P1: FYK Revised Pages Qu: 00, 00, 00, 00Encyclopedia of Physical Science and Technology EN001-05 May 25, 2001 16:7Acoustic ChaosWerner LauterbornUniversit at G ottingenI. The Problem of Acoustic Cavitation NoiseII. The Period-Doubling Noise SequenceIII. A Fractal Noise AttractorIV. Lyapunov AnalysisV. Period-Doubling Bubble OscillationsVI. Theory of Driven BubblesVII. Other SystemsVIII. Philosophical ImplicationsGLOSSARYBifurcation Qualitative change in the behavior of a sys-tem, when a parameter (temperature, pressure, etc.) isaltered (e.g., period-doubling bifurcation); related tophase change in thermodynamics.Cavitation Rupture of liquids when subject to tension ei-ther in owelds (hydraulic cavitation) or by an acous-tic wave (acoustic cavitation).Chaos Behavior (motion) with all signs of statistics de-spite an underlying deterministic law (often, determin-istic chaos).Fractal Object (set of points) that does not have a smoothstructure with an integer dimension (e.g., three dimen-sional). Instead, a fractal (noninteger) dimension mustbe ascribed to them.Period doubling Special way of obtaining chaotic (irreg-ular) motion; the period of a periodic motion doublesrepeatedly until in the limit of innite doubling aperi-odic motion is obtained.Phase space Space spanned by the dependent variables ofa dynamic system. Apoint in phase space characterizesa specic state of the system.Strange attractor In dissipative systems, the motiontends to certain limits forms (attractors). When the mo-tion comes to rest, this attractor is called a xed point.Chaotic motions run on a strange attractor, which hasinvolved properties (e.g., a fractal dimension).THE PAST FEWyears have seen a remarkable develop-ment in physics, which may be described as the upsurgeof chaos. Chaos is a term scientists have adapted fromcommon language to describe the motion or behavior ofa system (physical or biological) that, although governedby an underlying deterministic law, is irregular and, in thelong term, unpredictable.Chaotic motion seems to appear in any sufcientlycomplex dynamical system. Acoustics, that part ofphysics that descibes the vibration of usually larger en-sembles of molecules in gases, liquids, and solids, makesno exception. As a main necessary ingredient of chaotic117P1: FYK Revised PagesEncyclopedia of Physical Science and Technology EN001-05 May 8, 2001 14:48118 Acoustic Chaosdynamics is nonlinearity, acoustic chaos is closely relatedto nonlinear oscillations and waves in gases, liquids,and solids. It is the science of never-repeating soundwaves. This property it shares with noise, a term havingits origin in acoustics and formerly attributed to everysound signal with a broadband Fourier spectrum. ButFourier analysis is especially adapted to linear oscillatorysystems. The standard interpretation of the lines in aFourier spectrumis that each line corresponds to a (linear)mode of vibration and a degree of freedom of the system.However, as examples from chaos physics show, a broad-band spectrum can already be obtained with just three(nonlinear) degrees of freedom (that is, three dependentvariables). Chaos physics thus develops a totally newview of the noise problem. It is a deterministic view,but it is still an open question how far the new approachwill reach in explaining still unsolved noise problems(e.g., the 1/f -noise spectrum encountered so often). Thedetailed relationship between chaos and noise is still anarea of active research. An example, where the propertiesof acoustic noise could be related to chaotic dynamics, isgiven below for the case of acoustic cavitation noise.Acoustic chaos appears in an experiment when a liq-uid is irradiated with sound of high intensity. The liquidthen ruptures to form bubbles or cavities (almost emptybubbles). The phenomenon is known as acoustic cavita-tion and is accompanied by intense noise emissiontheacoustic cavitation noise. It has its origin in the bubbles setinto oscillation in the sound eld. Bubbles are nonlinearoscillators, and it can be shown both experimentally andtheoretically that they exhibit chaotic oscillations after aseries of period doublings. The acoustic emission fromthese bubbles is then a chaotic sound wave (i.e., irregularand never repeats). This is acoustic chaos.I. THE PROBLEM OF ACOUSTICCAVITATION NOISEThe projection of high-intensity sound into liquids hasbeen investigated since the application of sound to locateobjects under water became used. It was soon noticed thatat too high an intensity the liquid may rupture, giving riseto acoustic cavitation. This phenomenon is accompaniedby broadband noise emission, which is detrimental to theuseful operation of, for instance, a sonar device.The noise emission presents an interesting physicalproblem that may be formulated in the following way.A sound wave of a single frequency (a pure tone) is trans-formed into a broadband sound spectrum, consisting ofan (almost) innite number of neighboring frequencies.What is the physical mechanism that causes this transfor-mation? The question may even be shifted in its emphasisto ask what physical mechanisms are known to convert asingle frequency to a broadband spectrum? This could notbe answeredbefore chaos theorywas developed. However,although chaos theory is now well established, a physical(intuitive) understanding is still lacking.II. THE PERIOD-DOUBLINGNOISE SEQUENCETo investigate the sound emission from acoustic cavita-tion the experimental arrangement as depicted in Fig. 1is used. To irradiate the liquid (water) a piezoceramiccylinder (PZT-4) of 76-mm length, 76-mm inner diameter,and 5-mm wall thickness is used. When driven at its mainresonance, 23.56 kHz, a high-intensity acoustic eld isgenerated in the interior and cavitation is easily achieved.The noise is picked up by a broadband hydrophone anddigitized at rates up to 60 MHz after suitable lowpassltering (for correct analog-to-digital conversion for laterprocessing) and strong ltering of the driving frequency,which would otherwise dominate the noise output. Theexperiment is fully computer controlled. The amplitude ofthe driving sound eld can be made an arbitrary functionof time via a programmable synthesizer. In most cases,linear ramp functions are applied to study the buildup ofnoise when the driving pressure amplitude in the liquid isincreased.From the data stored in the memory of the transientrecorder, power spectra are calculated via the fast-Fourier-transform algorithm from usually 4096 samples out of the128 1024 samples stored. This yields about 1000 short-time spectra when the 4096 samples are shifted by 128samples from one spectrum to the next.Figure 2 shows four power spectra from one suchexperiment. Each diagram gives the excitation level atFIGURE 1 Experimental arrangement for measurements onacoustic cavitation noise (chaotic sound).P1: FYK Revised PagesEncyclopedia of Physical Science and Technology EN001-05 May 8, 2001 14:48Acoustic Chaos 119FIGURE 2 Power spectra of acoustic cavitation noise at different excitation levels (related to the pressure amplitudesof the driving sound wave). (From Lauterborn, W. (1986). Phys. Today 39, S-4.)the transducer in volts, the time since the experiment(irradiating the liquid with a linear ramp of increasingexcitation) has started in milliseconds, and the powerspectrum at this time. At the beginning of the experiment,at lowsoundintensity, onlythe drivingfrequency f0showsup. In the upper left diagram of Fig. 2 the third harmonic,3 f0, is present. When comparing both lines it shouldbe remembered that the driving frequency is stronglydamped by ltering. In the lower left-hand diagram, manymore lines are present. Of special interest is the spectralline at 12f0 (and their harmonics). Awell-known feature ofnonlinear systems is that they produce higher harmonics.Not yet widely known is that subharmonics can also beproduced by some nonlinear systems. These then seemto spontaneously divide the applied frequency f0 toyield, for example, exactly half that frequency (or exactlyone-third). This phenomenon has become known as aperiod-doubling (-tripling) bifurcation. A large class ofsystems has been found to show period doubling, amongthem driven nonlinear oscillators. A peculiar featureof the period-doubling bifurcation is that it occurs insequences; that is, when one period-doubling bifurcationhas occurred, it is likely that further period doubling willoccur upon altering a parameter of the system, and so on,often in an innite series. Acoustic cavitation has been oneof the rst experimental examples known to exhibit thisseries. In Fig. 2, the upper right-hand diagram shows thenoise spectrum after further period doubling to 14f0. Thedoubling sequence can be observed via 18f0 and 116f0 upto 132f0 (not shown here). It is obvious that the spectrumisrapidly lled with lines and gets more and more dense.The limit of the innite series yields an aperiodic motion,a densely packed power spectrum (not homogeneously),that is, broadband noise (but characteristically colored bylines). One such noise spectrum is shown in Fig. 2 (lowerright-hand diagram). Thus, at least one way of turningP1: FYK Revised PagesEncyclopedia of Physical Science and Technology EN001-05 May 8, 2001 14:48120 Acoustic Chaosa pure tone into broadband noise has been foundviasuccessive period doubling.This nding has a deeper implication. If a system be-comes aperiodic through the phenomenon of repeated pe-riod doubling, then this is a strong indication that the ir-regularity attained in this way is of simple deterministicorigin. This implies that acoustic cavitation noise is not abasically statistical phenomenon but a deterministic one. Italso implies that a description of the systemwith usual sta-tistical means may not be appropriate and that a successfuldescription by some deterministic theory may be feasible.III. A FRACTAL NOISE ATTRACTORIn Section II the sound signal has been treated by Fourieranalysis. Fourier analysis is a decomposition of a signalinto a sum of simple waves (normal modes) and is said togive the degrees of freedomof the describedsystem. Chaostheory shows that this interpretation must be abandoned.Broadband noise, for instance, is usually thought to be dueto a high (nearly innite) number of degrees of freedomthat superposed yield noise. Chaotic systems, however,have the ability to produce noise with only a few (nonlin-ear) degrees of freedom, that is, with only a fewdependentvariables. Also, it has been found that continuous systemswith only three dependent variables are capable of chaoticmotions and thus, producing noise. Chaos theory has de-veloped new methods to cope with this problem. One ofthese is phase-space analysis, which in conjunction withfractal dimension estimation is capable of yielding the in-trinsic degrees of freedom of the system. This method hasbeen applied to inspect acoustic cavitation noise. The an-swer it may give is the dimension of the dynamical systemproducing acoustic cavitation noise. See SERIES.The sampled noise data are rst used to construct anoise attractor in a suitable phase space. Then the (frac-tal) dimension of the attractor is determined. The pro-cedure to construct an attractor in a space of chosen di-mension n simply consists in combining n samples (notnecessarily consecutive ones) to an n-tuple, whose en-tries are interpreted as the coordinate values of a point inn-dimensional Euclidian space. An example of a noise at-tractor constructedinthis wayis giveninFig. 