% File: nlopt/errata/errata.tex [pure TeX code]
% Last change: March 16, 2003
%
% Summary of the errors I have found in the book ``The elements of nonlinear
% optics'', by P. N. Butcher and D. Cotter (Cambridge University Press, 1991),
% being the course litterature in the course ``Nonlinear optics'', held
% January-March 2003 at the Royal Institute of Technology, Stockholm, Sweden.
%
% Copyright (C) 1996-2003, Fredrik Jonsson
%
% Revision history of this document:
%
% [2002-11-17] Added the note regarding page 66, line 11.
% [2002-12-06] Added the note regarding page 72, last line.
% [2003-03-06] Added the note regarding page 240, line 6.
% [2003-03-12] Added the note regarding page 241, line 30.
% [2003-03-16] Added the note regarding page 159, line 13.
% [2003-03-16] Added the note regarding page 160, line 2.
%
\input amssym % to get the {\Bbb E} font (strikethrough E)
\hsize=150mm
\vsize=230mm
\topskip=0pt
\baselineskip 12pt
\parskip 0pt
\leftskip 0pt
\parindent 15pt
\font\twelvebf=cmbx12
\font\twelvebfsl=cmbxsl10 at 12 truept
\font\twelvesc=cmcsc10 at 12 truept
\def\sech{\mathop{\rm sech}\nolimits}
%
% Define a handy macro for typesetting items in the errata.
%
\def\erritem#1#2#3 {\goodbreak\noindent{\bf p.\hskip 3pt{#1}}%
\hskip 3pt[{\it {#2}\thinspace}]\hskip 4pt{#3}}
\centerline{\twelvebf Errors in {\twelvebfsl The Elements of
Nonlinear Optics} (1991)}
\centerline{\twelvebf by Paul Butcher and David Cotter}\bigskip
\centerline{Errata written by Fredrik Jonsson}\medskip
\centerline{Updated as of March 16, 2003}\medskip
\centerline{
Please send additions or corrections to {\tt fredrik.jonsson@proximion.com}}
\bigskip
{\narrower\noindent
Below follows a listing of errors found in P.~N.~Butcher and D.~Cotters
book {\sl The Elements of Nonlinear Optics} (Cambridge University Press,
Cambridge, 1991). Being a summary of the notes I have made in my personal
copy of the book since June 1996, this list should by no means be considered
as any kind of ``official'' list of errors, but rather as an attempt to
collect the (rather few) misprints in the text. In the list, not only
typographical misprints, but also some inconsequent notations -- which do
not alter the described theory -- are included.\par}
\bigskip
\erritem{15}{lines 14 and 16}
In order not to introduce any ambiguity of the multiple arguments
of the symmetric and antisymmetric parts, the arguments $(t;\tau_1,\tau_2)$
should be explicitly written in the left-hand sides of the equations.
\medskip
\erritem{15}{line 18}
``$\ldots$ dummy variables $\alpha\tau_1$ and $\beta\tau_2$.''
should be replaced by
``$\ldots$ dummy variables $(\alpha,\tau_1)$ and $(\beta,\tau_2)$.'',
following the notation as used later in, for example, \S2.3.2
and \S4.3.1.
\medskip
\erritem{49}{line 31}
$H_1(t)$ should be replaced by $H_{\rm I}(t)$.
\medskip
\erritem{54}{lines 3, 4, and 6}
In Eq.~(3.80), the upper limit of integration $t_1$ should be
replaced by $\tau_1$.
\medskip
\erritem{54}{lines 8 and 24}
In Eqs.~(3.81) and~(3.82), the upper limits of integration $t_1$
and $t_{n-1}$ should be replaced by $\tau_1$ and $\tau_{n-1}$,
respectively.
