photonic s spectra fast light slow light 2007

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to realize that there are many quan-tities that can be introduced to de-scribe the speed at which a lightpulse moves through a material sys-tem.2This confusing situation arisesfrom the fact that a pulse propagat-ing through any material system willexperience some level of distortion e.g., it spreads out in time and re-shapes and, hence, a single ve-locity cannot be used to describe themotion of the pulse.

Probably the most familiar suchquantity is the phase velocity of light.Consider a continuous-wave mono-chromatic (single frequency) beamof light of frequency

c; the electro-

magnetic field oscillates rapidly (theoscillation period is 1.8 fs for greenlight). The phase velocity

pdescribes

the speed at which the crests of theseoscillations propagate, as shown inFigure 1. In a material system char-acterized by the frequency-depen-dent index of refractions n(), thephase velocity is defined as:

The situation becomes more com-plicated when apulseof light prop-agates through a dispersive opticalmedium. According to Fouriers the-orem, a pulse of duration is neces-sarily composed of a range of fre-quencies. In a sense, a pulse can bethought of as resulting from con-structive and destructive interfer-ence among the various Fourier

(frequency) components. At the peakof the pulse, these components will

p= zt2t1

= cn()

Fast Light, Slow Light

and Optical Precursors:What Does It All Mean?

by Daniel J. Gauthier, Duke University, and Robert W. Boyd, University of Rochester

The speed of light in vacuum(c 3 108 m/s) is an im-portant physical constant that

appears in Maxwells theory of elec-tromagnetism. For this reason, sci-entists have endeavored to measureit with very high precision, making itone of the most accurately known ofall physical constants. The situationbecomes murkier for a slightly dif-ferent situation: Send a pulse of lightthrough a dispersive optical mater-ial rather than vacuum, and bizarrethings start to appear. For example,under conditions such that the dis-persion of the medium is anomalousover some spectral region,1 as de-scribed in greater detail below, it ispossible to observe the peak of apulse of light apparently leaving apiece of dispersive material before itenters.

Fast lightThe possibility of such fast light

behavior has been known for nearly

a century and has been the sourceof continued controversy and con-fusion. Some of the controversyarises because some people inter-pret Einsteins special theory of rel-ativity as placing a speed limit of con any sort of motion. Yet, a rathersimple mathematical proof showsthat fast light behavior is completelyconsistent with Maxwells equationsthat describe pulse propagationthrough a dispersive material and,hence, does not violateEinsteins

special theory of relativity, which isbased on Maxwells equations. Al-

though the proof is straightforward,great care must be taken in inter-preting the special theory of rela-tivity and in determining whetherexperimental observations are con-sistent with its predictions.

From our point of view, the spe-cial theory of relativity places a speedlimit on the transfer of informationbetween two parties, and all experi-ments performed to date are consis-tent with the speed limit being in-terpreted in this manner.

To understand the basics of fast-light pulse propagation, it is crucial

82 PHOTONICS SPECTRA JANUARY 2007

Figure 1.A monochromatic

electromagnetic wave propagates

through an optical material. Snapshots

of the electric field distribution are

shown at two different times. The

phase velocity describes the speed of

the crests of the wave. The distance

traveled by a crest in the time interval

t2t

1is denoted by Z.

How can the group velocity of a pulse of light propagating through a dispersive material exceed

the speed of light in vacuum without violating Einsteins special theory of relativity?


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tend to add up in phase, while in-terfering destructively in the tempo-ral wings of the pulse, as shown inFigure 2.

In a material with a frequency-de-

pendent refractive index, each fre-quency propagates with a differentphase velocity, thereby modifying thenature of the interference. If n()varies linearly with frequency , theeffect of the modified interference isto shift the peak of the pulse in time,but with the pulse shape staying thesame. The fact that the pulse is tem-porally shifted implies that it is trav-eling with a velocity different fromthe phase velocity. This new veloc-ity is known as the group velocity

and is defined as:

where cis the central frequency and

ngis the group indexof the material.

We see that ngdiffers from the phase

index by a term that depends on thedispersion dn/d of the refractiveindex. For slow light, which occursfor n

g>1, the point of constructive

interference occurs at a later time.For fast light, which occurs for n

g


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standing how fast light is consistentwith the special theory of relativity istied into the details of how suddenchanges in the pulse shape propa-gate through a dispersive material.

For achieving extremely fast lightpropagation, it is important to find amaterial system for which dn/d isas large and negative as possible. Adilute gas of atoms with a strong ab-sorption resonance is an interestingcandidate. The refractive index forsuch a sample is typically small (n 1 103), but it varies rapidly inthe region of the resonance so thatdn/d and, hence, n

gcan be large.