3. The attrac-tor has been obtained froma time series of pressure values{ p(kts); t =1, . . . , 4096; ts =1 sec} taken at a samplingfrequency of fs =1/ts =1 MHz by forming the three-tuples [ p(kts), p(kts +T), p(kts +2T)], k =1, . . . , 4086,with T = 5 sec. The frequency of the driving sound eldhas been 23.56 kHz. The attractor in Fig. 3 is shown fromdifferent views to demonstrate its nearly at structure. It ismost remarkable that not an unstructured cluster of pointsis obtainedas is expectedfor noise, but a quite well-denedFIGURE3 Strange attractor of acoustic cavitation noise obtainedby phasespace analysis of experimental data (a time series ofpressure values sampled at 1 MHz). The attractor is rotated tovisualize its three-dimensional structure. (Courtesy of J. Holzfuss.From Lauterborn, W. (1986). In Frontiers in Physical Acoustics(D. Sette, ed.), pp. 124144, North Holland, Amsterdam.)object. This suggests that the dynamical system produc-ing the noise has only a fewnonlinear degrees of freedom.The at appearance of the attractor in a three-dimensionalphase space (Fig. 3) suggests that only three essential de-grees are needed for the system. This is conrmed by afractal dimension analysis, which yields a dimension ofd =2.5 for this attractor. Unfortunately, a method has notyet been conceived of how to construct the equations ofmotion from the data.IV. LYAPUNOV ANALYSISChaotic systems exhibit what is called sensitive depen-dence on initial conditions. This expression has been intro-duced to denote the property of a chaotic systemthat smalldifferences in the initial conditions, however small, arepersistently magnied because of the dynamics of the sys-tem. This property is captured mathematically by the no-tion of Lyapunov exponents and Lyapunov spectra. Theirdenition can be illustrated by the deformation of a smallP1: FYK Revised PagesEncyclopedia of Physical Science and Technology EN001-05 May 8, 2001 14:48Acoustic Chaos 121FIGURE 4 Idea for dening Lyapunov exponents. A small spherein phase space is deformed to an ellipsoid, indicating expansionor contraction of neighboring trajectories.sphere of initial conditions along a ducial trajectory (seeFig. 4). The expansion or contraction is used to dene theLyapunov exponents i, i =1, 2, . . . , m, where m is thedimension of the phase space of the system. When, on theaverage, for example, r1(t ) is larger than r1(0), then 1 >0and there is a persistent magnication in the system. Theset {i, i =1, . . . , m}, whereby the i usually are ordered1 2 m, is called the Lyapunov spectrum.FIGURE 5 Acoustic cavitation bubble eld in water inside a cylin-drical piezoelectric transducer of about 7 cm in diameter. Twoplanes in depth are shown about 5 mmapart. The pictures are ob-tained by photographs from the reconstructed three-dimensionalimage of a hologram taken with a ruby laser.In dissipative systems, the nal motion takes place onattractors. Besides the fractal dimension, as discussed inthe previous section, the Lyapunov spectrum may serve tocharacterize these attractors. When at least one Lyapunovexponent is greater than zero, the attractor is said to bechaotic. Progress in the eld of nonlinear dynamics hasmade possible the calculation of the Lyapunov spectrumfrom a time series. It could be shown that acoustic cavita-tion in the region of broadband noise emission is charac-terized by one positive Lyapunov exponent.V. PERIOD-DOUBLING BUBBLEOSCILLATIONSThus far, only the acoustic signal has been investigated.An optic inspection of the liquid inside the piezoelectriccylinder (see Fig. 1) reveals that a highly structured cloudof bubbles or cavities is present (Fig. 5) oscillating andmoving in the sound eld. It is obviously these bubblesthat produce the noise. If this is the case, the bubbles mustFIGURE 6 Reconstructed images from (a) a holographic seriestaken at 23.100 holograms per second of bubbles inside a piezo-electric cylinder driven at 23.100 Hz and (b) the correspondingpower spectrum of the noise emitted. Two period-doublings havetaken place.P1: FYK Revised PagesEncyclopedia of Physical Science and Technology EN001-05 May 8, 2001 14:48122 Acoustic ChaosFIGURE 7 Period-doubling route to chaos for a driven bubble oscillator. Left column: radius-time solution curves;middle left column: trajectories in phase space; middle right column: Poincar e section plots: right column: powerspectra. Rn is the radius of the bubble at rest, Ps and v are the pressure amplitude and frequency of the driving soundeld, respectively. (From Lauterborn, W., and Parlitz, U. (1988). J. Acoust. Soc. Am. 84, 1975.)P1: FYK Revised PagesEncyclopedia of Physical Science and Technology EN001-05 May 8, 2001 14:48Acoustic Chaos 123FIGURE 7 (Continued)P1: FYK Revised PagesEncyclopedia of Physical Science and Technology EN001-05 May 8, 2001 14:48124 Acoustic Chaosmove chaotically and should show the period-doublingsequence encountered in the noise output. This has beenconrmed by holographic investigations where once perperiod of the driving sound eld a hologram of the bub-ble eld has been taken. Holograms have been taken be-cause the bubbles move in three dimensions, and it isdifcult to photograph them at high resolution when anextended depth of view is needed. In one experiment thedriving frequency was 23,100 Hz, which means 23,100holograms per second have been taken. The total num-ber of holograms, however, was limited to a few hundred.Figure 6a gives an example of a series of photographstaken from a holographic series. In this case, two period-doubling bifurcations have already taken place since theoscillations only repeat after four cycles of the drivingsound wave. The rst period doubling is strongly visible;the second one can only be seen by careful inspection.Figure 6b gives the noise power spectrum taken simulta-neously with the holograms. The acoustic measurementsshow both period doublings more clearly than the opticalmeasurement (documented in Fig. 6a) as the 14f0 ( f0 =23.1 kHz) spectral line is strongly present together with itsharmonics.VI. THEORY OF DRIVEN BUBBLESA theory has not yet been developed that can account forthe dynamics of a bubble eld as shown in Fig. 5. The mostadvanced theory is only able to describe the motion of asingle spherical bubble in a sound eld. Even with suitableneglections the model is a highly nonlinear ordinarydifferential equation of second order for the radius R ofthe bubble as a function of time. With a sinusoidal drivingterm (sound wave) the phase space is three dimensional,just sufcient for a dynamical system to show irregular(chaotic) motion. The model is an example of a drivennonlinear oscillator for which chaotic solutions in certainparameter regions are by now standard. However, perioddoubling and irregular motion were found in the late 1960sin numerical calculations when chaos theory was not yetavailable and thus the interpretation of the results difcult.The surprising fact is that already this simple model of apurely spherically oscillating bubble set into oscillationby a sound wave yields successive period doubling upto chaotic oscillations. Figure 7 demonstrates the period-doubling route to chaos in four ways. The leftmost columngives the radius of the bubble in the sound eld as a func-tion of time, where the dot on the curve indicates the lapseof a full period of the driving sound eld. The next columnshows the corresponding trajectories in the plane spannedby the radius of the bubble and its velocity. The dots againmark the lapse of a full period of the driving sound eld.The third column shows so-called Poincar e section plots.Here, only the dots after the lapse of one full period ofthe driving sound eld are plotted in the radiusvelocityplane of the bubble motion. Period doubling is seen mosteasily here and also the evolution of a strange (or chaotic)attractor. The rightmost column gives the power spectraof the radial bubble motion. The lling of the spectrumwith successive lines in between the old lines is evident,as is the ultimate lling when the chaotic motion isreached.A compact way to show the period-doubling route tochaos is by plotting the radius of the bubble as a func-tion of a parameter of the system that can be varied, e.g.,the frequency of the driving sound eld. Figure 8a givesan example for a bubble of radius at rest of Rn =10 m,driven by a sound eld of frequency between 390 kHzand 510 kHz at a pressure amplitude of Ps =290 kPa.The period-doubling cascade to chaos is clearly visible.In the chaotic region, windows of periodicity showFIGURE 8 (a) A period-doubling cascade as seen in the bifurca-tion diagram. (b) The corresponding largest Lyapunov exponentmax. (c) The winding number w. (From Parlitz, U. et al. (1990).J. Acoust. Soc. Am. 88, 1061.)P1: FYK Revised PagesEncyclopedia of Physical Science and Technology EN001-05 May 8, 2001 14:48Acoustic Chaos 125up as regularly experienced with other chaotic systems.In Fig. 8b the largest Lyapunov exponent max is plot-ted. It is seen that max >0 when the chaotic region isreached. Figure 8c gives a further characterization of thesystem by the winding number w. The winding numberdescribes the winding of a neighboring trajectory aroundthe given one per period of the bubble oscillation. It canbe seen that this quantity changes quite regularly in theperiod-doubling sequence, and rules can be given for thischange.The driven bubble system shows resonances at vari-ous frequencies that can be labeled by the ratio of thelinear resonance frequency of the bubble to the drivingfrequency of the sound wave. Figure 9 gives an exampleof the complicated response characteristic of a driven bub-ble. At somewhat higher driving than given in the gurethe oscillations start to become chaotic. A chaotic bubbleattractor is shown in Fig. 10. To better reveal its structure,it is not the total trajectory that is plotted but only thepoints in the velocityradius plane of the bubble wall at axed phase of the driving. These points hop around on theattractor in an irregular fashion. These chaotic bubble os-cillations must be considered as the source of the chaoticsound output observed in acoustic cavitation.FIGURE 9 Frequency response curves (resonance curves) for a bubble in water with a radius at rest of Rn =10 mfor different sound pressure amplitudes pA of 0.4, 0.5, 0.6, 0.7, and 0.8 bar. (From Lauterborn, W. (1976). J. Acoust.Soc. Am. 59, 283.)VII. OTHER SYSTEMSAre there other systems in acoustics with chaotic dynam-ics? The answer is surely yes, although the subtleties ofchaotic dynamics make it difcult to easily locate them.When looking for chaotic acoustic systems, the ques-tion arises as to what ingredients an oscillatory system, asan acoustic one, must possess to be susceptible to chaos.The full answer is not yet known, but some understandingis emerging. A necessary, but unfortunately not sufcient,ingredient is nonlinearity. Next, period doubling is knownto be a precursor of chaos. It is a peculiar fact that, whenone period doubling has occurred, another one is likely toappear, and indeed a whole series with slight alterations ofparameters. Further, the appearance of oscillations whena parameter is altered points to an intrinsic instability of asystem and thus to the possibility of becoming a chaoticone. After all, two distinct classes can be formulated: (1)periodically driven passive nonlinear systems (oscillators)and (2) self-excited systems (oscillators). Passive meansthat in the absence of any external driving the systemstays at rest as, for instance, a pendulum does. But apendulum has the potential to oscillate chaotically whenbeing driven periodically, for instance by a sinusoidallyP1: FYK Revised PagesEncyclopedia of Physical Science and Technology EN001-05 May 8, 2001 14:48126 Acoustic ChaosFIGURE 10 A numerically calculated strange bubble attractor(Ps =300 kPa, v =600 kHz). (Courtesy of U. Parlitz.)varying torque. This is easily shown experimentally bythe repeated period doubling that soon appears at higherperiodic driving. Self-excited systems develop sustainedoscillations from seemingly constant exterior conditions.One example is the Rayleigh-B enard convection, where aliquid layer is heated from below in a gravitational eld.The system goes chaotic at a high enough temperaturedifference between the bottom and surface of the liquidlayer. Self-excited systems may also be driven, givingan important subclass of this type. The simplest modelin this class is the driven van der Pol oscillator. A realphysical system of this category is the weather (theatmosphere). It is periodically driven by solar radiationwith the low period of 24 hr, and it is a self-excitedsystem, as already constant heating by the sun may lead toRayleigh-B enard convection as observed on a faster timescale.The rst reported period-doubled oscillation from a pe-riodically driven passive system dates back to Faraday in1831. Startingwiththe investigationof sound-emitting, vi-brating surfaces with the help of Chladni gures, Faradayused water instead of sand, resulting in vibrating a layerof liquid