Contents of file 'bcconven.tex':
1 % File: nlopt/bcconven/bcconven.tex [pure TeX code]
2 % Last change: January 26, 2003
3 %
4 % Notes on the ``Butcher and Cotter convention'' in nonlinear optics,
5 % and an example of its application to the formulation of the polarization
6 % density in the case of optical Kerr-effect. This memo is part of the
7 % material handed out in the course ``Nonlinear optics'', held January-March,
8 % 2003, at the Royal Institute of Technology, Stockholm, Sweden.
9 %
10 % This memo was in the 2003 course handed out during the third lecture.
11 %
12 % Copyright (C) 2003, Fredrik Jonsson
13 %
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23 \input amssym % to get \Bbb fonts
24 \input epsf
25 \font\ninerm=cmr9
26 \font\twelvebf=cmbx12
27 \def\Re{\mathop{\rm Re}\nolimits} % real part
28 \def\Im{\mathop{\rm Im}\nolimits} % imaginary part
29 \def\Tr{\mathop{\rm Tr}\nolimits} % quantum mechanical trace
30 %
31 % The following macro for generation of double column output is taken from
32 % Donald Knuths ``The TeXbook'' (Addison-Wesley, Massachusetts, 1990),
33 % Appendix D: Dirty Trics, page 387.
34 %
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54 \else\global\highwidth=\trialwidth\fi \advance\n by1 }
55 \def\section #1 {\medskip\goodbreak\noindent{\bf #1}
56 \par\nobreak\smallskip\noindent}
57 \def\subsection #1 {\medskip\goodbreak\noindent{\sl #1}
58 \par\nobreak\smallskip\noindent}
59 \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
60 \else\hfill\fi}
61 \def\rightheadline{\hfil{\it Nonlinear Optics 5A5513 (2003)}}
62 \def\leftheadline{{\it Nonlinear Optics 5A5513 (2003)}\hfil}
63 \voffset=2\baselineskip
64 \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps}
65 \vskip-18pt\hfill{\it Nonlinear Optics 5A5513 (2003)}
66 \vskip 36pt
67 \centerline{\twelvebf Notes on the ``Butcher and Cotter convention'' in
68 nonlinear optics}
69 \bigskip
70
71 \section{Convention for description of nonlinear optical polarization}
72 As a ``recipe'' in theoretical nonlinear optics, Butcher and Cotter
73 provide a very useful convention which is well worth to hold on to.
74 For a superposition of monochromatic waves, and by invoking the general
75 property of the intrinsic permutation symmetry, the monochromatic
76 form of the $n$th order polarization density can be written as
77 $$
78 (P^{(n)}_{\omega_{\sigma}})_{\mu}
79 =\varepsilon_0\sum_{\alpha_1}\cdots\sum_{\alpha_n}\sum_{\omega}
80 K(-\omega_{\sigma};\omega_1,\ldots,\omega_n)
81 \chi^{(n)}_{\mu\alpha_1\cdots\alpha_n}
82 (-\omega_{\sigma};\omega_1,\ldots,\omega_n)
83 (E_{\omega_1})_{\alpha_1}\cdots(E_{\omega_n})_{\alpha_n}.
84 \eqno{(1)}
85 $$
86 The first summations in Eq.~(1), over $\alpha_1,\ldots,\alpha_n$,
87 is simply an explicit way of stating that the Einstein convention
88 of summation over repeated indices holds.
89 The summation sign $\sum_{\omega}$, however, serves as a reminder
90 that the expression that follows is to be summed over
91 {\sl all distinct sets of $\omega_1,\ldots,\omega_n$}.
92 Because of the intrinsic permutation symmetry, the frequency arguments
93 appearing in Eq.~(1) may be written in arbitrary order.
94
95 By ``all distinct sets of $\omega_1,\ldots,\omega_n$'', we here mean
96 that the summation is to be performed, as for example in the case of
97 optical Kerr-effect, over the single set of nonlinear susceptibilities
98 that contribute to a certain angular frequency as
99 $(-\omega;\omega,\omega,-\omega)$ {\sl or} $(-\omega;\omega,-\omega,\omega)$
100 {\sl or} $(-\omega;-\omega,\omega,\omega)$.
101 In this example, each of the combinations are considered as {\sl distinct},
102 and it is left as an arbitary choice which one of these sets that are
103 most convenient to use (this is simply a matter of choosing notation,
104 and does not by any means change the description of the interaction).