This situation is shown in Figure 3.If the carrier frequency of the pulseis tuned to the resonance frequency,the anomalous dispersion becomesvery large. For easily attainable con-ditions,

gcan take on very large pos-

itive or even negative values.3,4 How-

ever, the absorption near a resonanceis very high. Essentially all of thelight is absorbed within a few mi-crons, severely limiting the attain-able pulse advancement.

To get around the problem of largeabsorption, recent fast-light researchhas taken advantage of anomalousdispersion between two gain lines5-7

or that due to coherent populationoscillations in a reverse saturableabsorber.8

An example of fast-light pulse prop-

agation is shown in Figure 4, whichdepicts the propagation of a smooth

Gaussian pulse through vacuum incomparison with propagation througha fast-light dispersive material (a dilutegas of potassium atoms pumped by abichromatic laser beam). The pulsepropagating through the fast-lightmedium is amplified by a factor ofseven, but it has been normalized tothe same height as the vacuum-prop-agated pulse for easier comparison ofthe relative pulse advancement.

These observations compare onlywhat happens at the output face ofthe medium. What occurs inside themedium is even stranger. A numer-ical simulation showing what hap-pens when a pulse of light passesthrough a medium with a negativegroup index is shown in Figure 5.We see that, within the materialmedium, the pulse does move in thebackward direction, correspondingto the negative value of the group ve-

locity. We also see that there is a neg-ative time delay; that is, the peak ofthe transmitted pulse leaves themedium before the peak of the inci-dent pulse enters the medium.

For illustration purposes, we havekept the pulse amplitude fixed; inreality, the pulse amplitude growsfrom left to right, as observed in arecent experiment.9This fast-lightexperiment was conducted using anoptical amplifier; for this reason,there is no concern with the appar-

ent lack of energy conservation seenin the figure because energy is ex-

changed between the field and themedium in a transitory manner.

To understand why the behaviorin Figures 4 and 5 seems so discon-certing, we need to briefly reviewsome aspects of Einsteins special

theory of relativity. This theoryconcerns the behavior of Maxwellsequations under coordinate trans-formations and has far-reaching con-sequences. In his public discussionsof the theory,10 Einstein focuses onthe concept of an event, such as aspark caused by a lightning bolt, andon how the event (or multiple events)would be observed by people at var-ious locations. He was especially in-terested in observers moving with re-spect to a coordinate system that is

stationary with respect to the events.A detailed description of his findingsis not needed for this discussion, asit is necessary to consider only theproperties of a single event in a sin-gle coordinate system.

Space-time plotsA convenient way to discuss the

flow of information from an event isto use a space-time diagram, wherethe horizontal axis is a single spa-tial coordinate and time is plottedalong the vertical axis (see Figure 6).According to the special theory of rel-ativity, the fastest way that knowl-edge of an event can reach an ob-server is if it travels at the speed oflight in vacuum. The lines that con-nect points in a space-time diagramthat follow vacuum speed-of-lightpropagation define the light cone the blue region in Figure 6a. The in-verse of the slope of lines drawn in aspace-time diagram is equal to thevelocity.

Observers at space-time points

within the blue cone (observer A inFigure 6b) can see the event andthose outside the cone cannot (ob-server B in Figure 6b). The flow ofinformation within the light cone issaid to be relativistically causal. Notethat the cone extending for timespreceding the event represents thespace-time regions where light couldreach the location of the event. Thatis, an observer (not shown) in thisregion can affect the event but can-not see it.

On the other hand, the hypothet-ical faster-than-light propagation of


Figure 4. Experimentally observed fast-light pulse propagation is shown here.

Pulse propagation through vacuum is represented by the black line, and througha gas of potassium atoms pumped by a bichromatic laser beam is shown by the

red line. Adapted from Reference 7.


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information is relativistically acausal,meaning that there is no direct time-ordered link between a cause andan effect. We are not aware of anyobservation of acausal communica-tion, but it is important to discusssuch a hypothetical behavior be-cause it can teach us more aboutour existing understanding of na-ture, and it clearly points out theconsequences of faster-than-c in-formation transmission.

An example of a hypothet icalfaster-than-light communicationscheme is shown in Figure 6c, wherewe assume that it is possible totransmit information at a speed thatis less than zero (negative velocity).If such a superluminal signal werepossible, information could be trans-mitted from the positive-time lightcone to a person at position D. Thisobserver could change the outcomeof the event (e.g., prevent it fromhappening) because she is located

within the light cone leading to theevent, but at a time before the event

happens. Thus, she can change theoutcome of the event. We are surethat some gamblers would pay us alot of money if we could constructsuch a relativistically acausal com-munications system.

Fronts and precursorsOne question that perplexed re-

searchers soon after Einstein pub-lished his theory is whether pulsepropagation in a dispersive medium

might allow for relativistically acaus-al communication. By the early1900s, it was known that some ofthe velocities describing pulse prop-agation through a dispersive mater-ial could attain values greater thanc what we now call fast light. Manyeminent scientists debated whethersuch behavior constitutes a violationof the special theory of relativity.

