Contents of file 'lect1/lect1.tex':
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4 % Lecture No 1 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
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16 \def\rightheadline{\tenrm{\it Lecture notes #1}
17 \hfil{\it Nonlinear Optics 5A5513 (2003)}}
18 \def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)}
19 \hfil{\it Lecture notes #1}}
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28
29 \lecture{1}
30 Nonlinear optics is the discipline in physics in which the electric
31 polarization density of the medium is studied as a nonlinear function
32 of the electromagnetic field of the light. Being a wide field of
33 research in electromagnetic wave propagation, nonlinear interaction
34 between light and matter leads to a wide spectrum of phenomena, such
35 as optical frequency conversion, optical solitons, phase conjugation,
36 and Raman scattering. In addition, many of the analytical tools applied
37 in nonlinear optics are of general character, such as the perturbative
38 techniques and symmetry considerations, and can equally well be applied
39 in other disciplines in nonlinear dynamics.
40
41 \section{The contents of this course}
42 This course is intended as an introduction to the wide field of
43 phenomena encountered in nonlinear optics.
44 The course covers:
45 \smallskip
46
47 \item{$\bullet$}{The theoretical foundation of nonlinear interaction
48 between light and matter.}
49 \item{$\bullet$}{Perturbation analysis of nonlinear interaction
50 between light and matter.}
51 \item{$\bullet$}{The Bloch equation and its interpretation.}
52 \item{$\bullet$}{Basics of soliton theory and the inverse
53 scattering transform.}
54 \smallskip
55
56 It should be emphasized that the course does not cover state-of-the-art
57 material constants of nonlinear optical materials, etc.~but rather
58 focus on the theoretical foundations and ideas of nonlinear optical
59 interactions between light and matter.
60
61 A central analytical technique in this course is the perurbation
62 analysis, with its foundation in the analytical mechanics.
63 This technique will in the course mainly be applied to the
64 quantum-mechanical description of interaction between light and
65 matter, but is central in a wide field of cross-disciplinary
66 physics as well.
67 In order to give an introduction to the analytical theory of
68 nonlinear systems, we will therefore start with the analysis
69 of the nonlinear equations of motion for the mechanical pendulum.
