REVIEW ARTICLE The role of dispersion in ultrafast opticslochi/ultra/Zusatz... · solids, the...

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 72, NUMBER 1 JANUARY 2001

REVIEW ARTICLE

The role of dispersion in ultrafast opticsIan Walmsleya)

The Institute of Optics, University of Rochester, Rochester, New York 14627

Leon WaxerLaboratory for Laser Energetics, University of Rochester, 250 East River Road, Rochester, New York 14623

Christophe DorrerThe Institute of Optics, University of Rochester, Rochester, New York 14627

~Received 22 February 2000; accepted for publication 12 October 2000!

In this article, we review the phenomenon of dispersion, paying particular attention to its impact inthe optics of ultrashort pulses, as well as its measurement and management. At present, lasersgenerating coherent bandwidths of several hundred nanometers have been demonstrated andcorrespondingly short pulses of 10 fs or so are quite usual. The limits to the breadth of opticalspectra and brevity of pulse durations that may be achieved are often set by the dispersive propertiesof the linear optical elements of which the source is constructed. Progress in ultrafast optics to datehas therefore relied extensively on the development of ways to characterize and manipulatedispersion. The means by which this can be accomplished are significantly different for laseroscillators and laser amplifiers, as well as for nonlinear interactions that are used to extend the rangeof frequencies at which short optical pulses are available, but in all cases it is this phenomenon thatdetermines the output of current optical sources. ©2001 American Institute of Physics.@DOI: 10.1063/1.1330575#

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I. INTRODUCTION

A. General considerations

The work of Edward Muybridge1 and Harold Edgerton2

is compelling, as art as well as science, because it providview into a world that is outside our everyday experienand in it reveals a drama that is unanticipated. It is perhnot too great a leap to argue that the continuation oflegacy of these past masters is to be found currently inscience of ultrafast phenomena. This is the business of loing into the realm of events that take place on femtosectime scales (10215s51 fs).

The technology required for this activity uses ideasdeveloped at all even at the time of Edgerton, but noneless the basic principle is still the same as that developedhim—scatter a short burst of light from the dynamic entand detect the reflection. The ability of modern scientistsobserve events such as the ballistic transport of electronsolids, the first events in photosynthesis, the primary dynics in the biology of the human visual response, andgyrations of a molecule undergoing a chemical reaction hall been predicated on the ability to manipulate short buof light. And this ability has largely been one of being ablemanage dispersion, since it is this ubiquitous phenomethat often limits the duration of the burst that can be genated or delivered effectively to the target.

a!Electronic mail: [email protected]

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Another, equally important, branch of ultrafast opticsthe generation of very large optical fields using lasersmodest scale. In this branch, too, dispersion plays a cenrole in both the amplification and frequency conversionultrashort pulses. But there are the added technological plems of damage of the optical elements, and the conseqneed for large-aperture optics.

In this article, we will discuss the role of dispersionultrafast optics in a broad way: how it arises, what its effeon optical pulse propagation is, how it is measured, and hit is manipulated. The subject divides quite naturally into ttechniques necessary for the management of dispersiolaser sources, in amplifiers, and in nonlinear optical fquency generation, each of which has special conditionssociated with them, such as the requirement for extremlow loss or high-peak power handling capability.

Dispersion is the phenomenon that the phase velocitya wave depends on its frequency.3 For optical fields, thebasic vacuum dispersion relation is quite simple:v5ck,wherev is the angular frequency of the radiation,k the wavenumber~the modulus of the wave vector!, andc a constantvelocity. In this case, the phase velocityv/k and the groupvelocity ]v/]k of a wave packet centered at frequencyv areboth constant, equal toc. In all optical media, the presence oabsorptive resonances modifies this dispersion relationso that the phase and group velocities are different, everegions of little absorption far away from resonances. Tmost important consequence of this is that both velocities

