bf M}<<{\bf H}$'' should be replaced by
321 ``$|{\bf M}|\ll|{\bf H}|$''.
322 \medskip
323
324 \erritem{318}{line 23}
325 ``${\bf e}_j\cdot e{\bf r}+i{\bf k}\cdot{\bf q}\cdot{\bf e}_j
326 +{\bf m}\cdot({\bf k}\times{\bf e}_j)/\omega$'' should be replaced by
327 the same expression, though with each term divided by $e$.
328 \medskip
329
330 \erritem{333}{line 5}
331 In reference Manley, J.~M. and Rowe, H.~E.~(1956), the page numbers
332 should yield 904 -- 14.
333 \medskip
334
335 \erritem{334}{line 44}
336 ``Terhune, R.~W.~and Weinburger, D.~A. $\ldots$'' should be replaced
337 by ``Terhune, R.~W.~and Weinberger, D.~A. $\ldots$''.
338 \medskip
339
340 \bye
341
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