70
71 \section{Examples of applications of nonlinear optics}
72 Some important applications in nonlinear optics:
73 \medskip
74 \item{$\bullet$}{Optical parametric amplification (OPA) and
75 oscillation (OPO), $\hbar\omega_{\rm p}\to\hbar\omega_{\rm s}
76 +\hbar\omega_{\rm i}$.}
77 \item{$\bullet$}{Second harmonic generation (SHG),
78 $\hbar\omega+\hbar\omega\to\hbar(2\omega)$.}
79 \item{$\bullet$}{Third harmonic generation (THG),
80 $\hbar\omega+\hbar\omega+\hbar\omega\to\hbar(3\omega)$.}
81 \item{$\bullet$}{Pockels effect, or the linear electro-optical
82 effect (applications for optical switching).}
83 \item{$\bullet$}{Optical bistability (optical logics).}
84 \item{$\bullet$}{Optical solitons (ultra long-haul communication).}
85 \vfill\eject
86
87 \section{A brief history of nonlinear optics}
88 Some important advances in nonlinear optics:
89 \medskip
90 \item{$\bullet$}{Townes et al. (1960), invention of the
91 laser.\footnote{${}^1$}{Charles H.~Townes was in 1964 awarded
92 with the Nobel Prize for the invention of the ammonia laser.}
93 }
94 \item{$\bullet$}{Franken et al. (1961), First observation ever of
95 nonlinear optical effects, second harmonic generation
96 (SHG).\footnote{${}^2$}{Franken et al. detected
97 ulvtraviolet light ($\lambda=347.1$ nm) at twice the frequency of a
98 ruby laser beam ($\lambda=694.2$ nm) when this beam traversed a
99 quartz crystal; P.~A.~Franken, A.~E.~Hill, C.~W. Peters, G.~Weinreich,
100 Phys.~Rev.~Lett. {\bf 7}, 118 (1961).
101 Second harmonic generation is also the first nonlinear effect ever
102 observed where a coherent input generates a coherent output.}
103 }
104 \item{$\bullet$}{Terhune et al. (1962), First observation of
105 third harmonic generation (THG).\footnote{${}^3$}{In their experiment,
106 Terhune et al. detected only about a thousand THG photons per pulse,
107 at $\lambda=231.3$ nm, corresponding to a conversion of one photon
108 out of about $10^{15}$ photons at the fundamental wavelength at
109 $\lambda=693.9$ nm; R.~W. Terhune, P.~D. Maker, and
110 C.~M. Savage, Phys. Rev. Lett. {\bf 8}, 404 (1962).}
111 }
112 \item{$\bullet$}{E.~J.~Woodbury and W.~K.~Ng (1962),
113 first demonstration of stimulated Raman
114 scattering.\footnote{${}^4$}{E.~J.~Woodbury and W.~K.~Ng,
115 Proc.~IRE {\bf 50}, 2347 (1962).}
116 }
117 \item{$\bullet$}{Armstrong et al. (1962), formulation of general
118 permutation symmetry relations in nonlinear
119 optics.\footnote{${}^5$}{The general permutation symmetry
120 relations of higher-order susceptibilities were published
121 by J.~A.~Armstrong, N.~Bloembergen, J.~Ducuing, and
122 P.~S. Pershan, Phys.~Rev.~{\bf 127}, 1918 (1962).}
123 }
124 \item{$\bullet$}{A.~Hasegawa and F.~Tappert (1973),
125 first theoretical prediction of soliton generation in optical
126 fibers.\footnote{${}^6$}{A.~Hasegawa and F.~Tappert,
127 ``Transmission of stationary nonliner optica pulses in dispersive
128 optical fibers: I, Anomalous dispersion; II Normal dispersion'',
129 Appl. Phys. Lett. {\bf 23}, 142--144 and 171--172
130 (August 1 and 15, 1973).}
131 }
132 \item{$\bullet$}{H.~M.~Gibbs et al. (1976), first demonstration
133 and explaination of optical
134 bistability.\footnote{${}^7$}{H.~M. Gibbs, S.~M. McCall,
135 and T.~N.~C. Venkatesan, Phys. Rev. Lett. {\bf 36}, 1135 (1976).}
136 }
137 \item{$\bullet$}{L.~F.~Mollenauer et al.~(1980),
138 first confirmation of soliton generation in optical
139 fibers.\footnote{${}^8$}{L.~F.~Mollenauer, R.~H. Stolen,
140 and J.~P. Gordon, ``Experimental observation of picosecond
141 pulse narrowing and solitons in optical fibers'',
142 Phys. Rev. Lett. {\bf 45}, 1095--1098 (September 29, 1980);
143 the first reported observation of solitons was though made in 1834 by
144 John Scott Russell, a Scottish scientist and later famous
145 Victorian engineer and shipbuilder, while studying water waves
146 in the Glasgow-Edinburgh channel.}
147 }
148 \smallskip
149 Recently, many advances in nonlinear optics has been made,
150 with a lot of efforts with fields of, for example, Bose-Einstein
151 condensation and laser cooling; these fields are, however,
152 a bit out of focus from the subjects of this course, which
153 can be said to be an introduction to the 1960s and 1970s
154 advances in nonlinear optics.
155 It should also be emphasized that many of the effects observed in
156 nonlinear optics, such as the Raman scattering, were observed
157 much earlier in the microwave range.
158
159 \section{Outline for calculations of polarization densities}
160 \subsection{Metals and plasmas}
161 From an all-classical point-of-view, the calculation of the
162 electric polarization density of metals and plasmas, containing
163 a free electron gas, can be performed using the model of free
164 charges acting under the Lorenz force of an electromagnetic field,
165 $$m_{\rm e}{{d^2{\bf r}_{\rm e}}\over{dt^2}}
166 =-e{\bf E}(t)-e{{d{\bf r}_{\rm e}}\over{dt}}\times{\bf B}(t),$$
167 where ${\bf E}$ and ${\bf B}$ are all-classical electric and magnetic
168 fields of the electromagnetic field of the light.
169 In forming the equation for the motion of the electron, the origin was
170 chosen to coincide with the center of the nucleus.
171
172 \subsection{Dielectrics}
173 A very useful model used by Drude and Lorentz\footnote{${}^9$}{R.~Becker,
174 {\it Elektronen Theorie}, (Teubner, Leipzig, 1933).} to calculate
175 the linear electric polarization of the medium describes the
176 electrons as harmonically bound particles.