© 2001 American Institute of Physics

IP license or copyright, see http://ojps.aip.org/rsio/rsicr.jsp

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2 Rev. Sci. Instrum., Vol. 72, No. 1, January 2001 Walmsley, Waxer, and Dorrer

functions of frequency, having the effect that a short puwill change shape as it propagates. One can think of a spulse as being made up of a set of much longer wave pacof narrow spectrum, all added together coherently. In apersive system, these wave packets all travel at differentlocities, and consequently the initial short pulse mbroaden. Of course, in reality, things are more complicathan this simple picture: Both diffraction of finite beams atheir behavior at interfaces between media of different dpersion, to say nothing of nonlinear effects, are importan

Nonetheless, it is instructive to look at a particular eample. The shortest pulses that can be generated by opsources to date are in the region of 4–6 fs in duration.4,5 Forthese pulses with spectra centered near 800 nm, this meduration of a couple of optical cycles, which is close to tphysical limit for a propagating wave with finite spatial etent. It has been conjectured that much shorter pulsespossible at much shorter wavelengths, in the vacuum uviolet and x-ray regions of the spectrum. In this regime, eva 1 fs duration pulse contains many cycles. However theygenerated, a common theme is the necessity to carefullythe relative phases of each of the spectral componentachieve the shortest temporal pulse duration for a given strum. The durationDt of an optical pulse~in the sense of themean square! can be expressed as

Dt25^~ t2^t&!2&5E ~ t2^t&!2I ~ t !dt

5E u~ t2^t&!E~ t !u2dt. ~1!

Using Parseval’s theorem, this can be rewritten as

Dt25E U]~E~v!eiv^t&!

]vU2

dv

5E U]r~v!

]v1 ir~v!

]f~v!

v U2

dv

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]v U2

dv1E Ur~v!]f

]vU2

dv, ~2!

whereE(v)5r(v)eif0(v) and f(v)5f0(v)1v^t&. Con-sider the final expression in Eq.~2!. The square root of thefirst term on the right-hand side is the pulse duration fopulse whose phase is constant across its spectrum (]f/]v50). The second term on the right-hand side is alwagreater than or equal to zero, and therefore the shortesration of an optical pulse for a given spectrum occurs whthe pulse has a constant spectral phase functionf~v!.

As a consequence, one seeks optical componentspreserve this condition, and this can be quite difficultpulses with large bandwidths. For instance, if a pulse witflat spectral phase and a Gaussian temporal field with awidth at half-maximum~FWHM! duration of 6 fs propagatethrough a window of standard flint glass that is 500mmthick, its duration will increase to 8 fs. One can infer froeven this simple situation that keeping a short pulse localiin time at a target, especially when it must be focused othat target, is not an easy task.


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Despite its detrimental effects when using short lapulses for experiments, dispersion has a very importantto play in actually generating such pulses. The primasource for ultrashort optical pulses is the mode-locked laoscillator. In this device, the many longitudinal modes of tlaser resonator must be kept phase locked and equally spin frequency. Dispersion causes the modes to be spacedequally in frequency, and leads to output pulse durationsare longer than the minimum that can be supported byamplifier bandwidth. An equivalent picture is that a pulpropagating around the laser cavity must have the sashape after one round trip. Clearly dispersion will causepulse to broaden. The nonlinear mode-locking mechanismsome extent balances this broadening tendency, though mmizing linear dispersion is still necessary. Although it hproven possible to generate very short pulses using molocked lasers with a variety of gain media, by far the mocommon source of such pulses are lasers which use titansapphire~Ti:sapphire!.6 This is for purely technological reasons, which are compelling enough to warrant a technolbased around a wavelength of 800 nm. However, it is ofdesirable to generate wavelengths away from this region,this is the domain of ultrafast nonlinear optics. In this aretoo, dispersion plays a central role. The bandwidth ofnonlinear process in phase matched configurations, sucharmonic generation and other parametric interactionslimited by the dispersive properties of the nonlinear medior the geometry of the interaction, or some combinationthese.