vertically. He was very astonished about the re-sult: regular spatial patterns of a different kinds appearedand, above all, these patterns were oscillating at half thefrequency of the vertical motion of the plate. Photographywas not yet invented to catch the motion, but Faraday maywell have seen chaotic motion without knowing it. It is in-teresting to note that there is a connection to the oscillationof bubbles as considered before. Besides purely sphericaloscillations, bubbles are susceptible to surface oscillationsas are drops of liquid. The Faraday case of a vibrating atsurface of a liquid may be considered as the limiting caseof either a bubble of larger and larger size or a drop oflarger and larger size, when the surface is bent around upor down. Today, the Faraday patterns and Faraday oscil-lations can be observed better, albeit still with difcultiesas it is a three-dimensional (space), nonlinear, dynamical(time) system; that is, it requires three space coordinatesand one time coordinate to be followed. This is at theborder of present-day technology both numerically andexperimentally. The latest measurements have singled outmode competition as the mechanism underlying the com-plex dynamics. Figure 11 gives two examples of oscilla-tory patterns: a periodic hexagonal structure (Fig. 11a) andabFIGURE 11 Two patterns appearing on the surface of a liq-uid layer vibrated vertically in a cylindrical container: (a) regularhexagonal pattern at low amplitude, and (b) pattern when ap-proaching chaotic vibration. (Courtesy of Ch. Merkwirth.)P1: FYK Revised PagesEncyclopedia of Physical Science and Technology EN001-05 May 8, 2001 14:48Acoustic Chaos 127its dissolution on the way to chaotic motion (Fig. 11b) atthe higher vertical driving oscillation amplitude of a thinliquid layer.The other class of self-excited systems in acousticsis quite large. It comprises (1) musical instruments, (2)thermoacoustic oscillators as used today for cooling withsound waves, and (3) speech production via the vocalfolds. Period doubling could be observed in most of thesesystems; however, very fewinvestigations have been doneso far concerning their chaotic properties.VIII. PHILOSOPHICAL IMPLICATIONSThe results of chaos physics have shed new light on therelation between determinism and predictability and onhow seemingly random (irregular) motion is produced. Ithas been found that deterministic laws do not imply pre-dictability. The reason is that there are deterministic lawswhich persistently show a sensitive dependence on initialconditions. This means that in a nite, mostly short timeany signicant digit of a measurement has been lost, andanother measurement after that time yields a value thatappears to come from a random process. Chaos physicshas thus shown a way of howrandom(seemingly random,one must say) motion is produced out of determinism andhas developed convincing methods (some of them exem-plied in the preceding sections on acoustic chaos) to clas-sify such motion. Random motion is thereby replaced bychaotic motion. Chaos physics suggests that one shouldnot resort too quickly to statistical methods when facedwith irregular data but instead should try a deterministicapproach. Thus, chaos physics has sharpened our viewconsiderably on how nature operates.But, as always in physics, when progress has been madeon one problemother problems pile up. Quantummechan-ics is thought to be the correct theory to describe nature.It contains true randomness. But, what then about therelationship between classical deterministic physics andquantum mechanics? Chaos physics has revived interestinthese questions andformulatednewspecic ones, for in-stance, on how chaotic motion crosses the border to quan-tum mechanics. What is the quantum mechanical equiva-lent to sensitive dependence on initial conditions?The exploration of chaos physics, including its relationto quantum mechanics, is therefore thought to be one ofthe big scientic enterprises of the newcentury. It is hopedthat acoustic chaos will accompany this enterprise furtheras an experimental testing ground.SEE ALSO THE FOLLOWING ARTICLESACOUSTICAL MEASUREMENT CHAOS FOURIER SERIES FRACTALS QUANTUM MECHANICSBIBLIOGRAPHYLauterborn, W., and Holzfuss, J. (1991). Acoustic chaos. Int. J. Bifur-cation and Chaos 1, 1326.Lauterborn, W., and Parlitz, U. (1988). Methods of chaos physics andtheir application to acoustics. J. Acoust. Soc. Am. 84, 19751993.Parlitz, U., Englisch, V., Scheffezyk, C., and Lauterborn, W. (1990).Bifurcation structure of bubble oscillators. J. Acoust. Soc. Am. 88,10611077.Ruelle, D. (1991). Chance andChaos, PrincetonUniv. Press, Princeton,NJ.Schuster, H. G. (1995). Deterministic Chaos: An Introduction, Wiley-VCH, Weinheim.P1: FVZ Revised Pages Qu: 00, 00, 00, 00Encyclopedia of Physical Science and Technology EN001-08 May 25, 2001 16:4Acoustical MeasurementAllan J. ZuckerwarNASA Langley Research CenterI. Instruments for Measuringthe Properties of SoundII. Instruments for Processing Acoustical DataIII. Examples of Acoustical MeasurementsGLOSSARYAnechoic Having no reections or echoes.Audio Pertaining to sound within the frequency range ofhuman hearing, nominally 20 Hz to 20 kHz.Coupler Small leak-tight enclosure into which acousticdevices are inserted for the purpose of calibration, mea-surement, or testing.Diffuse eld Region of uniform acoustic energy density.Free eld Region where sound propagation is unaffectedby boundaries.Harmonic Pertaining to a pure tone, that is, a sinusoidalwave at a single frequency: an integral multiple of afundamental tone.Infrasonic Pertaining to sound at frequencies below thelimit of human hearing, nominally 20 Hz.Reverberant Highly reecting.Ultrasonic Pertaining to sound at frequencies above thelimit of human hearing, nominally 20 kHz.A SOUND WAVE propagating through a medium pro-duces deviations in pressure and density about their meanor static values. The deviation in pressure is called theacoustic or sound pressure, which has standard interna-tional (SI) units of pascal (Pa) or newton per square meter(N/m2). Because of the vast range of amplitude coveredin acoustic measurements, the sound pressure is conve-niently represented on a logarithmic scale as the soundpressure level (SPL). The SPL unit is the decibel (dB),dened asSPL(dB) = 20 log( p/p0)in which p is the root mean square (rms) sound pressureamplitude and p0 the reference pressure of 20 106Pa.The equivalent SPLs of some common units are thefollowing:pascal (Pa) 93.98 dB psi (lb/in.2) 170.75 dBatmosphere (atm) 194.09 torr (mm Hg) 136.48bar 193.98 dyne/cm273.98The levels of some familiar sound sources and environ-ments are listed in Table I.The displacement per unit time of a uid particle due tothe sound wave, superimposed on that due to its thermalmotion, is called the acoustic particle velocity, in units ofmeters per second. Determination of the sound pressureand acoustic particle velocity at every point completelyspecies an acoustic eld, just as the voltages and currentscompletely specify an electrical network. Thus, acousticalinstrumentation serves to measure one of these quanti-ties or both. Since in most cases the relationship between 91P1: FVZ Revised PagesEncyclopedia of Physical Science and Technology EN001-08 April 20, 2001 12:4592 Acoustical MeasurementTABLE I Representative Sound Pressure Levels ofFamiliar Sound Sources and EnvironmentsSource or environment Level (dB)Concentrated sources: re 1 mFour-jet airliner 155Pipe organ, loudest 125Auto horn, loud 115Power lawnmower 100Conversation 60Whisper 20Diffuse environmentsConcert hall, loud orchestra 105Subway interior 95Street corner, average trafc 80Business ofce 60Library 40Bedroom at night 30Threshold levelsOf pain 130Of hearing impairment, continous exposure 90Of hearing 0Of detection, good microphone 2sound pressure and particle velocity is known, it is suf-cient to measure only one quantity, usually the soundpressure. The scope of this article is to describe instru-mentation for measuring the properties of sound in uids,primarily in air and water, and in the audio (20 Hz20 kHz)and infrasonic (3, two correspond to the period-2 points at x(1)and x(2). The remaining 12 period-4 points can form threedifferent period-4 cycles that appear for different values ofa. Figure 7 shows a graph of F(4)(xn) for a =3.2, wherethe period-2 cycle is still stable, and for a =3.5, wherethe unstable period-2 cycle has bifurcated into a period-4 cycle. (The other two period-4 cycles are only brieystable for other values of a >a)We could repeat the same arguments to describe the ori-gin of period 8; however, now the graph of the return mapof the corresponding polynomial of degree 32 would be-gin to tax the abilities of our graphics display terminal aswell as our eyes. Fortunately, the slaving of the stabilityproperties of each periodic point via the chain-rule argu-ment (described previously for the period-2 cycle) meansthat we only have to focus on the behavior of the succes-sive iterates of the map in the vicinity of the periodic pointclosest to x = 0.5. In fact, a close examination of Figs. 4,5, and 7 reveals that the bifurcation process for each F(n) issimply a miniature replica of the original period-doublingbifurcation from the period-1 cycle to the period-2 cy-cle. In each case, the return map is locally described bya parabolic curve (although it is not exactly a parabolabeyond the rst iteration and the curve is ipped over forevery other F(N).Because each successive period-doubling bifurcationis described by the xed points of a return mapxn+N= F(N)(Xn) with ever greater oscillations on the unitinterval, the amount the parameter a must increase beforethe next bifurcation decreases rapidly, as shown in the bi-furcation diagram in Fig. 6. The differences in the changesin the control parameter for each succeeding bifurcation,an+1an, decreases at a geometric rate that is found torapidly converge to a value of: =anan1an+1an= 4.6692016 . . . (9)Inaddition, the maximumseparationof the stable daughtercycles of each pitchfork bifurcation also decreases rapidly,as shown in Fig. 6, by a geometric factor that rapidlyconverges to: = 2.502907875 . . . (10)2. UniversalityThe fact that each successive period doubling is controlledby the behavior of the iterates of the map, F(N)(x), nearx =0.5, lies at the root of a very signicant propertyof nonlinear dynamical systems that exhibit sequencesof period-doubling bifurcations called universality. In theprocess of developing a quantitative description of perioddoubling in the logistic map, Feigenbaum discovered thatthe precise functional formof the map did not seemto mat-ter. For example, he found that a map on the unit intervaldescribed by F(x) =a sin x gave a similar sequence ofperiod-doubling bifurcations. Although the values of thecontrol parameter a at which each period-doubling bifur-cation occurs are different, he found that both the ratiosof the changes in the control parameter and the separa-tions of the stable daughter cycles decreased at the samegeometrical rates and as the logistic map.P1: GPJ 2nd Revised PagesEncyclopedia of Physical Science and Technology EN002E-94 May 19, 2001 20:28648 ChaosFIGURE 7 The appearance of the period-4 cycle as a is increased from 3.2 to 3.5 is illustrated by these graphs ofthe return maps for the fourth iterate of the logistic map, F(4). For a=3.2, there are only four period-4 xed pointsthat correspond to the two unstable period-1 points and the two stable period-2 points. However, when a is increasedto 3.5, the same process that led to the birth of the period-2 xed points is repeated again in miniature. Moreover,the similarity of the portion of the map near xn=0.5 to the original map indicates how this same bifurcation processoccurs again as a is increased.This observation ultimately led to a rigorous proof,using the mathematical methods of the renormalizationgroup borrowed from the theory of critical phenomena,that these geometrical ratios were universal numbers thatwould apply to the quantitative description of any period-doubling sequence generated by nonlinear maps with asingle quadratic extremum. The logistic map and the sinemap are just two examples of this large universality class.The great signicance of this result is that the global