177
178 For dielectrics in the nonlinear optical regime, as being the focus
179 of our attention in this course, the calculation of the electric
180 polarization density is instead performed using a nonlinear spring
181 model of the bound charges, here quoted for one-dimensional motion as
182 $$m_{\rm e}{{d^2{x}_{\rm e}}\over{dt^2}}
183 +\Gamma_{\rm e}{{d{x}_{\rm e}}\over{dt}}+\alpha^{(1)} x_{\rm e}
184 +\alpha^{(2)} x^2_{\rm e}+\alpha^{(3)} x^3_{\rm e}+\ldots
185 =-eE_x(t).$$
186 As in the previous case of metals and plasmas, in forming the equation
187 for the motion of the electron, the origin was also here chosen to
188 coincide with the center of the nucleus.
189
190 This classical mechanical model will later in this lecture be applied
191 to the derivation of the second-order nonlinear polarzation density
192 of the medium.
193
194 \section{\bf Introduction to nonlinear dynamical systems}
195 In this section we will, as a preamble to later analysis of
196 quantum-mechanical systems, apply perturbation analysis to a
197 simple mechanical system.
198 Among the simplest nonlinear dynamical systems is the pendulum,
199 for which the total mechanical energy of the system, considering the
200 point of suspension as defining the level of zero potential energy,
201 is given as the sum of the kinetic and potential energy as
202 $$E=T+V={1\over2m}|{\bf p}|^2 - mgl(\cos\vartheta-1),\eqno{(1)}$$
203 where $m$ is the mass, $g$ the gravitation constant, $l$ the length,
204 and $\vartheta$ the angle of deflection of the pendulum, and where ${\bf p}$
205 is the momentum of the point mass.
206 \medskip
207 \centerline{\epsfxsize=50mm\epsfbox{../images/pendulum/pendulum.1}}
208 \centerline{Figure 1. The mechanical pendulum.}
209 \medskip
210 \noindent
211 From the total mechanical energy~(1) of the system, the equations of
212 motion for the point mass is hence given by Lagranges
213 equations,\footnote{${}^{10}$}{Herbert Goldstein, {\it Classical Mechanics},
214 2nd ed.~(Addison-Wesley, Massachusetts, 1980).}
215 $$
216 {{d}\over{dt}}\bigg({{\partial L}\over{\partial p_j}}\bigg)
217 -{{\partial L}\over{\partial q_j}}=0,\qquad j=x,y,z,\eqno{(2)}
218 $$
219 where $q_j$ are the generalized coordinates, $p_j=\dot{q}_j$ are the
220 components of the generalized momentum, and $L=T-V$ is the Lagrangian of
221 the mechanical system. In spherical coordinates $(\rho,\varphi,\vartheta)$,
222 the momentum for the mass is given as its mass times the velocity,
223 $$
224 {\bf p}=ml(\dot{\vartheta}{\bf e}_{\vartheta}
225 +\dot{\varphi}\sin\vartheta{\bf e}_{\varphi})
226 $$
227 and the Lagrangian for the pendulum is hence given as
228 $$
229 \eqalign{
230 L&={{1}\over{2m}}(p^2_{\varphi}+p^2_{\varphi})+mgl(\cos\vartheta-1)\cr
231 &={{ml^2}\over{2}}(\dot{\vartheta}^2
232 +\dot{\varphi}^2\sin^2\vartheta)
233 +mgl(\cos\vartheta-1).\cr
234 }
235 $$
236 As the Lagrangian for the pendulum is inserted into Eq.~(2),
237 the resulting equations of motion are for $q_j=\varphi$ and $\vartheta$
238 obtained as
239 $$
240 {{d^2\varphi}\over{dt^2}}
241 +\bigg({{d\varphi}\over{dt}}\bigg)^2\sin\varphi\cos\varphi=0,
242 \eqno{(3{\rm a})}
243 $$
244 and
245 $$
246 {{d^2\vartheta}\over{dt^2}}+(g/l)\sin\vartheta=0,
247 \eqno{(3{\rm b})}
248 $$
249 respectively, where we may notice that the motion in ${\bf e}_{\varphi}$ and
250 ${\bf e}_{\vartheta}$ directions are decoupled.
251 We may also notice that the equation of motion for $\varphi$ is
252 independent of any of the physical parameters involved in te Lagrangian,
253 and the evolution of $\varphi(t)$ in time is entirely determined by
254 the initial conditions at some time $t=t_0$.
255
256 In the following discussion, the focus will be on the properties of the
257 motion of $\vartheta(t)$.