B. Linear systems in ultrafast optics

The purview of optics as specified by the wavelengranges it supposedly incorporates currently extends fromnear ultraviolet, around 200 nm, to the mid-infrared, arou2 mm, give or take. Some would include the x-ray and Tehertz regimes. Even optical materials that are transpaover large fractions of the optical region of the spectrugenerally have absorption resonances in the near ultravand far infrared, and therefore the change of phase velowith frequency cannot be ignored for pulses whose speare broad enough. In this region of the spectrum, the realmthe ultrafast is similarly malleable, but is generally takenmean pulses in the picosecond (10212s), femtosecond(10215s), or perhaps in the near future attosecond (10218s)range of durations, with spectra correspondingly of 1 nmmore than 200 nm width. Over such bandwidths, neither mterial nor other sources of dispersion can be ignored.

It is common to consider a rather simple picture of pupropagation in first-order analysis of ultrafast optical sytems. That is, the electric fieldEout(t) at the output of alinear optical system is related to the input fieldEin(t) via

Eout~ t !5E2`

`

dt8 S~ t,t8!Ein~ t8!, ~3!

whereS(t,t8) is the impulse response function of the systeDispersive elements in ultrafast optical systems are usutaken as phase-only linear filters with time-stationary


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3Rev. Sci. Instrum., Vol. 72, No. 1, January 2001 Dispersion in ultrafast optics

sponse functions. In that case, taking the Fourier transfof Eq. ~3!, the input and output field spectra are related b

Eout~v!5S~v!Ein~v!. ~4!

Note that, unlike the case of continuous quasimonochmatic fields, there is no difficulty in defining the spectruman optical pulse, which is, by definition, taken to have finsupport in the time domain. Nonetheless, there are difficties in defining the ensemble and time averages in a content manner when this approach is taken. The issue is nocompletely resolved formally, but for most practical puposes, the approach taken here appears to give satisfaresults.

The system transfer functionS(v) is defined by

S~v!d~v2v8!5E2`

`

dt8E2`

`

dt S~ t,t8!eivte2 iv8t8. ~5!

The argument of the transfer functionS(v) is the spectralphase transfer functionf~v!. This is the phase accumulateby the spectral component of the pulse at frequencyv uponpropagation between the input and output reference plathat define the optical system. This function plays a cenrole in ultrafast optics.

It is important to understand the limitations of thmodel. It derives from the Helmholtz equation under tcondition that the system response function does not cothe spatial and frequency variables. This means thattransverse structure of the pulse may be completely spressed. Thus the possibility that each frequency coulddisplaced or imaged in a different way by the optical systhas been ignored. This might seem a rather drastic reducist stance, especially when one learns that a common wacontrol the dispersion of a system is to make use ofspatiotemporal coupling inherent in refraction and diffration, but it is also understood that the vast majority of useultrafast optical systems have the feature that even thothe pulse may be spatially dispersed inside the system, ainput and output all frequencies are spatially overlappSystems that do not satisfy this criterion are said to gene‘‘spatial chirp,’’ one of the banes of current ultrafast tecnology. Such a simple model should be used only to impment a first-order design or analysis of a system for the pposes of evaluating dispersion. Despite this limitation,model works remarkably well in specifying the dispersicharacter of a system.

Usually a system is designed to first order using a stdard set of equations that suppose ideal conditions. Nexwell-known connection between paraxial optics and Gauian beams is used to evaluate, and perhaps optimize,parameters of an optical system. Beyond this, more sopticated ray-tracing methods are used to evaluate aberratThe conclusion of this design is usually to provide a spefication for the layout and elements of a system operawithin the paraxial regime, whose dispersive propertiesgiven by the coefficients of a polynomial approximationthe spectral phase functionf~v!:


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n!f~n!~v0!~v2v0!n, ~6!

wheref (n)(v0) is the nth derivative of the phase functiowith respect to frequency evaluated at a reference angfrequency v0 . This quantity is specified in units obrad12n fsnc, most of the time simplified tobfsnc. The termshave the following interpretation

The absolute phasef (0)(v0) measures the phase accmulated at the reference frequencyv0 . In terms of opticalpulse propagation, it specifies, for simple pulse shapwhere the optical reference is located with respect topeak of the envelope of the pulse. It is usually ignored indesign of a system because it does not play a role ininteraction of pulses with matter at durations longer than oor two optical cycles. Its measurement is currently a fieldintense research.7–9