detailsof the dynamical system do not matter. A thorough under-standingof the simple logistic mapis sufcient for describ-ing both qualitatively and, to a large extent, quantitativelythe period-doubling route to chaos in a wide variety ofnonlinear dynamical systems. In fact, we will see that thisuniversality class extends beyond one-dimensional mapsto nonlinear dynamical systems described by more real-istic physical models corresponding to two-dimensionalmaps, systems of ordinary differential equations, and evenpartial differential equations.3. ChaosOf course, these stable periodic cycles, described byFeigenbaums universal theory, are not chaotic. Even thecycle with an innite period at the period-doubling accu-mulation point a has a zero average Lyapunov exponent.However, for many values of a above a, the time se-quences generated by the logistic map have a positive aver-age Lyapunov exponent and therefore satisfy the denitionof chaos. Figure 8 plots the average Lyapunov exponentcomputed numerically using Eq. (3) for the same rangeof values of a, as displayed in the bifurcation diagramin Fig. 6.FIGURE 8 The values of the average Lyapunov exponent, com-puted numerically using Eq. (3), are displayed for the same valuesof a shown in Fig. 6. Positive values of correspond to chaotic dy-namics, while negative values represent regular, periodic motion.P1: GPJ 2nd Revised PagesEncyclopedia of Physical Science and Technology EN002E-94 May 19, 2001 20:28Chaos 649Wherever the trajectory appears to wander chaoticallyover continuous intervals, the average Lyapunov expo-nent is positive. However, embedded in the chaos fora kc, MacKay et al. (1987)have shown that this last conning curve breaks up into aso-called cantorus, which is a curve lled with gaps resem-bling a Cantor set. These gaps allowchaotic trajectories toleak through so that single orbits can wander throughoutlarge regions of the phase space, as shown in Fig. 14 fork =2.3. Chaotic DiffusionBecause of the intrinsic nonlinearity of Eq. (17b), the re-striction of the map to the 2 square was only a graphicalconvenience that exploited the natural periodicities of themap. However, in reality, both the angle variable and theP1: GPJ 2nd Revised PagesEncyclopedia of Physical Science and Technology EN002E-94 May 19, 2001 20:28Chaos 659angular velocity of a real physical system described bythe standard map can take on all real values. In particular,when the golden mean KAM torus is destroyed, the angu-lar velocity associated with the chaotic orbits can wanderto arbitrarily large positive and negative values.Because the chaotic evolution of both the angle andangular velocity appears to execute a random walk in thephase space, it is natural to attempt to describe the dynam-ics using a statistical description despite the fact that theunderlying dynamical equations are fully deterministic.In fact, when k kc, careful numerical studies show thatthe evolution of an ensemble of initial conditions can bewell described by a diffusion equation. Consequently, thissimple deterministic dynamical system provides an inter-esting model for studying the problem of the microscopicfoundations of statistical mechanics, which is concernedwith the question of how the reversible and deterministicequations of classical mechanics can give rise to the ir-reversible and statistical equations of classical statisticalmechanics and thermodynamics.C. The H enonHeiles ModelOur third example of a Hamiltonian system that exhibitsa transition from regular behavior to chaos is describedby a system of four coupled, nonlinear differential equa-tions. It was originally introduced by Michel H enon andCarl Heiles in 1964 as a model of the motion of a starin a nonaxisymmetric, two-dimensional potential corre-sponding to the mean gravitational eld in a galaxy. Theequations of motion for the two components of the posi-tion and momentum,dx/dt = px (18a)dy/dt = py (18b)dpx/dt = x 2xy (18a)dpy/dt = y + y2 x2(18b)are generated by the HamiltonianH(x, y, px, py) =p2x2 +p2y2 +12(x2+ y2) + x2y 13y3(19)where the mass is taken to be unity. Equation 19 cor-responds to the Hamiltonian of two uncoupled harmonicoscillators H0=( p2x/2) +( p2y/2) +12(x2+y2) (consistingof the sum of the kinetic and a quadratic potential energy)plus a cubic perturbation H1=x2y 13y3, which providesa nonlinear coupling for the two linear oscillators.Since the Hamiltonian is independent of time, it is aconstant of motion that corresponds to the total energy ofthe system E = H(x, y, px, py). When E is small, boththe values of the momenta ( px, py) and the positions (x, y)must remainsmall. Therefore, inthe limit E 1, the cubicperturbation can be neglected and the motion will be ap-proximately described by the equations of motion for theunperturbed Hamiltonian, which are easily integrated an-alytically. Moreover, the application of the KAM theoremto this problem guarantees that as long as E is sufcientlysmall the motion will remain regular. However, as E isincreased, the solutions of the equations of motion, likethe orbits generated by the standard map, will become in-creasingly complicated. First, nonlinear resonances willappear from the coupling of the motions in the x and the ydirections. As the energy increases, the effect of the non-linear coupling grows, the sizes of the resonances grow,and, when they begin to overlap, the orbits begin to exhibitchaotic motion.1. Poincar e SectionsAlthough Eq. (18) can be easily integrated numericallyfor any value of E, it is difcult to graphically displaythe transition from regular behavior to chaos because theresulting trajectories move in a four-dimensional phasespace spanned by x, y, px, and py. Although we can usethe constancy of the energy to reduce the dimension ofthe accessible phase space to three, the graphs of the re-sulting three-dimensional trajectories would be even lessrevealing than the three-dimensional graphs of the Lorenzattractor since there is no attractor to consolidate the dy-namics. However, we can simplify the display of the tra-jectories by exploiting the same device used to relate theH enon map to the Lorenz model. If we plot the value of pxversus x every time the orbit passes through y =0, thenwe can construct a Poincar e section of the trajectory thatprovides a very clear display of the transition fromregularbehavior to chaos.Figure 15 displays these Poincare sections for a num-ber of different initial conditions corresponding to threedifferent energies, E = 112, 18, and 16. For very small E,most of the trajectories lie on an ellipsoid in four-dimensional phase space, so the intersection of the orbitswith the pxx plane traces out simple ellipses centeredat (x, px) =(0, 0). For E = 112, these ellipses are distortedand island chains associated with the nonlinear resonancesbetween the coupled motions appear; however, most or-bits appear to remain on smooth, regular curves. Finally,as E is increased to 18 and 16, the Poincar e sections reveala transition from ordered motion to chaos, similar to thatobserved in the standard map.In particular, when E =16, a single orbit appears to uni-formly cover most of the accessible phase space dened bythe surface of constant energy in the full four-dimensionalP1: GPJ 2nd Revised PagesEncyclopedia of Physical Science and Technology EN002E-94 May 19, 2001 20:28660 ChaosFIGURE 15 Poincar e sections for a number of different orbitsgenerated by the H enonHeiles equations are plotted for threedifferent values of the energy E. These gure were created byplotting the position of the orbit in the xpx plane each time thesolutions of the H enonHeiles equations passed through y =0with positive, py. For E = 112, the effect of the perturbation is smalland the orbits resemble the smooth but distorted curves observedin the standard map for small k, with resonance islands associ-ated with coupling of the x and y oscillations. However, as theenergy increases and the effects of the nonlinearities becomemore pronounced, large regions of chaotic dynamics become vis-ible and grow until most of the accessible phase space appearsto be chaotic for E = 16. (These gures can be compared with theless symmetrical Poincar e sections plotted in the ypy plane thatusually appear in the literature).phase space. Although the dynamics of individual trajec-tories is very complicated in this case, the average prop-erties of an ensemble of trajectories generated by this de-terministic but chaotic dynamical system should be welldescribed using the standard methods of statistical me-chanics. For example, we may not be able to predict whena star will move chaotically into a particular region ofthe galaxy, but the average time that the star spends inthat region can be computed by simply measuring the rel-ative volume of the corresponding region of the phasespace.D. ApplicationsThe earliest applications of the modern ideas of nonlineardynamics and chaos to Hamiltonian systems were in theeld of accelerator design starting in the late 1950s. Inorder to maintain a beam of charged particles in an ac-celerator or storage ring, it is important to understand thedynamics of the corresponding Hamiltonian equations ofmotion for very long times (in some cases, for more than108revolutions). For example, the nonlinear resonancesassociated with the coupling of the radial and vertical os-cillations of the beam can be described by models similarto the H enonHeiles equations, and the coupling to eldoscillations around the accelerator can be approximatedby models related to the standard map. In both cases, ifthe nonlinear coupling or perturbations are too large, thechaotic orbits can cause the beam to defocus and run intothe wall.Similar problems arise in the description of magneti-cally conned electrons and ions in plasma fusion devices.The densities of these thermonuclear plasmas are suf-ciently low that the individual particle motions are effec-tively collisionless on the time scales of the experiments,so dissipation can be neglected. Again, the nonlinear equa-tions describing the motion of the plasma particles can ex-hibit chaotic behavior that allows the particles to escapefrom the conning elds. For example, electrons circu-lating along the guiding magnetic eld lines in a toroidalconnement device called a TOKAMAK will feel a peri-odic perturbation because of slight variations in magneticelds, which can be described by a model similar to thestandard map. When this perturbation is sufciently large,electron orbits can become chaotic, which leads to ananomalous loss of plasma connement that poses a seriousimpediment to the successful design of a fusion reactor.The fact that a high-temperature plasma is effectivelycollisionless also raises another problem in which chaosactually plays a benecial role and which goes right to theroot of a fundamental problem of the microscopic foun-dations of statistical mechanics. The problem is how doyou heat a collisionless plasma? How do you make anirreversible transfer of energy from an external source,P1: GPJ 2nd Revised PagesEncyclopedia of Physical Science and Technology EN002E-94 May 19, 2001 20:28Chaos 661such as the injection of a high-energy particle beam orhigh-intensity electromagnetic radiation, to a reversible,Hamiltonian system? The answer is chaos. For example,the application of intense radio-frequency radiation in-duces a strong periodic perturbation on the natural oscil-latory motion of the plasma particles. Then, if the pertur-bation is strong enough, the particle motion will becomechaotic. Although the motion remains deterministic andreversible, the chaotic trajectories associated with the en-semble of particles can wander over a large region of thephase space, in particular to higher and lower velocities.Since the temperature is a measure of the range of possiblevelocities, this process causes the plasma temperature toincrease.Progress in the understanding of chaotic behavior hasalso caused a revival of interest in a number of problemsrelated to celestial mechanics. In addition to H enon andHeiles work on stellar dynamics described previously,Jack Wisdom at MIT has recently solved several old puz-zles relating to the origin of meteorites and the presenceof gaps in the asteroid belt by invoking chaos. Each timean asteroid that