258 The equation of motion for the $\vartheta$ coordinate is here
259 described by the so-called Sine-Gordon equation.\footnote{${}^{11}$}{The
260 term ``Sine-Gordon equation'' has its origin as an allegory over the
261 similarity between the time-dependent (Sine-Gordon) equation
262 $${{\partial^2\varphi}\over{\partial z^2}}-{{1}\over{c^2}}
263 {{\partial^2\varphi}\over{\partial t^2}}=\mu^2_0\sin\varphi,$$
264 appearing in, for example, relativistic field theories, as compared
265 to the time-dependent Klein-Gordon equation, which takes the form
266 $${{\partial^2\varphi}\over{\partial z^2}}-{{1}\over{c^2}}
267 {{\partial^2\varphi}\over{\partial t^2}}=\mu^2_0\varphi.$$
268 The Sine-Gordon equation is sometimes also called ``pendulum equation''
269 in the terminology of classical mechanics.}
270 This nonlinear differential equation is hard\footnote{$\dagger$}{Impossible?}
271 to solve analytically, but if the nonlinear term is expanded as a
272 Taylor series around $\vartheta=0$,
273 $$
274 {{d^2\vartheta}\over{dt^2}}+(g/l)
275 (\vartheta-{{\vartheta^3}\over{3!}}+{{\vartheta^5}\over{5!}}+\ldots)=0,
276 $$
277 Before proceeding further with the properties of the solutions to
278 the approximative Sine-Gordon, including various orders of nonlinearities,
279 the general properties will now be illustrated.
280 In order to illustrate the behaviour of the Sine-Gordon equation,
281 we may normalize it by using the normalized time $\tau=(g/l)^{1/2}t$,
282 giving the Sine-Gordon equation in the normalized form
283 $${{d^2\vartheta}\over{d\tau^2}}+\sin\vartheta=0.\eqno{(4)}$$
284 The numerical solutions to the normalized Sine-Gordon
285 equation are in Fig.~2 shown for initial conditions (a) $y(0)=0.1$,
286 (b) $y(0)=2.1$, and (c) $y(0)=3.1$, all cases with $y'(0)=0$.
287 \medskip
288 \centerline{\epsfxsize=45mm\epsfbox{sg010.eps}
289 \hskip 7.5mm\epsfxsize=45mm\epsfbox{sg210.eps}
290 \hskip 7.5mm\epsfxsize=45mm\epsfbox{sg310.eps}}
291 \centerline{Figure 2. Numerical solutions to the normalized Sine-Gordon
292 equation.}
293 \medskip
294 \noindent
295 First of all, we may consider the linear case, for which the approximation
296 $\sin\vartheta\approx\vartheta$ holds. For this case, the Sine-Gordon
297 equation~(4) hence reduces to the one-dimensional linear wave-equation,
298 with solutions $\vartheta=A\sin((g/l)^{1/2}(t-t_0))$.
299 As seen in the frequency domain, this solution gives a delta peak
300 at $\omega=(g/l)^{1/2}$ in the power spectrum $|\tilde{\vartheta}(\omega)|^2$,
301 with no other frequency components present.
302 However, if we include the nonlinearities, the previous sine-wave
303 solution will tend to flatten at the peaks, as well as increase in
304 period, and this changes the power spectrum to be broadened as well
305 as flattened out.
306 In other words, the solution to the Sine-Gordon give rise to a wide
307 spectrum of frequencies, as compared to the delta peaks of the
308 solutions to the linearized, approximative Sine-Gordon equation.
309
310 From the numerical solutions, we may draw the conclusion that whenever
311 higher order nonlinear restoring forces come into play, even such a
312 simple mechanical system as the pendulum will carry frequency components
313 at a set of frequencies differing from the single frequency given
314 by the linearized model of motion.
315
316 More generally, hiding the fact that for this particular case the
317 restoring force is a simple sine function, the equation of motion
318 for the pendulum can be written as
319 $${{d^2\vartheta}\over{dt^2}}+a^{(0)}+a^{(1)}\vartheta+a^{(2)}\vartheta^2
320 +a^{(3)}\vartheta^3+\ldots=0.\eqno{(5)}$$
321 This equation of motion may be compared with the nonlinear wave
322 equation for the electromagnetic field of a travelling optical
323 wave of angular frequency $\omega$, of the form
324 $$
325 {{\partial^2 E}\over{\partial z^2}}
326 +{{\omega^2}\over{c^2}}E
327 +{{\omega^2}\over{c^2}}(\chi^{(1)}E+\chi^{(2)}E^2+\chi^{(3)}E^3+\ldots)=0,
328 $$
329 which clearly shows the similarity between the nonlinear wave propagation
330 and the motion of the nonlinear pendulum.