105
106 In Eq.~(1), the degeneracy factor $K$ is formally described as
107 $$K(-\omega_{\sigma};\omega_1,\ldots,\omega_n)=2^{l+m-n}p$$
108 where
109 $$
110 \eqalign{
111 p&={\rm the\ number\ of\ {\sl distinct}\ permutations\ of}
112 \ \omega_1,\omega_2,\ldots,\omega_1,\cr
113 n&={\rm the\ order\ of\ the\ nonlinearity},\cr
114 m&={\rm the\ number\ of\ angular\ frequencies}\ \omega_k
115 \ {\rm that\ are\ zero,\ and}\cr
116 l&=\bigg\lbrace\matrix{1,\qquad{\rm if}\ \omega_{\sigma}\ne 0,\cr
117 0,\qquad{\rm otherwise}.}\cr
118 }
119 $$
120 In other words, $m$ is the number of DC electric fields present,
121 and $l=0$ if the nonlinearity we are analyzing gives a static, DC,
122 polarization density, such as in the previously (in the spring model)
123 described case of optical rectification in the presence of second
124 harmonic fields (SHG).
125
126 A list of frequently encountered nonlinear phenomena in nonlinear
127 optics, including the degeneracy factors as conforming to the above
128 convention, is given in Butcher and Cotters book, Table 2.1, on page 26.
129
130 \section{Note on the complex representation of the optical field}
131 Since the observable electric field of the light, in Butcher and
132 Cotters notation taken as
133 $$
134 {\bf E}({\bf r},t)={{1}\over{2}}\sum_{\omega_k\ge 0}
135 [{\bf E}_{\omega_k}\exp(-i\omega_k t)+{\bf E}^*_{\omega_k}\exp(i\omega_k t)],
136 $$
137 is a real-valued quantity, it follows that negative frequencies in the
138 complex notation should be interpreted as the complex conjugate of the
139 respective field component, or
140 $$
141 {\bf E}_{-\omega_k}={\bf E}^*_{\omega_k}.
142 $$
143
144 \section{Example: Optical Kerr-effect}
145 Assume a monochromatic optical wave (containing forward and/or backward
146 propagating components) polarized in the $xy$-plane,
147 $$
148 {\bf E}(z,t)=\Re[{\bf E}_{\omega}(z)\exp(-i\omega t)]\in{\Bbb R}^3,
149 $$
150 with all spatial variation of the field contained in
151 $$
152 {\bf E}_{\omega}(z)={\bf e}_x E^x_{\omega}(z)
153 +{\bf e}_y E^y_{\omega}(z)\in{\Bbb C}^3.
154 $$
155 Optical Kerr-effect is in isotropic media described by the third order
156 susceptibility
157 $$
158 \chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega),
159 $$
160 with nonzero components of interest for the $xy$-polarized beam given in
161 Appendix 3.3 of Butcher and Cotters book as
162 $$
163 \chi^{(3)}_{xxxx}=\chi^{(3)}_{yyyy},\quad
164 \chi^{(3)}_{xxyy}=\chi^{(3)}_{yyxx}
165 =\bigg\{\matrix{{\rm intr.\ perm.\ symm.}\cr
166 (\alpha,\omega)\rightleftharpoons(\beta,\omega)\cr}\bigg\}=
167 \chi^{(3)}_{xyxy}=\chi^{(3)}_{yxyx},\quad
168 \chi^{(3)}_{xyyx}=\chi^{(3)}_{yxxy},
169 $$
170 with
171 $$
172 \chi^{(3)}_{xxxx}=\chi^{(3)}_{xxyy}+\chi^{(3)}_{xyxy}+\chi^{(3)}_{xyyx}.