initially lies in an orbit between Mars andJupiter passes the massive planet Jupiter, it feels a gravi-tational tug. This periodic perturbation on small orbitingasteroids results in a strong resonant interaction when thetwo frequencies are related by low-order rational numbers.As in the standard map and the H enonHeiles model, ifthis resonant interaction is sufciently strong, the aster-oid motion can become chaotic. The ideal Kepler ellipsesbegin to precess and elongate until their orbits cross theorbit of Earth. Then, we see them as meteors and mete-orites, and the depletion of the asteroid belts leaves gapsthat correspond to the observations.The study of chaotic behavior in Hamiltonian systemshas also found many recent applications in physical chem-istry. Many models similar to the H enonHeiles modelhave been proposed for the description of the interactionof coupled nonlinear oscillators that correspond to atomsin a molecule. The interesting questions here relate to howenergy is transferred from one part of the molecule to theother. If the classical dynamics of the interacting atoms isregular, then the transfer of energy is impeded by KAMsurfaces, such as those in Figs. 14 and 15. However, ifthe classical dynamics is fully chaotic, then the moleculemay exhibit equipartition of energy as predicted by statis-tical theories. Even more interesting is the common casewhere some regions of the phase space are chaotic andsome are regular. Since most realistic, classical models ofmolecules involve more than two degrees of freedom, theunraveling of this complex phase-space structure in six ormore dimensions remains a challenging problem.Finally, most recently there has been considerable in-terest in the classical Hamiltonian dynamics of electronsin highly excited atoms in the presence of strong magneticelds and intense electromagnetic radiation. The studiesof the regular and chaotic dynamics of these strongly per-turbed systems have provided a new understanding of theatomic physics in a realm in which conventional meth-ods of quantum perturbation theory fail. However, thesestudies of chaos in microscopic systems, like those ofmolecules, have also raised profound, new questions re-lating to whether the effects of classical chaos can survivein the quantum world. These issues will be discussed inSection V.V. QUANTUM CHAOSThe discovery that simple nonlinear models of classicaldynamical systems can exhibit behavior that is indistin-guishable from a random process has naturally raised thequestion of whether this behavior persists in the quantumrealm where the classical nonlinear equations of motionare replaced by the linear Schrodinger equation. This iscurrently a lively area of research. Although there is gen-eral consensus on the key problems, the solutions remaina subject of controversy. In contrast to the subject of clas-sical chaos, there is not even agreement on the denitionof quantum chaos. There is only a list of possible symp-toms for this poorly characterized disease. In this section,we will briey discuss the problem of quantum chaos anddescribe some of the characteristic features of quantumsystems that correspond to classically chaotic Hamilto-nian systems. Some of these features will be illustratedusing a simple model that corresponds to the quantizeddescription of the kicked rotor described in Section IV.B.Then, we will conclude with a description of the compari-son of classical and quantumtheory with real experimentson highly excited atoms in strong elds.A. The Problem of Quantum ChaosGuided by Bohrs correspondence principle, it might benatural to conclude that quantum mechanics should agreewith the predictions of classical chaos for macroscopicsystems. In addition, because chaos has played a funda-mental role in improving our understanding of the micro-scopic foundations of classical statistical mechanics, onewould hope that it would play a similar role in shoring upthe foundations of quantum statistical mechanics. Unfor-tunately, quantum mechanics appears to be incapable ofexhibiting the strong local instability that denes classicalchaos as a mixing systemwith positive KolmogorofSinaientropy.One way of seeing this difculty is to note that theSchrodinger equation is a linear equation for the waveP1: GPJ 2nd Revised PagesEncyclopedia of Physical Science and Technology EN002E-94 May 19, 2001 20:28662 Chaosfunction, and neither the wave function nor any observ-able quantities (determined by taking expectation valuesof self-adjoint operators) can exhibit extreme sensitivityto initial conditions. In fact, if the Hamiltonian system isbounded (like the H enonHeiles Model), then the quan-tum mechanical energy spectrum is discrete and the timeevolution of all quantum mechanical quantities is doomedto quasiperiodic behavior, such as that Eq. (1).Although the question of the existence of quantumchaos remains a controversial topic, nearly everyoneagrees that the most important questions relate to howquantum systems behave when the corresponding clas-sical Hamiltonian systems exhibit chaotic behavior. Forexample, how does the wave function behave for stronglyperturbed oscillators, such as those modeled by the clas-sical standard map, and what are the characteristics of theenergy levels for a system of strongly coupled oscillators,such as those described by the H enonHeiles model?B. Symptoms of Quantum ChaosEven though the Schr odinger equation is a linear equa-tion, the essential nonintegrability of chaotic Hamilto-nian systems carries over to the quantum domain. Thereare no known examples of chaotic classical systems forwhich the corresponding wave equations can be solvedanalytically. Consequently, theoretical searches for quan-tumchaos have also relied heavily on numerical solutions.These detailed numerical studies by physical chemists andphysicists studying the dynamics of molecules and the ex-citation and ionization of atoms in strong elds have ledto the identication of several characteristic features ofthe quantum wave functions and energy levels that revealthe manifestation of chaos in the corresponding classicalsystems.One of the most studied characteristics of nonintegrablequantum systems that correspond to classically chaoticHamiltonian systems is the appearance of irregular energyspectra. The energy levels in the hydrogen atom, describedclassically by regular, elliptical Kepler orbits, form an or-derly sequence, En=1/(2n2), where n =1, 2, 3, . . . isthe principal quantum number. However, the energy lev-els of chaotic systems, such as the quantumH enonHeilesmodel, do not appear to have any simple order at large en-ergies that can be expressed in terms of well-dened quan-tum numbers. This correspondence makes sense since thequantum numbers that dene the energy levels of inte-grable systems are associated with the classical constantsof motion (such as angular momentum), which are de-stroyed by the nonintegrable perturbation. For example,Fig. 16 displays the calculated energy levels for a hydro-gen atom in a magnetic eld that shows the transitionfrom the regular spectrum at low magnetic elds to an ir-FIGURE 16 The quantum mechanical energy levels for a highlyexcited hydrogen atomin a strong magnetic eld are highly irregu-lar. This gure shows the numerically calculated energy levels as afunction of the square of the magnetic eld for a range of energiescorresponding to quantumstates with principal quantumnumbersn4050. Because the magnetic eld breaks the natural spher-ical and Coulomb symmetries of the hydrogen atom, the energylevels and associated quantum states exhibit a jumble of multipleavoided crossings caused by level repulsion, which is a commonsymptom of quantum systems that are classically chaotic. [FromDelande, D. (1988). Ph. D. thesis, Universit e Pierre & Marie Curie,Paris.]regular spectrum (spaghetti) at high elds in which themagnetic forces are comparable to the Coulomb bindingelds.This irregular spacing of the quantum energy levels canbe conveniently characterized in terms of the statistics ofthe energy level spacings. For example, Fig. 17 shows ahistogram of the energy level spacings, s = Ei +1 Ei, forthe hydrogen atomin a magnetic eld that is strong enoughto make most of the classical electron orbits chaotic. Re-markably, this distribution of energy level spacings, P(s),is identical to that found for a much more complicatedquantum system with irregular spectracompound nuclei.Moreover, both distributions are well described by thepredictions of random matrix theory, which simply re-places the nonintegrable (or unknown) quantum Hamil-tonian with an ensemble of large matrices with randomvalues for the matrix elements. In particular, this distribu-tion of energy level spacings is expected to be given by theWignerDyson distribution, P(s) s exp(s2), displayedin Fig. 17. Although these random matrices cannot predictthe location of specic energy levels, they do account formany of the statistical features relating to the uctuationsin the energy level spacings.Despite the apparent statistical character of the quan-tum energy levels for classically chaotic systems, theselevel spacings are not completely random. If they wereP1: GPJ 2nd Revised PagesEncyclopedia of Physical Science and Technology EN002E-94 May 19, 2001 20:28Chaos 663FIGURE 17 The repulsion of the quantum mechanical energylevels displayed in Fig. 16 results in a distribution of energy levelspacings, P(s), in which accidental degeneracies (s=0) are ex-tremely rare. This gure displays a histogram of the energy levelspacings for 1295 levels, such as those in Fig. 16. This distribu-tion compares very well with the WignerDyson distribution (solidcurve), which is predicted for the energy level spacing for randommatrices. If the energy levels were uncorrelated randomnumbers,then they would be expected to have a Poisson distribution indi-cated by the dashed curve. [From Delande. D., and Gay, J. C.(1986). Phys. Rev. Lett. 57, 2006.]completely uncorrelated, then the spacings statisticswould obey a Poison distribution, P(s) exp(s), whichwould predict a much higher probability of nearly degen-erate energy levels. The absence of degeneracies in chaoticsystems is easily understood because the interaction of allthe quantum states induced by the nonintegrable pertur-bation leads to a repulsion of nearby levels. In addition,the energy levels exhibit an important long-range correla-tion called spectral rigidity, which means that uctuationsabout the average level spacing are relatively small overa wide energy range. Michael Berry has traced this spec-tral rigidity in the spectra of simple chaotic Hamiltoniansto the persistence of regular (but not necessarily stable)periodic orbits in the classical phase space. Remarkably,these sets of measure-zero classical orbits appear to have adominant inuence on the characteristics of the quantumenergy levels and quantum states.Experimental studies of the energy levels of Rydbergatoms in strong magnetic elds by Karl Welge and col-laborators at the University of Bielefeld appear to haveconrmed many of these theoretical and numerical pre-dictions. Unfortunately, the experiments can only resolvea limited range of energy levels, which makes the con-rmation of statistical predictions difcult. However, theexperimental observations of this symptom of quantumchaos are very suggestive. In addition, the experimentshave provided very striking evidence for the important roleof classical regular orbits embedded in the chaotic sea oftrajectories in determining gross features in the uctua-tions in the irregular spectrum. In particular, there appearsto be a one-to-one correspondence between regular oscil-lations in the spectrumand the periods of the shortest peri-odic orbits in the classical Hamiltonian system. Althoughthe corresponding classical dynamics of these simple sys-tems is fully chaotic, the quantum mechanics appears tocling to the remnants of regularity.Another symptom of quantum chaos that is more directis to simply look for quantum behavior that resembles thepredictions of classical chaos. In the cases of atoms ormolecules in strong electromagnetic elds where classi-cal chaos predicts ionization or dissociation, this symptomis unambiguous. (The patient dies.) However, quantumsystems appear to