331
332 Having solved the particular problem of the nonlinear pendulum,
333 we may ask ourselves if the equations of motion may be altered
334 in some way in order to give insight in other areas of nonlinear
335 physics as well.
336 For example, the series~(5) that define the feedback that tend to
337 restore the mechanical pendulum to its rest position clearly defines
338 equations of motion that conserve the total energy of the
339 mechanical system.
340 This, however, in generally not true for an arbitrary series
341 of terms of various power for the restoring force.
342 As we will later on see, in nonlinear optics we generally have a
343 complex, though in many cases most predictable, transfer of energy
344 between modes of different frequencies and directions of propagation.
345
346 \section{The anharmonic oscillator}
347 Among the simplest models of interaction between light and matter
348 is the all-classical one-electron oscillator, consisting of a
349 negatively charged particle (electron) with mass $m_{\rm e}$, mutually
350 interacting with a positively charged particle (proton)
351 with mass $m_{\rm p}$, through attractive Coulomb forces.
352 \medskip
353 \centerline{\epsfxsize=90mm\epsfbox{../images/spring/udspring.1}}
354 \smallskip
355 \centerline{Figure 5. Setup of the one-dimensional undamped spring model.}
356 \medskip
357
358 In the one-electron oscillator model, several levels of approximations
359 may be applied to the problem, with increasing algebraic complexity.
360 At the first level of approximation, the proton is assumed to
361 be fixed in space, with the electron free to oscillate around
362 the proton.
363 Quite generally, at least within the scope of linear optics,
364 the restoring spring force which confines the electron
365 can be assumed to be linear with the displacement distance
366 of the electron from the central position.
367 Providing the very basic models of the concept of refractive index and
368 optical dispersion, this model has been applied by numerous authors,
369 such as Feynman~[R.~P.~Feynman, {\sl Lectures on Physics} (Addison-Wesley,
370 Massachusetts, 1963)], and Born and Wolf~[M.~Born and E.~Wolf,
371 {\sl Principles of Optics} (Cambridge University Press, Cambridge, 1980)].
372
373 Moving on to the next level of approximation, the bound proton-electron
374 pair may be considered as constituting a two-body central force
375 problem of classical mechanics, in which one may assume a fixed center
376 of mass of the system, around which the proton as well as the electron
377 are free to oscillate.
378 In this level of approximation, by introducing the concept of reduced
379 mass for the two moving particles, the equations of motion for the
380 two particles can be reduced to one equation of motion, for the
381 evolution of the electric dipole moment of the system.
382
383 The third level of approximation which may be identified is when
384 the center of mass is allowed to oscillate as well, in which case
385 an equation of motion for the center of mass appears in addition to
386 the one for the evolution of the electric dipole moment.
387
388 In each of the models, nonlinearities of the restoring central force
389 field may be introduced as to include nonlinear interactions as well.
390 It should be emphasized that the spring model, as now will be introduced,
391 gives an identical form of the set of nonzero elements of the
392 susceptibility tensors, as compared with those obtained using a
393 quantum mechanical analysis.
394
395 Throughout this analysis, the wavelength of the electromagnetic
396 field will be assumed to be sufficiently large in order to neglect
397 any spatial variations of the fields over the spatial extent of the
398 oscillator system.
399 In this model, the central force field is modelled by a mechanical
400 spring force with spring constant $k_{\rm e}$, as shown schematically
401 in Fig.~5, and the all-classical Newton's equations
402 of motion for the electron and nucleus are
403 $$
404 \eqalign{
405 m_{\rm e}{{\partial^2 x_{\rm e}}\over{\partial t^2}}
406 &=\underbrace{-eE(t)}_{\rm optical}
407 -\underbrace{k_0(x_{\rm e}-x_{\rm n})
408 +k_1(x_{\rm e}-x_{\rm n})^2}_{\rm spring},\cr
409 m_{\rm n}{{\partial^2 x_{\rm n}}\over{\partial t^2}}
410 &=\underbrace{+eE(t)}_{\rm optical}
411 +\underbrace{k_0(x_{\rm e}-x_{\rm n})
412 -k_1(x_{\rm e}-x_{\rm n})^2}_{\rm spring},\cr
413 }
414 $$
415 corresponding to a system of two particles connected by a spring with
416 spring ``constant''
417 $$
418 k=-{{\partial F^{({\rm spring})}_{\rm e}}
419 \over{\partial (x_{\rm e}-x_{\rm n})}}
420 ={{\partial F^{({\rm spring})}_{\rm n}}
421 \over{\partial (x_{\rm e}-x_{\rm n})}}
422 =k_0-2k_1(x_{\rm e}-x_{\rm n}).