The group delayf (1)(v0) is the time taken for a wavepacket centered at the reference frequency to propagatetween the reference planes. It is likewise not consideredportant. All of the present techniques for the characterizatof ultrashort pulses that are self-referencing~such as FROG10

and SPIDER11! cannot measure the absolute phase orgroup delay at the reference frequency, since this requprecise knowledge of a reference. The required referenceformation is usually another pulse that has not traversedoptical system under study, or the precise positions ofinput and output reference planes.

The lowest order term that is considered usefulf (2)(v0), known as the group delay dispersion~GDD! or thequadratic phase. The terms ‘‘positive dispersion’’ a‘‘negative dispersion’’ are defined regarding the sign of thquantity. It is specified in units of@ f s2# and measures therate at which a pulse centered at the reference frequencyincrease in duration upon propagation through the systemtransform-limited Gaussian pulse with durationt i ~specifiedby the FWHM of the intensity! centered at frequencyv0 ischanged to a Gaussian pulse with durationt0 such as

t05t iA1116~ ln 2!2$f~2!~v0!%2

t i4 ~7!

by a system with group delay dispersion only. The pulsealso said to be ‘‘chirped.’’ That is, its instantaneous frquency, defined as] arg$E(t)%/]t, is a function of time.

Higher-order coefficients are known as the cubic, qutic, quintic or, in general, the prosaic but accuratenth-orderspectral phase.

In fact the polynomial approximation to the spectrphase function is rather too simple, in a strict sense,proposals have been made for the tests of its inadequacoptical materials.12 Despite this, designs for dispersive elments based on the natural dispersion of optical materialsoften based on this approximation, with the group delay dpersion and higher order spectral phase often derived froSellmeier equation for the refractive index of the materiOne-dimensional models have proven effective in bothsign and in assessing the effects of system parameter chaon the pulse propagation.


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Finally, it is worth noting that the specification of thdispersion given here is unique to the ultrafast optical comunity. In fiber optics, for example, where the quadraphase is the dominant factor, other terms are used. Inticular theb parameter of a fiber is related to the group dedispersion via

b5f~2!~v0!

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whereL is the length of the fiber, and 1 ps51 picosecond510212s.

Likewise the fiberD coefficient measures the groupdelay dispersion per unit bandwidth and is related tomodulus ofb by

D5v0

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nm•kmG . ~9!

II. PHYSICAL ORIGINS OF DISPERSION

There are three sources of dispersion in optics, arisfrom refraction, diffraction, and interference. The firstthese is the most pervasive, and the best known: the variaof the refractive index of matter with frequency. The secoarises from the coupling of the wave vector direction afrequency that occurs at dielectric interfaces at non-norincidence or upon diffraction. The third has its origin in thinterference of waves in periodic structures. Although this a little flexibility in choosing the first of these, often thdispersion arising from materials is set by other constrain the system: the damage threshold, or whether the matis an appropriate gain medium, for example. The secondthird mechanisms are usually used to counteract certain pof the dispersion arising from materials, since they canadjusted. The price for this adjustability is often increascomplexity in fabrication or alignment.

A. Material dispersion

When an electromagnetic wave is incident on an atothe atom is polarized.13 The coherent superposition of mansuch dipoles in a crystalline or amorphous material leadthe linear response from which is defined the bulk refractindex. The Lorentz atom provides a simple model for teffect. Consider an electron bound to a heavy nucleus.electron is taken to behave as if bound to its equilibriuposition with respect to the nucleus in a harmonic potenof frequencyv0 . When driven with an optical field at frequencyv, a polarization is induced. In the Lorentz modthis polarization is always proportional to the applied fiestrength, with the proportionality constant being the linesusceptibility. The susceptibility of the medium consistinga dilute mixture of such noninteracting atoms is then relato the strength of the field~assuming that the density opolarizable atoms in the sample isN!

x~1!~v!5Ne2

m

1

~v022v2!1 ivG

, ~10!

wherev is the frequency of the applied optical field,m is thereduced mass of the electron,e its charge, andG a phenom-


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enological damping rate for the electronic vibrational mtion. The real and imaginary parts of the susceptibility aplotted in Fig. 1~a!. The refractive indexn is related to thesusceptibility by the Lorentz–Lorenz formula, which aproximates for weakly absorptive media as

n2~v!2154px~1!~v!. ~11!