\medskip
\erritem{60}{line 24}
The sentence ``To achieve this end we $\ldots$'' should be
replaced by ``To achieve this we $\ldots$''
\medskip
\erritem{66}{line 11}
In the right hand side of Eq.~(4.49), one should in order not to cause
confusion with the Einstein convention of summation over repeated indices
explicitly state that no summation is implied, and hence the equation
should be written as
$$H_0 u_i(\Theta)={\Bbb E}_i u_i(\Theta).\quad({\rm no\ sum})\eqno{(4.49)}$$
using the common notation as used in tensor calculus.
\medskip
\erritem{67}{line 6}
``$\ldots$ express the the unperturbed $\ldots$'' should be
replaced by ``$\ldots$ express the unperturbed $\ldots$''
\medskip
\erritem{72}{line 15}
``$(\alpha,\omega_{1)}$'' should be replaced by ``$(\alpha,\omega_1)$''.
\medskip
\erritem{72}{last line}
``$\ldots$ of this type, one of which is an identity.'' should be replaced by
``$\ldots$ of this type, of which one is an identity.''
\medskip
\erritem{86}{line 4}
In Eq.~(4.103), ``$\cdots{\bf f}_t(\omega_n\cdot{\bf e}_n\rangle$''
should be replaced by ``$\cdots{\bf f}_t(\omega_n)\cdot{\bf e}_n\rangle$''.
\medskip
\erritem{93}{lines 12--13}
Strictly speaking, the real part of the susceptibility
$\chi^{(1)}(-\omega_{\sigma};\omega)$ is not proportional to the
refractive index $n(\omega)$, but rather to $n^2(\omega)-1$.
\medskip
\erritem{97}{lines 15 and 20}
Strictly speaking, $\Omega_{fg}$ is the transition angular frequency,
and does not have the physical dimension of energy; therefore replace
``$\Omega_{fg}$'' in lines 15 and 20 by ``$\hbar\Omega_{fg}$''.
\medskip
\erritem{106}{line 9}
In the first term of Eq.~(4.128), the summation should be performed
over index $j$ rather than index $i$, i.~e.~replace $\sum_i{\bf p}_j$
by $\sum_j{\bf p}_j$.
\medskip
\erritem{132}{line 8}
In Eq.~(5.30), ``$E_i u_i(\Theta)$'' should be replaced by
``${\Bbb E}_i u_i(\Theta)$ (no sum)''.
\medskip
\erritem{132}{line 25}
``$\rho_0(a)=\eta\exp(-E_a/kT)$'' should be replaced by
``$\rho_0(a)=\eta\exp(-{\Bbb E}_a/kT)$''.
\medskip
\erritem{136}{line 19}
In Eq.~(5.43), ``$\chi_{ua}(1)(-\omega,\omega)$''
should be replaced by ``$\chi_{ua}^{(1)}(-\omega,\omega)$''.
\medskip
\erritem{159}{line 13}
In the left-hand side of Eq.~(6.33), the Hamiltonian $H_0$ describing the
system is a quantum-mechanical operator, while in the right-hand side, the
matrix representation of the corresponding elements
$\langle m|H_0|n\rangle=\delta_{mn}{\Bbb E}_{n}$
in the energy representation appears. In order to overcome this inconsistency,
Eq.~(6.33) should (in analogy with, for example, Eq.~(6.31) for the matrix
elements of the density operator) be replaced by either
$$
\pmatrix{\langle a|H_0|a\rangle&\langle a|H_0|b\rangle\cr
\langle b|H_0|a\rangle&\langle b|H_0|b\rangle\cr}
=\pmatrix{{\Bbb E}_a&0\cr
0&{\Bbb E}_b\cr},
$$
or
$$
\pmatrix{[H_0]_{aa}&[H_0]_{ab}\cr
[H_0]_{ba}&[H_0]_{bb}\cr}
=\pmatrix{{\Bbb E}_a&0\cr
0&{\Bbb E}_b\cr}.
$$
\medskip
\erritem{160}{line 2}
In the left-hand side of Eq.~(6.34), the Hamiltonian $H_{\rm I}(t)$ is a
quantum-mechanical operator, while in the right-hand side, the matrix
representation of its scalar elements $\langle a|H_{\rm I}(t)|b\rangle$
appears. (The same inconsistency appear in Eq.~(6.33).)