Arnold Sommerfeld and his stu-dent, Lon Brillouin, took up thechallenge in 1907 to see whether they

could prove that information travelsat subluminal speeds.11 Sommerfeld

JANUARY 2007

Region of NegativeGroup Index

FieldAmplitude

Propagation Distance

Figure 5.A time sequence of frames shows how a pulse of light propagates

through a material with a negative value of the group velocity. Note that the peak

of the exiting pulse leaves before the peak of the incident pulse enters, and that

the pulse appears to move backward within the medium. This sort of behavior was

seen in a recent laboratory experiment, although for broader pulses.



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realized that the proof had to startwith the definition of a signal, wherea signal is a modulation of an elec-tromagnetic wave that allows twoparties to transmit information. Inits essence, the simplest signal is one

in which the wave is initially zeroand suddenly turns on to a finitevalue a so-called step-modulatedpulse. In terms of our discussion ofrelativity in the preceding section,the moment that the wave turns oncorresponds to the event.

In their analysis, Sommerfeld andBrillouin used Maxwells equationsto predict the propagation of the elec-tromagnetic wave, which was cou-pled to a set of equations that de-scribed how it modifies the dispersive

material. For the dispersive mater-ial, they assumed that it consistedof a collection of Lorentz oscillators,simple models for an atom that de-scribe its resonant behavior wheninteracting with light. Each oscillatorconsists of a massive (immovable)positive core and a light negativecharge that experiences a restoringforce obeying Hookes law (a simplemodel for the force generated by aspring). They also assumed that thenegative charge experiences a veloc-ity-dependent damping, so that anyoscillations set in motion will even-tually decay in the absence of an ap-plied field.

Conceptually, an incident electro-magnetic wave polarizes the mater-ial causes a displacement of thenegative charges away from theirequilibrium position acting backon the electromagnetic field to change

its properties (e.g., amplitude andphase). The coupled Maxwell-Lorentzoscillator model possesses spectralregions of anomalous dispersion,where

gtakes on strange values and,

thus, should be able to address thecontroversy. The model is so goodthat it is still in use today for de-scribing the linear optical responseof dispersive materials.

Using fairly sophisticated mathe-matical methods of the time, Som-merfeld was able to predict what

happens to the propagated field fortimes immediately following the sud-den turn-on of the wave what hecalled the front of the pulse. Heshowed that the velocity of the frontalways equals c. In other words, thefront of the pulse coincides with theboundary of the light cone shown inFigure 6.

He gave an intuitive explanationfor his prediction. When the electro-magnetic field first starts to interactwith the oscillators, the oscillatorscannot immediately act back on thefield via the induced polarization be-cause they have a finite responsetime. Thus, for a brief moment after

Figure 6. The lines that connect points in a space-time diagram that follow

vacuum speed-of-light propagation define the light cone (a). Observers are at

space-time points A and B. In a world that is relativistically causal, A observes the

event, but B does not (b). If relativistic causality could be violated, a person at C

could observe the event and transmit information to a person at D using a

superluminal communication channel. The person at D could then change theoutcome of the event (c).


See us at Photonics West, Booth #1138


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the front passes, the dispersive ma-terial behaves as if there is nothingthere as if it were vacuum. Fromthe point of view of information prop-agation, one should be able to de-tect the field immediately following

the front and, hence, observe infor-mation traveling precisely at c.

After the front passes, mathemat-ical predictions are very difficult tomake because of the complexity ofthe problem. Brillouin extended Som-

merfelds work to show that the ini-tial step-modulated pulse, after prop-agating far into a medium with abroad resonance line, transforms intotwo wave packets (now known as op-tical precursors) and is followed by

the bulk of the wave (what Brillouincalled the main signal). The scien-tists found that the precursors tendto be very small in amplitude and,thus, it would be difficult to measureinformation transmitted at c; rather,

it would be easier to detect at the ar-rival of the main signal, which theyfound travels slower than c. The termprecursor is somewhat confusingbecause it implies that the wavepacket comes before something; in

this usage, the precursors come be-fore the main signal but afterthepulse front.

One aspect of Sommerfeld andBrillouins result that can lead toconfusion is the possible situationwherein one or more of these wavepackets travels faster than c. What isimplied here is that a velocity can beassigned to the precursors and tothe main signal to the extent thatthey do not distort, and that thesevelocities can all take on different

values. In a situation where the ve-locity of a wavepacket exceeds c, itwill eventually approach the pulsefront (which travels at c), becomemuch distorted (so that assigning ita velocity no longer makes sense)and either disappear or pile up atthe front.