The square root of this function near an absorption renance is plotted in Fig. 1~b!. For most optical materials, thresonance frequencyv0 , at which the material is absorptivis in the ultraviolet region of the spectrum, correspondingwavelengths in the range 150–300 nm. Thus the interesportion of the curve in many applications in ultrafast optiis the region of positive group delay dispersion for frequecies lower than resonance. Typical values of GDD are ab500 fs2/cm.

The Lorentz model is easily extended to the case of seral resonance frequencies, which is more common. Moover, the Lorentz–Lorenz formula has the same form as~10! even in strongly absorbing~but nonetheless linear! me-dia provided the resonance frequencies are modified inmanner prescribed by Sellmeier. It is usual to use Sellmeiformula

n2~v!215(j

Aj

v j22v2 ~12!

to determine the refractive index of material, and valuesthe parametersAj and v j are tabulated for many opticaglasses and crystals.

We point out that another way of describing the indvariation of a material with the wavelength is by the usethe Schott coefficients. These coefficients, which are theefficients of a power series, are calculated to give the baccuracy when fitting an experimental determination ofindex with the calculated index on a broad range of walengths. They thus do not take explicitly into account proerties of the material’s resonance frequencies.

An important limitation of the Lorentz model atom ithat an optical material can act only as an absorber or sterer, never as an amplifier. For this, one must use a qutized model of the atom. The simplest such model represthe atom as a two-level system for purposes of its interacwith laser light. In the linear regime, in which the atompopulations of the two states are fixed, the polarizability osample of atoms of this type is

x~1!~v!52d12

2

\

~N22N1!

~v02v!1 iG, ~13!

whered12 is the quantum mechanical dipole matrix elemeN2 is the density of atoms in the upper state, andN1 thedensity in the lower state. The difference in these numberthe inversion density, and is negative for absorbers, posifor amplifiers. Importantly, the sign of the GDD changcompared to that of an absorber when the material is usean amplifier. The gainG0 of a linear amplifier is related tothe inversion density via

ln~G0!5s~N22N1!L, ~14!


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wheres is the stimulated emission cross section andL thelength of the amplifier~a similar relation holds for the lineaabsorption!.

When a pulse traverses a sample of material, say a bof glass used at normal incidence, the phase transfer funcis

f~v!5v

cn~v!L ~15!

with the refractive index usually evaluated from the susctibility using the approximate Lorentz–Lorenz or Sellmeformula.

One way to effect minor adjustments of dispersion isuse materials that exhibit birefringence. These materials hdifferent refractive indices for waves polarized in the placontaining the symmetry axis of the material and the wavector of the light~the extraordinary wave! and waves polar-ized orthogonally to this plane. While this phenomenoncritical for ultrafast nonlinear optics, it is not commonly usfor the linear systems.

B. Interference

A second source of dispersion is the interferencewaves in periodic or waveguiding structures. Significanthe dispersion depends on the topology of the periodic stture and can therefore sometimes be specified over a wrange, including the region of negative dispersion. This flibility of design makes such interferometrically induced dpersion a very useful tool in ultrafast optics.

The underlying principle by which dispersion occurs vinterference is this: A periodic structure will transmit wavof certain frequencies, and reflect others. This occurs usufor wavelengths comparable to the periodicity of the strture, which undergo strong Bragg-type scattering. Thusperiodicity effectively induces a resonance in the transfunction of the system, which has dispersion associated wit.