be only capable of mimicking classi-cal chaotic behavior for nite times determined by thedensity of quantum states (or the size of the quantumnumbers). In the case of as few as 50 interacting parti-cles, this break time may exceed the age of the universe,however, for small quantum systems, such as those de-scribed by the simple models of Hamiltonian chaos, thistime scale, where the Bohr correspondence principle forchaotic systems breaks down, may be accessible to exper-imental measurements.C. The Quantum Standard MapOne model system that has greatly enhanced our under-standing of the quantum behavior of classically chaoticsystems is the quantum standard map, which was rst in-troduced by Casati et al. in 1979. The Schrodinger equa-tion for the kicked rotor described in Section IV.B alsoreduces to a map that describes howthe wave function (ex-pressed in terms of the unperturbed quantum eigenstatesof the rotor) spreads at each kick. Although this map isformally described by an innite system of linear differ-ence equations, these equations can be solved numericallyto good approximation by truncating the set of equationsto a large but nite number (typically, 1000 states).The comparison of the results of these quantum calcu-lations with the classical results for the evolution of thestandard map over a wide range of parameters has re-vealed a number of striking features. For short times, thequantum evolution resembles the classical dynamics gen-erated by evolving an ensemble of initial conditions withthe same initial energy or angular momenta but differentinitial angles. In particular, when the classical dynamicsis chaotic, the quantum mechanical average of the kineticenergy also increases linearly up to a break time where theclassical dynamics continue to diffuse in angular velocitybut the quantum evolution freezes and eventually exhibitsquasi-periodic recurrences to the initial state. Moreover,when the classical mechanics is regular the quantumwavefunction is also conned by the KAM surfaces for shorttimes but may eventually tunnel or leak through.P1: GPJ 2nd Revised PagesEncyclopedia of Physical Science and Technology EN002E-94 May 19, 2001 20:28664 ChaosThis relatively simple example shows that quantumme-chanics is capable of stabilizing the dynamics of the clas-sically chaotic systems and destabilizing the regular clas-sical dynamics, depending on the system parameters. Inaddition, this dramatic quantum suppression of classicalchaos in the quantum standard map has been related tothe phenomenon of Anderson localization in solid-statephysics where an electron in a disordered lattice will re-main localized (will not conduct electricity) through de-structive quantum interference effects. Although there isno randomdisorder in the quantumstandard map, the clas-sical chaos appears to play the same role.D. Microwave Ionization of HighlyExcited Hydrogen AtomsAs a consequence of these suggestive results for the quan-tum standard map, there has been a considerable effort tosee whether the manifestations of classical chaos and itssuppression by quantum interference effects could be ob-served experimentally in a real quantumsystemconsistingof a hydrogen atom prepared in a highly excited state thatis then exposed to intense microwave elds.Since the experiments can be performed with atoms pre-pared in states with principal quantum numbers as high asn = 100, one could hope that the dynamics of this elec-tron with a 0.5-m Bohr radius would be well describedby classical dynamics. In the presence of an intense oscil-lating eld, this classical nonlinear oscillator is expectedto exhibit a transition to global chaos such as that exhib-ited by the classical standard map at k 1. For example,Fig. 18 shows a Poincar e section of the classical action-angle phase space for a one-dimensional model of a hydro-gen atom in an oscillating eld for parameters that corre-spond closely to those of the experiments. For small valuesof the classical action I , which correspond to low quan-tum numbers by the BohrSomerfeld quantization rule,the perturbing eld is much weaker than the Coulombbinding elds and the orbits lie on smooth curves that arebounded by invariant KAM tori. However, for larger val-ues of I , the relative size of the perturbation increases andthe orbits become chaotic, lling large regions of phasespace and wandering to arbitrarily large values of the ac-tion and ionizing. Since these chaotic orbits ionize, theclassical theory predicts an ionization mechanism that de-pends strongly on the intensity of the radiation and onlyweakly on the frequency, which is just the opposite of thedependence of the traditional photoelectric effect.In fact, this chaotic ionization mechanism was rst ex-perimentally observed in the pioneering experiments ofJim Bayeld and Peter Koch in 1974, who observed thesharp onset of ionization in atoms prepared in the n 66state, when a 10-GHz microwave eld exceeded a criticalthreshold. Subsequently, the agreement of the predictionsFIGURE 18 This Poincar e section of the classical dynamics ofa one-dimensional hydrogen atom in a strong oscillating electriceld was generated by plotting the value of the classical action Iand angle once every period of the perturbation with strengthI4F =0.03 and frequency I3=1.5. In the absence of the pertur-bations, the action (which corresponds to principal quantum num-ber n by the BohrSommerfeld quantization rule) is a constantof motion. In this case, different initial conditions (correspondingto different quantum states of the hydrogen atom) would traceout horizontal lines in the phase space, such as those in Fig. 14,for the standard map at k =0. Since the Coulomb binding elddecreases as 1/I4(or 1/n4), the relative strength of the pertur-bation increases with I . For a xed value of the perturbing eldF, the classical dynamics is regular for small values of I with aprominent nonlinear resonance below I =1.0. A prominent pair ofislands also appears near I =1.1, but it is surrounded by a chaoticsea. Since the chaotic orbits can wander to arbitrarily high valuesof the action, they ultimately led to ionization of the atom.of classical chaos on the quantum measurements has beenconrmed for a wide range of parameters corresponding toprincipal quantum numbers from n = 32 to 90. Figure 19shows the comparison of the measured thresholds for theonset of ionization with the theoretical predictions for theonset of classical chaos in a one-dimensional model ofthe experiment.Moreover, detailed numerical studies of the solution ofthe Schr odinger equation for the one-dimensional modelhave revealed that the quantum mechanism that mimicsthe onset of classical chaos is the abrupt delocalization ofthe evolving wave packet when the perturbation exceedsa critical threshold. However, these quantum calculationsalso showed that in a parameter range just beyond thatstudied in the original experiments the threshold eldsfor quantum delocalization would become larger than theclassical predictions for the onset of chaotic ionization.This quantum suppression of the classical chaos wouldbe analogous to that observed in the quantum standardmap. Very recently, the experiments in this new regimeP1: GPJ 2nd Revised PagesEncyclopedia of Physical Science and Technology EN002E-94 May 19, 2001 20:28Chaos 665FIGURE 19 A comparison of the threshold eld strengths for theonset of microwave ionization predicted by the classical theoryfor the onset of chaos (solid curve) with the results of experi-mental measurements on real hydrogen atoms with n=32 to 90(open squares) and with estimates from the numerical solution ofthe corresponding Schr odinger equation (crosses). The thresh-old eld strengths are conveniently plotted in terms of the scaledvariable n4F = I4F, which is the ratio of the perturbing eld Fto the Coulomb binding eld 1/n4versus the scaled frequencyn3=l3, which is the ratio of the microwave frequency to theKepler orbital frequency 1/n3. The prominent features near ratio-nal values of the scaled frequency, n3=1, 12, 13, and 14, whichappear in both the classical and quantum calculations as well asthe experimental measurements, are associated with the pres-ence of nonlinear resonances in the classical phase space.have been performed, and the experimental evidence sup-ports the theoretical prediction for quantumsuppression ofclassical chaos, although the detailed mechanisms remaina topic of controversy.These experiments and the associated classical andquantum theories are parts of the exploration of the fron-tiers of a new regime of atomic and molecular physics forstrongly interacting and strongly perturbed systems. Asour understanding of the dynamics of the simplest quan-tum systems improves, these studies promise a number ofimportant applications to problems in atomic and molec-ular physics, physical chemistry, solid-state physics, andnuclear physics.SEE ALSO THE FOLLOWING ARTICLESACOUSTIC CHAOS ATOMIC AND MOLECULAR COLLI-SIONS COLLIDER DETECTORS FOR MULTI-TEV PARTI-CLES FLUID DYNAMICS FRACTALS MATHEMATICALMODELING MECHANICS, CLASSICAL NONLINEAR DY-NAMICS QUANTUM THEORY TECTONOPHYSICS VI-BRATION, MECHANICALBIBLIOGRAPHYBaker, G. L., and Gollub, J. P. (1990). Chaotic Dynamics: An Introduc-tion, Cambridge University Press, New York.Berry, M. V. (1983). Semi-classical mechanics of regular and irregularmotion, In Chaotic Behavior of Deterministic Systems (G. Iooss,R. H. G. Helleman, and R. H. G. Stora, eds.), p. 171. North-Holland,Amsterdam.Berry, M. V. (1985). Semi-classical theory of spectral rigidity, Proc.R. Soc. Lond. A 400, 229.Bohr, T., Jensen, M. H., Paladin, G., and Vulpiani, A. (1998). DynamicalSystems Approach to Turbulence, Cambridge University Press, NewYork.Campbell, D., ed. (1983). Order in Chaos, Physica 7D, Plenum, NewYork.Casati, G., ed. (1985). Chaotic Behavior in QuantumSystems, Plenum,New York.Casati, G., Chirikov, B. V., Shepelyansky, D. L., and Guarneri, I. (1987).Relevance of classical chaos in quantum mechanics: the hydrogenatom in a monochromatic eld, Phys. Rep. 154, 77.Crutcheld, J. P., Farmer, J. D., Packard, N. H., and Shaw, R. S. (1986).Chaos, Sci. Am. 255, 46.Cvitanovic, P., ed. (1984). Universality in Chaos, Adam Hilger, Bris-tol. (This volume contains a collection of the seminal articles by M.Feigenbaum, E. Lorenz, R. M. May, and D. Ruelle, as well as anexcellent review by R. H. G. Helleman.)Ford, J. (1983). How random is a coin toss? Phys. Today 36, 40.Giannoni, M.