173 $$
174 The degeneracy factor $K(-\omega;\omega,\omega,-\omega)$ is calculated as
175 $$
176 K(-\omega;\omega,\omega,-\omega)=2^{l+m-n}p=2^{1+0-3}3=3/4.
177 $$
178 From this set of nonzero susceptibilities, and using the calculated
179 value of the degeneracy factor in the convention of Butcher and Cotter,
180 we hence have the third order electric polarization density at
181 $\omega_{\sigma}=\omega$ given as ${\bf P}^{(n)}({\bf r},t)=
182 \Re[{\bf P}^{(n)}_{\omega}\exp(-i\omega t)]$, with
183 $$
184 \eqalign{
185 {\bf P}^{(3)}_{\omega}
186 &=\sum_{\mu}{\bf e}_{\mu}(P^{(3)}_{\omega})_{\mu}\cr
187 &=\{{\rm Using\ the\ convention\ of\ Butcher\ and\ Cotter}\}\cr
188 &=\sum_{\mu}{\bf e}_{\mu}
189 \bigg[\varepsilon_0{{3}\over{4}}\sum_{\alpha}\sum_{\beta}\sum_{\gamma}
190 \chi^{(3)}_{\mu\alpha\beta\gamma}(-\omega;\omega,\omega,-\omega)
191 (E_{\omega})_{\alpha}(E_{\omega})_{\beta}(E_{-\omega})_{\gamma}\bigg]\cr
192 &=\{{\rm Evaluate\ the\ sums\ over\ } (x,y,z)
193 {\rm\ for\ field\ polarized\ in\ the\ }xy{\rm\ plane}\}\cr
194 &=\varepsilon_0{{3}\over{4}}\{
195 {\bf e}_x[
196 \chi^{(3)}_{xxxx} E^x_{\omega} E^x_{\omega} E^x_{-\omega}
197 +\chi^{(3)}_{xyyx} E^y_{\omega} E^y_{\omega} E^x_{-\omega}
198 +\chi^{(3)}_{xyxy} E^y_{\omega} E^x_{\omega} E^y_{-\omega}
199 +\chi^{(3)}_{xxyy} E^x_{\omega} E^y_{\omega} E^y_{-\omega}]\cr
200 &\qquad\quad
201 +{\bf e}_y[
202 \chi^{(3)}_{yyyy} E^y_{\omega} E^y_{\omega} E^y_{-\omega}
203 +\chi^{(3)}_{yxxy} E^x_{\omega} E^x_{\omega} E^y_{-\omega}
204 +\chi^{(3)}_{yxyx} E^x_{\omega} E^y_{\omega} E^x_{-\omega}
205 +\chi^{(3)}_{yyxx} E^y_{\omega} E^x_{\omega} E^x_{-\omega}]\}\cr
206 &=\{{\rm Make\ use\ of\ }{\bf E}_{-\omega}={\bf E}^*_{\omega}
207 {\rm\ and\ relations\ }\chi^{(3)}_{xxyy}=\chi^{(3)}_{yyxx},
208 {\rm\ etc.}\}\cr
209 &=\varepsilon_0{{3}\over{4}}\{
210 {\bf e}_x[
211 \chi^{(3)}_{xxxx} E^x_{\omega} |E^x_{\omega}|^2
212 +\chi^{(3)}_{xyyx} E^y_{\omega}{}^2 E^{x*}_{\omega}
213 +\chi^{(3)}_{xyxy} |E^y_{\omega}|^2 E^x_{\omega}
214 +\chi^{(3)}_{xxyy} E^x_{\omega} |E^y_{\omega}|^2]\cr
215 &\qquad\quad
216 +{\bf e}_y[
217 \chi^{(3)}_{xxxx} E^y_{\omega} |E^y_{\omega}|^2
218 +\chi^{(3)}_{xyyx} E^x_{\omega}{}^2 E^{y*}_{\omega}
219 +\chi^{(3)}_{xyxy} |E^x_{\omega}|^2 E^y_{\omega}
220 +\chi^{(3)}_{xxyy} E^y_{\omega} |E^x_{\omega}|^2]\}\cr
221 &=\{{\rm Make\ use\ of\ intrinsic\ permutation\ symmetry}\}\cr
222 &=\varepsilon_0{{3}\over{4}}\{
223 {\bf e}_x[
224 (\chi^{(3)}_{xxxx} |E^x_{\omega}|^2
225 +2\chi^{(3)}_{xxyy} |E^y_{\omega}|^2) E^x_{\omega}
226 +(\chi^{(3)}_{xxxx}-2\chi^{(3)}_{xxyy})
227 E^y_{\omega}{}^2 E^{x*}_{\omega}\cr
228 &\qquad\quad
229 {\bf e}_y[
230 (\chi^{(3)}_{xxxx} |E^y_{\omega}|^2
231 +2\chi^{(3)}_{xxyy} |E^x_{\omega}|^2) E^y_{\omega}
232 +(\chi^{(3)}_{xxxx}-2\chi^{(3)}_{xxyy})
233 E^x_{\omega}{}^2 E^{y*}_{\omega}.\cr
234 }
235 $$
236 For the optical field being linearly polarized, say in the $x$-direction,
237 the expression for the polarization density is significantly simplified,
238 to yield
239 $$
240 {\bf P}^{(3)}_{\omega}=\varepsilon_0(3/4){\bf e}_x
241 \chi^{(3)}_{xxxx} |E^x_{\omega}|^2 E^x_{\omega},
242 $$
243 i.~e.~taking a form that can be interpreted as an intensity-dependent
244 ($\sim|E^x_{\omega}|^2$) contribution to the refractive index
245 (cf.~Butcher and Cotter \S 6.3.1).
246 \bye
247
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