423 $$
424 By introducing the reduced mass\footnote{${}^{12}$}{Herbert Goldstein,
425 {\it Classical Mechanics}, 2nd ed.~(Addison-Wesley, Massachusetts, 1980).}
426 $m_{\rm r} = m_{\rm e} m_{\rm n} / (m_{\rm e} + m_{\rm n})$ of the system,
427 the equation of motion for the electric dipole moment
428 $p=-e(x_{\rm e}-x_{\rm n})$ is then obtained as
429 $$
430 {{\partial^2 p}\over{\partial t^2}}+{{k_0}\over{m_{\rm r}}}p
431 +{{k_1}\over{e m_{\rm r}}}p^2
432 ={{e^2}\over{m_{\rm r}}}E(t).\eqno(6)
433 $$
434 This inhomogeneous nonlinear ordinary differential equation for the
435 electric dipole moment is the primary interest in the discussion
436 that now is to follow.
437
438 The electric dipole moment of the anharmonic oscillator is now
439 expressed in terms of a perturbation series as
440 $$
441 p(t)=p^{(0)}(t)+\underbrace{p^{(1)}(t)}_{\propto E(t)}
442 +\underbrace{p^{(2)}(t)}_{\propto E^2(t)}
443 +\underbrace{p^{(3)}(t)}_{\propto E^3(t)}
444 +\ldots,
445 $$
446 where each term in the series is proportional to the applied electrical
447 field strength to the power as indicated in the superscript of repective
448 term, and formulate the system of $n+1$ equations for $p^{(k)}$,
449 $k=0,1,2,\ldots,n$, that define the time evolution of the electric
450 dipole. By inserting the perturbation series into Eq.~(6), we hence
451 have the equation
452 $$
453 \eqalign{
454 {{\partial^2 p^{(0)}}\over{\partial t^2}}&
455 +{{\partial^2 p^{(1)}}\over{\partial t^2}}
456 +{{\partial^2 p^{(2)}}\over{\partial t^2}}
457 +{{\partial^2 p^{(3)}}\over{\partial t^2}}
458 +\ldots\cr
459 &+{{k_0}\over{m_{\rm r}}}p^{(0)}
460 +{{k_0}\over{m_{\rm r}}}p^{(1)}
461 +{{k_0}\over{m_{\rm r}}}p^{(2)}
462 +{{k_0}\over{m_{\rm r}}}p^{(3)}
463 +\ldots\cr
464 &+{{k_1}\over{e m_{\rm r}}}
465 (p^{(0)}+p^{(1)}+p^{(2)}+\ldots)(p^{(0)}+p^{(1)}+p^{(2)}+\ldots)
466 ={{e^2}\over{m_{\rm r}}}E(t).\cr
467 }
468 $$
469 Since this equation is to hold for an arbitrary electric field $E(t)$,
470 that is to say, at least within the limits of the validity of the
471 perturbation analysis, each set of terms with equal power dependence
472 of the electric field must individually satisfy the relation.
473 By sorting out the various powers and identifying terms in the left
474 and right hand sides of the equation, we arrive at the system
475 of equations
476 $$
477 \eqalign{
478 &{{\partial^2 p^{(0)}}\over{\partial t^2}}
479 +{{k_0}\over{m_{\rm r}}}p^{(0)}
480 +{{k_1}\over{e m_{\rm r}}}p^{(0)}{}^2=0,\cr
481 &{{\partial^2 p^{(1)}}\over{\partial t^2}}
482 +{{k_0}\over{m_{\rm r}}}p^{(1)}
483 +{{k_1}\over{e m_{\rm r}}}2p^{(0)}p^{(1)}
484 ={{e^2}\over{m_{\rm r}}}E(t),\cr
485 &{{\partial^2 p^{(2)}}\over{\partial t^2}}
486 +{{k_0}\over{m_{\rm r}}}p^{(2)}
487 +{{k_1}\over{e m_{\rm r}}}(2p^{(0)}p^{(2)}+p^{(1)}{}^2)=0,\cr
488 &{{\partial^2 p^{(3)}}\over{\partial t^2}}
489 +{{k_0}\over{m_{\rm r}}}p^{(3)}
490 +{{k_1}\over{e m_{\rm r}}}(2p^{(0)}p^{(3)}+2p^{(1)}p^{(2)})=0,\cr
491 }
492 $$
493 where we kept terms with powers of the electric field up to and including
494 order three.
