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% File: nlopt/lect1/lect1.tex [pure TeX code]2% Last change: January 7, 20033%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 Jonsson8%9 \input epsf 10 \font\ninerm=cmr9 11 \font\twelvesc=cmcsc10 12 \def\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm 13 \topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt 14 \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi 15 \else\hfill\fi} 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}} 20 \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps} 21 \vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}} 22 \hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1} 23 \vskip 24pt\noindent} 24 \def\section #1 {\medskip\goodbreak\noindent{\bf #1} 25 \par\nobreak\smallskip\noindent} 26 \def\subsection #1 {\smallskip\goodbreak\noindent{\it #1} 27 \par\nobreak\smallskip\noindent} 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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