-J., Voros, A., and Zinn-Justin, J., eds. (1990). Chaos andQuantum Physics, Elsevier Science, London.Gleick, J. (1987). Chaos: Making of a NewScience, Viking, NewYork.Gutzwiller, M. C. (1990). Choas in Classical and QuantumMechanics,Springer-Verlag, New York. (This book treats the correspondence be-tween classical chaos and relevant quantum systems in detail, on arather formal level.)Jensen, R. V. (1987a). Classical chaos, Am. Sci. 75, 166.Jensen, R. V. (1987b). Chaos in atomic physics, In Atomic Physics10 (H. Narami and I. Shimimura, eds.), p. 319, North-Holland,Amsterdam.Jensen, R. V. (1988). Chaos in atomic physics, Phys. Today 41, S-30.Jensen, R. V., Susskind, S. M., and Sanders, M. M. (1991). Chaoticionization of highly excited hydrogen atoms: comparison of classicaland quantum theory with experiment, Phys. Rep. 201, 1.Lichtenberg, A. J., andLieberman, M. A. (1983). Regular andStochasticMotion, Springer-Verlag, New York.MacKay, R. S., and Meiss, J. D., eds. (1987). Hamiltonian DynamicalSystems, Adam Hilger, Bristol.Mandelbrot, B. B. (1982). The Fractal Geometry of Nature, Freeman,San Francisco.Ott, E. (1981). Strange attractors and chaotic motions off dynamicalsystems, Rev. Mod. Phys. 53, 655.Ott, E. (1993). Chaos in Dynamical Systems, Cambridge UniversityPress, New York. (This is a comprehensive, self-contained introduc-tiontothe subject of chaos, presentedat a level appropriate for graduatestudents and researchers in the physical sciences, mathematics, andengineering.)Physics Today (1985). Chaotic orbits and spins in the solar system,Phys. Today 38, 17.Schuster, H. G. (1984). Deterministic Chaos, Physik-Verlag, Wein-heim, F. R. G.P1: FLV 2nd Revised Pages Qu: 00, 00, 00, 00Encyclopedia of Physical Science and Technology EN002-95 May 19, 2001 20:57Charged-Particle OpticsP. W. HawkesCNRS, Toulouse, FranceI. IntroductionII. Geometric OpticsIII. Wave OpticsIV. Concluding RemarksGLOSSARYAberration A perfect lens would produce an image thatwas a scaled representation of the object; real lensessuffer fromdefects known as aberrations and measuredby aberration coefcients.Cardinal elements The focusing properties of opticalcomponents such as lenses are characterized by a setof quantities known as cardinal elements; the most im-portant are the positions of the foci and of the principalplanes and the focal lengths.Conjugate Planes are said to be conjugate if a sharp im-age is formed in one plane of an object situated in theother. Corresponding points in such pairs of planes arealso called conjugates.Electron lens A region of space containing a rotationallysymmetric electric or magnetic eld created by suit-ably shaped electrodes or coils and magnetic materialsis known as a round (electrostatic or magnetic) lens.Other types of lenses have lower symmetry; quadrupolelenses, for example, have planes of symmetry orantisymmetry.Electron prism A region of space containing a eld inwhich a plane but not a straight optic axis can be denedforms a prism.Image processing Images can be improved in variousways by manipulation in a digital computer or by op-tical analog techniques; they may contain latent infor-mation, which can similarly be extracted, or they maybe so complex that a computer is used to reduce thelabor of analyzing them. Image processing is conve-niently divided into acquisition and coding; enhance-ment; restoration; and analysis.Optic axis In the optical as opposed to the ballistic studyof particle motion in electric and magnetic elds, thebehavior of particles that remain in the neighborhoodof a central trajectory is studied. This central trajectoryis known as the optic axis.Paraxial Remaining in the close vicinity of the optic axis.In the paraxial approximation, all but the lowest orderterms in the general equations of motion are neglected,and the distance from the optic axis and the gradient ofthe trajectories are assumed to be very small.Scanning electron microscope (SEM) Instrument inwhich a small probe is scanned in a raster over the sur-face of a specimen and provokes one or several signals,which are then used to create an image on a cathoderaytube or monitor. These signals may be X-ray inten-sities or secondary electron or backscattered electroncurrents, and there are several other possibilities. 667P1: FLV 2nd Revised PagesEncyclopedia of Physical Science and Technology EN002-95 May 19, 2001 20:57668 Charged-Particle OpticsScanning transmission electron microscope (STEM)As in the scanning electron microscope, a small probeexplores the specimen, but the specimen is thin and thesignals used to generate the images are detected down-stream. The resolution is comparable with that of thetransmission electron microscope.Scattering When electrons strike a solid target or passthrough a thin object, they are deected by the lo-cal eld. They are said to be scattered, elasticallyif the change of direction is affected with negligibleloss of energy, inelastically when the energy loss isappreciable.Transmission electron microscope (TEM) Instrumentclosely resembling a light microscope in its generalprinciples. A specimen area is suitably illuminated bymeans of condenser lenses. An objective close to thespecimen provides the rst stage of magnication, andintermediate and projector lens magnify the image fur-ther. Unlike glass lenses, the lens strength can be variedat will, and the total magnication can hence be variedfrom a few hundred times to hundreds of thousands oftimes. Either the object plane or the plane in which thediffraction pattern of the object is formed can be madeconjugate to the image plane.OF THE MANY PROBES used to explore the structureof matter, charged particles are among the most versa-tile. At high energies they are the only tools availableto the nuclear physicist; at lower energies, electrons andions are used for high-resolution microscopy and manyrelated tasks in the physical and life sciences. The behav-ior of the associated instruments can often be accuratelydescribed in the language of optics. When the wavelengthassociated with the particles is unimportant, geometricoptics are applicable and the geometric optical proper-ties of the principal optical componentsround lenses,quadrupoles, and prismsare therefore discussed in de-tail. Electron microscopes, however, are operated closeto their theoretical limit of resolution, and to understandhow the image is formed a knowledge of wave optics isessential. The theory is presented and applied to the twofamilies of high-resolution instruments.I. INTRODUCTIONCharged particles in motion are deected by electric andmagnetic elds, and their behavior is described either bythe Lorentz equation, which is Newtons equation of mo-tion modied to include any relativistic effects, or bySchr odingers equation when spin is negligible. Thereare many devices in which charged particles travel in arestricted zone in the neighborhood of a curve, or axis,which is frequently a straight line, and in the vast major-ity of these devices, the electric or magnetic elds exhibitsome very simple symmetry. It is then possible to describethe deviations of the particle motion by the elds in thefamiliar language of optics. If the elds are rotationallysymmetric about an axis, for example, their effects areclosely analogous to those of round glass lenses on lightrays. Focusing can be described by cardinal elements, andthe associated defects resemble the geometric and chro-matic aberrations of the lenses used in light microscopes,telescopes, and other optical instruments. If the elds arenot rotationally symmetric but possess planes of symme-try or antisymmetry that intersect along the optic axis, theyhave an analog in toric lenses, for example the glass lensesin spectacles that correct astigmatism. The other importanteld conguration is the analog of the glass prism; herethe axis is no longer straight but a plane curve, typicallya circle, and such elds separate particles of different en-ergy or wavelength just as glass prisms redistribute whitelight into a spectrum.In these remarks, we have been regarding charged par-ticles as classical particles, obeying Newtons laws. Themention of wavelength reminds us that their behavior isalso governed by Schr odingers equation, and the resultingdescription of the propagation of particle beams is neededto discuss the resolution of electron-optical instruments,notably electron microscopes, and indeed any physical ef-fect involving charged particles in which the wavelengthis not negligible.Charged-particle optics is still a young subject. Therst experiments on electron diffraction were made in the1920s, shortly after Louis de Broglie associated the notionof wavelength with particles, and in the same decade HansBusch showed that the effect of a rotationally symmet-ric magnetic eld acting on a beam of electrons travelingclose to the symmetry axis could be described in opticalterms. The rst approximate formula for the focal lengthwas given by Busch in 19261927. The fundamental equa-tions and formulas of the subject were derived during the1930s, with Walter Glaser and Otto Scherzer contribut-ing many original ideas, and by the end of the decade theGerman Siemens Company had put the rst commercialelectron microscope with magnetic lenses on the market.The latter was a direct descendant of the prototypes builtby Max Knoll, Ernst Ruska, and Bodo von Borries from1932 onwards. Comparable work on the development ofan electrostatic instrument was being done by the AEGCompany.Subsequently, several commercial ventures werelaunched, and French, British, Dutch, Japanese, Swiss,P1: FLV 2nd Revised PagesEncyclopedia of Physical Science and Technology EN002-95 May 19, 2001 20:57Charged-Particle Optics 669American, Czechoslovakian, and Russian electron micro-scopes appeared on the market as well as the Germaninstruments. These are not the only devices that dependon charged-particle optics, however. Particle acceleratorsalso use electric and magnetic elds to guide the parti-cles being accelerated, but in many cases these elds arenot static but dynamic; frequently the current density inthe particle beam is very high. Although the traditionaloptical concepts need not be completely abandoned, theydo not provide an adequate representation of all the prop-erties of heavy beams, that is, beams in which the cur-rent density is so high that interactions between individualparticles are important. The use of very high frequencieslikewise requires different methods and a new vocabularythat, although known as dynamic electron optics, is farremoved from the optics of lenses and prisms. This ac-count is conned to the charged-particle optics of staticelds or elds that vary so slowly that the static equationscan be employed with negligible error (scanning devices);it is likewise restricted to beams in which the current den-sity is so low that interactions between individual parti-cles can be neglected, except in a few local regions (thecrossover of electron guns).New devices that exploit charged-particle optics areconstantly being added to the family that began with thetransmission electron microscope of Knoll and Ruska.Thus, in 1965, the Cambridge Instrument Co. launchedthe rst commercial scanning electron microscope aftermany years of development under Charles Oatley in theCambridge University Engineering Department. Here, theimage is formed by generating a signal at the specimen byscanning a small electron probe over the latter in a regu-lar pattern and using this signal to modulate the intensityof a cathode-ray tube. Shortly afterward, Albert Crewe ofthe Argonne National Laboratory and the University ofChicago developed the rst scanning transmission elec-tron microscope, which combines all the attractions of ascanning device with the very high resolution of a con-ventional electron microscope. More recently still, neelectron beams have been used for microlithography, forin the quest for microminiaturization of circuits, the wave-length of light set a lower limit on the dimensions attain-able. Finally, there are, many devices in which the chargedparticles are ions of one or many species. Some of theseoperate on essentially the same principles as their electroncounterparts; in others, such as mass spectrometers, thepresence of several ion species is intrinsic. The laws thatgovern the motion of all charged particles are essentiallythe same, however, and we shall consider mainly electronoptics; the equations are applicable to any charged par-ticle, provided that the appropriate mass and charge areinserted.II. GEOMETRIC OPTICSA. Paraxial EquationsAlthough it is, strictly speaking, true that any beam ofcharged particles that remains in the vicinity of an arbi-trary curve in space can be described in optical language,this is far too general a starting point for our present pur-poses. Even for light, the optics of systems in which theaxis is a skew curve in space, developed for the study ofthe eye by Allvar Gullstrand and pursued by ConstantinCarath eodory, are little known and rarely used. The sameis true of the corresponding theory for particles, devel-oped by G. A. Grinberg and Peter Sturrock. We shall in-stead consider the other extreme case, in which the axisis straight and any magnetic and electrostatic elds arerotationally symmetric about this axis.1. Round LensesWe introduce a Cartesian coordinate systemin which the zaxis coincides with the symmetry axis, and we provision-ally denote the transverse axes X and Y. The motion of acharged particle of rest mass m0 and charge Q in an elec-trostatic eld E and a magnetic eld B is then determinedby the differential equation(d/dt)( m0v) = Q(E +v B) = (1 v2/c2)1/2, (1)which represents Newtons second law modied forrelativistic effects (Lorentz equation); v is the veloc-ity. For electrons, we have e =Q 1.6 1019C ande/m0176 C/g. Since we are concerned with staticelds, the time of arrival of the particles is often of nointerest, and it is then preferable to differentiate not withrespect to time but with respect to the axial coordinate z.Afairly lengthy calculation yields the trajectory equationsd2Xdz2 = 2g_ g X X

