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    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   %
   20   \input amssym % to get the {\Bbb E} font (strikethrough E)
   21   \hsize=150mm
   22   \vsize=230mm
   23   \topskip=0pt
   24   \baselineskip 12pt
   25   \parskip 0pt
   26   \leftskip 0pt
   27   \parindent 15pt
   28   \font\twelvebf=cmbx12
   29   \font\twelvebfsl=cmbxsl10 at 12 truept
   30   \font\twelvesc=cmcsc10 at 12 truept
   31   \def\sech{\mathop{\rm sech}\nolimits}
   32   %
   33   % Define a handy macro for typesetting items in the errata.
   34   %
   35   \def\erritem#1#2#3 {\goodbreak\noindent{\bf p.\hskip 3pt{#1}}%
   36      \hskip 3pt[{\it {#2}\thinspace}]\hskip 4pt{#3}}
   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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