In order to overcome this inconsistency, Eq.~(6.34) should (in analogy
with, for example, Eq.~(6.31) for the matrix elements of the density
operator) be replaced by either
$$
\pmatrix{\langle a|H_{\rm I}(t)|a\rangle&\langle a|H_{\rm I}(t)|b\rangle\cr
\langle b|H_{\rm I}(t)|a\rangle&\langle b|H_{\rm I}(t)|b\rangle\cr}
=\pmatrix{\delta{\Bbb E}_a&-e{\bf r}_{ab}\cdot{\bf E}(t)\cr
-e{\bf r}_{ba}\cdot{\bf E}(t)&\delta{\Bbb E}_b\cr},
$$
or
$$
\pmatrix{[H_{\rm I}(t)]_{aa}&[H_{\rm I}(t)]_{ab}\cr
[H_{\rm I}(t)]_{ba}&[H_{\rm I}(t)]_{bb}\cr}
=\pmatrix{\delta{\Bbb E}_a&-e{\bf r}_{ab}\cdot{\bf E}(t)\cr
-e{\bf r}_{ba}\cdot{\bf E}(t)&\delta{\Bbb E}_b\cr}.
$$
\medskip
\erritem{164}{line 15}
In Eq.~(6.49), ``$i\hbar(1-\rho_{bb})/T_b$'' should be replaced by
``$i\hbar(1-\rho_{aa})/T_b$''.\footnote{$\dagger$}{Cf.~M.~D.~Levenson
and S.~S.~Kano, {\sl Introduction to Nonlinear Laser Spectroscopy}
(Academic Press, New York, 1988), p.~33, Eqs.~(2.3.1)--(2.3.2).}
\medskip
\erritem{203}{lines 32--33}
``Fig.~4.3'' should be replaced by ``Fig.~4.4(a)''.
\medskip
\erritem{215}{line 11}
In Eq.~(7.14),
$$
\ldots=\mu_0\int^{\infty}_{-\infty}\,d\omega'(\omega+\omega')^2\cdots
$$
should be replaced by
$$
\ldots={{1}\over{c^2}}
\int^{\infty}_{-\infty}\,d\omega'(\omega+\omega')^2\cdots
$$
\medskip
\erritem{220}{section 7.2.1}
In the example of second harmonic generation, the wave equation~(7.26) is given
without any explanation of which point symmetry classes of media it applies to,
and hence it is from the text virtually impossible to relate the effective
nonlinear parameters to the elements of
$\chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega,\omega)$.
\medskip
\erritem{234}{line 31}
In Eq.~(7.45), ``$\ldots=iq^*\hat{E}^*_3>$'' should be replaced by
``$\ldots=iq^*\hat{E}^*_3$''.
\medskip
\erritem{240}{line 6}
In the first line of Eq.~(7.55), there is an ambiguity of the denominator,
as well as an erroneous dispersion term, and the equation
$$u=\tau\sqrt{n_2\omega/c|d^2 k/d\omega^2|^2}\widehat{E}$$
should be replaced by
$$u=\tau\sqrt{n_2\omega/(c|d^2 k/d\omega^2|)}\widehat{E}.$$
(The other lines of Eq.~(7.55) are correct.)
\medskip
\erritem{241}{line 30}
The fundamental bright soliton solution to the nonlinear Schr\"{o}dinger
equation should yield ``$u(\zeta,s)=\sech(s)\exp(i\zeta/2)$'', that is to say,
{\sl without} any minus sign in the exponential.