gz_+Qg_Y

(Bz+ X

BX) BY(1 + X2)_d2Ydz2 = 2g_gY Y

gz_+Qg_X

(Bz+Y

BY) + BX(1 +Y2)_ (2)in which 2=1 + X2+Y2and g = m0v.By specializing these equations to the various cases ofinterest, we obtain equations from which the optical prop-erties can be derived by the trajectory method. It is wellP1: FLV 2nd Revised PagesEncyclopedia of Physical Science and Technology EN002-95 May 19, 2001 20:57670 Charged-Particle Opticsknown that equations such as Eq. (1) are identical with theEulerLagrange equations of a variational principle of theformW =_t1t0L(r, v, t) dt = extremum (3)provided that t0, t1, r(t0), and r(t1) are held constant. TheLagrangian L has the formL = m0c2[1 (1 v2/c2)1/2] + Q(v A ) (4)in which and A are the scalar and vector potentialscorresponding to E, E=grad and to B, B=curl A.For static systems with a straight axis, we can rewriteEq. (3) in the formS =_z1z0M(x, y, z, x

, y

) dz, (5)whereM = (1 + X2+Y2)1/2g(r)+Q(X

AX +Y

AY + Az). (6)The EulerLagrange equations,ddz_MX

_= MX ;ddz_MY

_= MY (7)again dene trajectory equations. Avery powerful methodof analyzing optical properties is based on a study of thefunction M and its integral S; this is known as the methodof characteristic functions, or eikonal method.We now consider the special case of rotationally sym-metric systems in the paraxial approximation; that is, weexamine the behavior of charged particles, specicallyelectrons, that remain very close to the axis. For such par-ticles, the trajectory equations collapse to a simpler form,namely,X

+

2 X

+

4 X + B1/2Y

+ B

2 1/2Y = 0(8)Y

+

2 Y

+

4 Y B1/2X

B

2 1/2X = 0in which (z) denotes the distribution of electrostaticpotential on the optic axis, (z) =(0, 0, z); (z) =(z)[1 +e(z)/2m0c2]. Likewise, B(z) denotes the mag-netic eld distribution on the axis. These equations arecoupled, in the sense that X and Y occur in both, but thiscan be remedied by introducing new coordinate axes x,y, inclined t