Still being debatedSommerfeld and Brillouins re-

search appeared to satisfy scientistsin the early 1900s that fast lightdoes not violate the special theory ofrelativity. Yet there continue to beresearchers who question aspects oftheir work. One point of contentionfor some people is their belief that itis impossible to generate a waveformin the lab that has a truly discon-tinuous jump (i.e., there is no elec-tromagnetic field before a particulartime and then a field appears).

Yet, having something appear at aparticular space-time point is pre-cisely what is meant by an event, asdescribed above. Thus, if one does

not believe in discontinuous wave-forms, then the very conceptualframework of the special theory ofrelativity and the associated lightcone shown in Figure 6 would needto be thrown out. Many scientistsare unwilling to do so. Also, the ex-istence of optical precursors has beenquestioned, because Sommerfeld andBrillouin made some mathematicalerrors in their analysis concerningthe propagated field for times wellbeyond the front although recent

research suggests that precursorscan be observed readily in setups

Transmitter

Transmitter

Receiver

Receiver

ReceiverTransmitter

Emitted Waveform

Vacuum Propagation

Fast-Light Propagation

Figure 7. In a typical experiment, the emitted waveform has the shape of a

well-defined pulse. Nonetheless, the waveform has a front, the moment of time

when the intensity first becomes nonzero. When such a waveform passes through

a fast-light medium, the peak of the pulse can move forward with respect to the

front, but it can never precede the front. Thus, even though the group velocity is

superluminal, no information can be transmitted faster than the front velocity,

which is always equal to c.

Figure 8. Pulse propagation in a fast-light medium with a negative group velocity

is shown in this space-time diagram. The peak is advanced as it passes through

the medium, but the pulse front is unaffected. The opening angle of the light coneis drawn differently from that in Figure 4 for illustration purposes only.



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similar to that used in recent fast-light research.12

So how can the data shown in Fig-ure 4 be consistent with the specialtheory of relativity? To answer thisquestion, we must make a connec-

tion between Sommerfelds idea of asignal and the data shown in the fig-ure. In the experiment, a pulse wasgenerated by opening a variable-transmission shutter (an acousto-optic modulator); only a segment ofthe pulse is shown. At an earlier time,not shown in the figure, the light wasturned from the off state to the onstate, but with very low amplitude.The moment the light first turns oncoincides with the pulse front (theevent). At a later time, the pulse am-

plitude grows smoothly to the peakof the pulse and then decays.As far as information transmission

is concerned, all the information en-coded on the waveform is available tobe detected at the pulse front (al-though it might be difficult to mea-sure in practice). The peak of thepulse shown in Figure 4 contains nonew information. Thus, the fact thatthe peak of the pulse is advanced intime is not a violation of the specialtheory of relativity so long as itnever advances beyond the pulse

front. Figure 8 shows a schematic ofthe light cone for just such a fast-light experiment.

In our opinion, all experiments todate are consistent with the specialtheory of relativity, even though itmay be difficult to show this. In someexperiments, the pulse shape is suchthat it is exceedingly difficult to de-tect the pulse front and, hence, itmay appear that the special theoryhas been violated. In other experi-ments especially designed to accen-

tuate the pulse front, it has beenshown that the information velocityis equal to cwithin the experimentaluncertainties in both fast- and slow-light regimes.7,13 In other words, donot yet invest in a faster-than-lightcommunications system.

AcknowledgmentsThe authors gratefully acknowl-

edge the financial support of theDARPA Defense Sciences OfficeSlow-Light Program, and DJG, the

hospitality of professor JrgenKurths of Universitt Potsdam in

Germany, where his portion of thearticle was written.

Meet the authorsDaniel J. Gauthier is the Anne T. and

Robert M. Bass professor of physics and

chair of the physics department at DukeUniversity in Durham, N.C.; e-mail: gau

[email protected].

Robert W. Boyd is the M. Parker Givens

professor of optics and a professor of

physics at the University of Rochester in

New York; e-mail: robert.boyd@rochester.

edu.

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Slow and fast light. In: Progress in

Optics, Vol. 43. E. Wolf, ed. Elsevier,

pp. 497-530.

2. R.L. Smith (August 1970). The veloci-ties of light.AM J PHYS, Vol. 38, pp.

978-984.

3. S. Chu and S. Wong (March 1982). Lin-

ear pulse propagation in an absorb-

ing medium. PHYS REV LETT, Vol. 48,

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4. B. Sgard and B. Macke (May 1985).

Observation of negative velocity pulse

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5. A.M. Steinberg and R.Y. Chiao (March

1994). Dispersionless, highly super-

luminal propagation in a medium with

a gain doublet. PHYS REV A, Vol. 49,

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sisted superluminal light propagation.

NATURE, Vol. 406, pp. 277-279.

7. M.D. Stenner et al (October 2003). The

speed of information in a fast-light

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8. M.S. Bigelow et al (July 2003). Super-

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10. A. Einstein (1988). Relativity: The Spe-

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