495 At a first glance, this system seem to suggest that only the first
496 order of the perturbation series depends on the applied electric
497 field of the light; however, taking a closer look at the system,
498 one can easily verify that all orders of the dipole moment
499 is coupled directly to the lower order terms.
500 The system of equations for $p^{(k)}$ can now be solved for
501 $k=0,1,2,\ldots$, in that order, to successively provide the
502 basis of solutions for higher and higher order terms, until reaching
503 some $k=n$ after which we may safely neglect the reamaining terms,
504 hence providing an approximate solution.\footnote{${}^{13}$}{It should
505 though be emphasized that in the limit $n\to\infty$, the described
506 theory still is an exact description of the motion of the electric
507 dipole moment within this model of interaction between light and matter.}
508
509 The zeroth order term in the perturbation series is decribed
510 by a nonlinear ordinary differential equation of order two,
511 a so-calles {\sl Riccati equation}, which analytically can be
512 solved exactly, either by directly applying the theory of Jacobian
513 elliptic integrals of by applying the Riccati
514 transormation.\footnote{${}^{14}$}{For examples of the application of the
515 Riccati transformation, see Zwillinger, {\it Handbook of Differential
516 Equations}, 2nd ed.~(Academic Press, Boston, 1992).}
517 However, by considering a system starting from rest, at a state of
518 equilibrium, we can immediately draw the conclusion that $p^{(0)}(t)$
519 must be identically zero for all times $t$.
520 This, of course, only holds for this particular model; in many
521 molecular systems, such in water, a permanent static dipole moment
522 is present, something that is left out in this particular spring model
523 of ours. (Not to be confused with the static polarization induced
524 by the electric field, which by definition of the terms in the
525 perturbation series is included in higher order terms, depending
526 on the power of the electric field.)
527
528 The first order term in the perturbation series is the first and only
529 one with an explicit dependence of the electric field of the light.
530 Since the zeroth order perturbation term is zero, the differential
531 equation for the first order term is linear, which simplifies the
532 calculus.
533 However, since it is an inhomogeneous differential equation, we must
534 generally look for a total solution to the equation as a sum of
535 a homogeneous solution (with zero right hand side) and a particular
536 solution (with the electric field in the right hand side present).
537 The homogeneous solution, which will contain two constants of integration
538 (since we are considering second-order ordinary differential equation)
539 will though only give the part of the solution which depend on initial
540 conditions, that is to say, in this case a harmonic natural oscillation
541 of the spring system which in the presence of damping terms rapidly
542 would decrease to zero.
543 This implies that in order to find steady-state solutions, in which
544 the oscillation of the dipole moment directly follows the oscillation
545 of the electric field of the light, we may directly start looking for
546 the particular solution.
547 For a time harmonic electric field, here taken as
548 $$E(t)=E_{\omega}\sin(\omega t),$$
549 the particular solution for the first order term is after some
550 straightforward algebra given as\footnote{${}^{15}$}{For the sake
551 of self consistency, the general solution for the first order
552 term is given as
553 $$p^{(1)}=A\cos((k_0/m_{\rm r})^{1/2}t)+B\sin((k_0/m_{\rm r})^{1/2}t)
554 +{{e^2/m_{\rm r}}\over{k_0/m_{\rm r}-\omega^2}}E_{\omega}\sin(\omega t),$$
555 where $A$ and $B$ are constants of integration, determined by initial
556 conditions.}
557 $$
558 p^{(1)}={{(e^2/m_{\rm r})}\over{(k_0/m_{\rm r}-\omega^2)}}E_{\omega}
559 \sin(\omega t),\qquad\omega^2\ne(k_0/m_{\rm r}).
560 $$
561 For a material consisting of $N$ dipoles per unit volume, and by
562 following the conventions for the linear electric susceptibility
563 in SI units, this corresponds to a first order electric polarization
564 density of the form
565 $$
566 \eqalign{
567 P^{(1)}(t)
568 &=P^{(1)}_{\omega}\sin(\omega t)\cr
569 &=\varepsilon_0\chi^{(1)}(\omega)E_{\omega}\sin(\omega t),\cr
570 }
571 $$
572 with the {\sl first order (linear) electric susceptibility} given as
573 $$
574 \chi^{(1)}(\omega)
575 =\chi^{(1)}(-\omega;\omega)
576 ={{N}\over{\varepsilon_0}}{{(e^2/m_{\rm r})}\over{(\Omega^2-\omega^2)}},
577 $$
578 where the resonance frequency $\Omega^2=k_0/m_{\rm r}$ was introduced.
