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