\medskip
\erritem{251}{line 6}
In Eq.~(8.5), there is parenthesis mismatch in both right- and
lefthand sides;
$$
f_0[({\Bbb E}_n({\bf k})]
=\{\exp[{\Bbb E}_n({\bf k})-{\Bbb E}_{\rm F}]/kT+1\}^{-1}
$$
should be replaced by
$$
f_0[{\Bbb E}_n({\bf k})]
=\{\exp[({\Bbb E}_n({\bf k})-{\Bbb E}_{\rm F})/kT]+1\}^{-1}.
$$
\medskip
\erritem{252}{line 6}
In Eq.~(8.7), $n_{\rm ph}(\omega_s({\bf q}))$ should be replaced
by $n_{\rm ph}[\omega_s({\bf q})]$, in order to follow the
functional style of notation as used in, for example, Eq.~(8.5).
\medskip
\erritem{253}{line 20}
In Eq.~(8.11), insert a ``$]$'' after ${\Bbb E}_n({\bf k})$.
\medskip
\erritem{298}{Table A3.2}
``$\ldots$ no centre of symmetry $\ldots$'' should be replaced
by ``$\ldots$ no centre of inversion $\ldots$''.
\medskip
\erritem{317}{lines 1, 9, and 24}
In Appendix 9, there is an inconsistency in the notation for
the polarisation density and the electric dipole operator, as
compared to the one used in Chapters 3 and 4.
While ${\bf P}^{\rm D}$, ${\bf P}^{\rm Q}$, and ${\bf M}^{\rm D}$
in Eq.~(A9.1) (and in line 9 on the same page)
denote the all-classical electric dipolar, electric quadrupolar
and magnetic dipolar polarization densities of the medium, they
in Eqs.~(A9.6) and~(A9.7) clearly denote quantum-mechanical operators.
In order to overcome this inconsistency in notation, which in
addition gives a wrong answer if properly inserted into the
perturbation calculus etc., one should chose either of the
conventions. By choosing ${\bf P}^{\rm D}$, ${\bf P}^{\rm Q}$,
and ${\bf M}^{\rm D}$ to denote the corresponding quantum-mechanical
operators, which seem to be the easiest way of correcting this
inconsistency, the following corrections to the text should be made:
{[{\it line 1}]}
$$
{\bf P}={\bf P}^{\rm D}+{\bf P}^{\rm Q},\qquad
{\bf M}={\bf M}^{\rm D},
\eqno{({\rm A}9.1)}
$$
should be replaced by
$$
{\bf P}=\langle{\bf P}^{\rm D}\rangle+\langle{\bf P}^{\rm Q}\rangle,\qquad
{\bf M}=\langle{\bf M}^{\rm D}\rangle,
\eqno{({\rm A}9.1)}
$$
{[{\it line 9}]}
Remove ``${\bf P}^{\rm D}$'' or replace with
``$\langle{\bf P}^{\rm D}\rangle$''.
{[{\it line 24}]} Somewhere in Appendix 9, there should be
a clarifying statement that the nabla operator appearing in Eq.~(A9.7)
only is operating on the all-classical, macroscopic electric field
of the light, and hence should be regarded as a classical vector
when evaluating the quantum-mechanical trace that is involved in
the expectation value of, for example, the corrected form of Eq.~(A9.1).
\medskip
\erritem{318}{line 12}
``${\bf M}<<{\bf H}$'' should be replaced by
``$|{\bf M}|\ll|{\bf H}|$''.
\medskip
\erritem{318}{line 23}
``${\bf e}_j\cdot e{\bf r}+i{\bf k}\cdot{\bf q}\cdot{\bf e}_j
+{\bf m}\cdot({\bf k}\times{\bf e}_j)/\omega$'' should be replaced by
the same expression, though with each term divided by $e$.
\medskip
\erritem{333}{line 5}
In reference Manley, J.~M. and Rowe, H.~E.~(1956), the page numbers
should yield 904 -- 14.
\medskip
\erritem{334}{line 44}
``Terhune, R.~W.~and Weinburger, D.~A. $\ldots$'' should be replaced
by ``Terhune, R.~W.~and Weinberger, D.~A. $\ldots$''.
\medskip
\bye