579 The Lorenzian shape of the frequency dependence is shown in Fig.~6.
580 \vfill\eject
581
582 \medskip
583 \centerline{\epsfxsize=90mm\epsfbox{chi1.eps}}
584 \smallskip
585 \centerline{Figure 6. Lorenzian shape of the linear susceptibility
586 $\chi^{(1)}(-\omega;\omega)$.}
587 \medskip
588 Continuing with the second order perturbation term, some straightforward
589 algebra gives that the particular solution for the second order term
590 of the electric dipole moment becomes
591 $$
592 \eqalign{
593 p^{(2)}(t)&=-{{k_1e^3}\over{2k_0 m^2_{\rm r}}}
594 {{1}\over{(\Omega^2-\omega^2)}}E^2_{\omega}
595 +{{k_1e^3}\over{2m^3_{\rm r}}}
596 {{1}\over{(\Omega^2-\omega^2)(\Omega^2-4\omega^2)}}
597 E^2_{\omega}\cr
598 &\qquad\qquad+{{k_1e^3}\over{m^3_{\rm r}}}
599 {{1}\over{(\Omega^2-\omega^2)(\Omega^2-4\omega^2)}}
600 E^2_{\omega}\sin^2(\omega t).\cr
601 }
602 $$
603 In terms of the polarization density of the medium, still with $N$
604 dipoles per unit volume and following the conventions in regular
605 SI units, this can be written as
606 $$
607 \eqalign{
608 P^{(2)}(t)
609 &=P^{(0)}_{0}+P^{(0)}_{2\omega}\sin(2\omega t)\cr
610 &=\underbrace{\varepsilon_0\chi^{(2)}(0;\omega,-\omega)
611 E_{\omega}E_{\omega}}_{\rm DC\ polarization}
612 +\underbrace{\varepsilon_0\chi^{(2)}(-2\omega;\omega,\omega)
613 E_{\omega}E_{\omega}\sin(2\omega)}_{\rm second
614 \ harmonic\ polarization}\cr
615 }
616 $$
617 with the {\sl second order (quadratic) electric susceptibility} given as
618 $$
619 \eqalign{
620 \chi^{(2)}(0;\omega,-\omega)&=
621 {{N}\over{\varepsilon_0}}{{k_1e^3}\over{2m^3_{\rm r}}}
622 \bigg[{{1}\over{(\Omega^2-\omega^2)(\Omega^2-4\omega^2)}}
623 -{{1}\over{\Omega^2(\Omega^2-\omega^2)}}\bigg],\cr
624 \chi^{(2)}(2\omega;\omega,\omega)&=
625 {{N}\over{\varepsilon_0}}{{k_1e^3}\over{m^3_{\rm r}}}
626 {{1}\over{(\Omega^2-\omega^2)(\Omega^2-4\omega^2)}}.\cr
627 }
628 $$
629 From this we may notice that for one-photon resonances, the nonlinearities
630 are enhanced whenever $\omega\approx\Omega$ or $2\omega\approx\Omega$,
631 for the induced DC as well as the second harmonic polarization density.
632
633 The explicit frequency dependencies of the susceptibilities
634 $\chi^{(2)}(-2\omega;\omega,\omega)$ and $\chi^{(2)}(0;\omega,-\omega)$
635 are shown in Figs.~7 and 8.
636 \vfill\eject
637
638 \medskip
639 \centerline{\epsfxsize=90mm\epsfbox{chi2a.eps}}
640 \smallskip
641 \centerline{Figure 7. Lorenzian shape of the linear susceptibility
642 $\chi^{(2)}(-2\omega;\omega,\omega)$ (SHG).}
643 \bigskip
644 \centerline{\epsfxsize=90mm\epsfbox{chi2b.eps}}
645 \smallskip
646 \centerline{Figure 8. Lorenzian shape of the linear susceptibility
647 $\chi^{(2)}(0;\omega,-\omega)$ (DC).}
648 \medskip
649
650 A well known fact in electromagnetic theory is that an electric dipole
651 that oscillates at a certain angular frequency, say at $2\omega$,
652 also emits electromagnetic radiation at this frequency.
653 In particular, this implies that the term described by the
654 susceptibility $\chi^{(2)}(-2\omega;\omega,\omega)$ will generate
655 light at twice the angular frequency of the light, hence generating
656 a second harmonic light wave.
657 \bye
658
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