Perhaps the simplest of such devices will serve to illtrate the point. This is the Gires–Tournois interferome~GTI! as shown in Fig. 2~a!.14 It is a variant of the betterknown Fabry–Perot interferometer, consisting of two parlel, planar mirrors. One of these has a reflectivity of 100and the other is partially reflective with reflectivityR. Theseparation of the reflective surfaces isL. Clearly the modulusof the transfer function is unity for this device, yet the phatransfer function is not constant.

The output field can easily be written in terms of tinput field and the parameters of the interferometer, leadto a transfer function of the formH(v)5eiw(v), where

w~v!52tan21H ~12R!sin~c2vT!

22AR1~11R!cos~c2vT!J ~16!

is the phase transfer function. In this formula,c is the phasechange of the field at the highly reflective surface, andT52L/c the round-trip time for the light in the GTI. A sketcof the function is shown in Fig. 2~b!, indicating that foroptical frequencies near the resonance frequencies of thterferometer numbered by an integer


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there can be a significant phase accumulation. This caninterpreted as an extra path length that fields at thesequencies experience—they are effectively trapped instructure for several round trips, whereas waves at frequcies between the resonances escape the device after a sround trip.

The GTI has few design parameters, so it is not possto fabricate interferometers with complicated phase tranfunctions using only two reflective surfaces. It is possibleadd several more interfaces, however, and to fabricate aror that will have almost arbitrary dispersion. The designsuch multilayer interferometers is rather involved, nonetless, the same principle is at work as in the GTI—each fquency is constrained by interference to spend a differtime in the structure. Commonly these consist of several lers of thin inorganic films on a glass or crystal substrateare designed so that waves at some frequencies are reflfrom the surface layers, whereas those at other frequenmust penetrate the structure more deeply before they areflected. This geometrical picture works quite well for esmating the GDD experienced by each frequency.

Dispersion in a waveguide can be introduced usingexample of an optical fiber as shown Fig. 3~a!, consisting ofa thin cylinder of glass a few microns in diameter, the cosurrounded by a thin tube of glass of lower refractive ind

FIG. 1. ~a! Real ~continuous line! and imaginary~dashed line! part of thenormalized linear susceptibility derived from the Lorentz model of Eq.~10!.Thex axis is graduated in units of the damping factor and has for originresonance frequency.~b! The refractive index of an optical material near aabsorption resonance, calculated using the previous model. The refraindex is identical to that calculated using a quantum-mechanical modelresonant medium in the limit of weak excitation. The refractive index olinear optical amplifier near resonance has the opposite sign to that sh


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the cladding. The dispersive properties of such a struchave three components:modal dispersionand chromatic dis-persion arising frommaterial dispersionandwaveguide dis-persion.

The fiber supports several modes, whose propagaconstants~analogous to wave numbers! are different. This isthe origin ofmodal dispersion. A geometrical picture similarto that suggested for interferometers can be used to unstand this source of dispersion. In terms of ray trajectorone can imagine that the lowest order mode correspondthe propagation of light along a trajectory through the cenof the fiber. A higher order mode follows a trajectory thbounces off the ‘‘walls’’ of the fiber core. Increasinglhigher order modes have larger and larger numbersbounces, meaning that their optical path length is actumuch longer than the physical fiber length.

Chromatic dispersion in fibers has two sources; the mterial of which the fiber is fabricated, and the details ofconstruction. The effects of material dispersion are the saas for bulk materials. In the visible and near infrared,material dispersion is positive. This can be counteractedsome extent by negative dispersion due to waveguiding.confinement of the fundamental mode of the fiber depeon the difference in refractive index between core and clding, and the boundary conditions which lead to this acause the dispersion to be negative near the waveguide cfrequency. This is exactly opposite to the case in bulk marials, andwaveguide dispersioncan lead to dispersion flattened or dispersion compensated fibers at telecommuntions wavelengths@see Fig. 3~b!#. A structure having a zerogroup-velocity dispersion wavelength close to 800 nmrecently been reported, using the cancellation of the matedispersion due to silica by waveguide dispersion.15

The details of the waveguide dispersion are criticadependent upon the way in which the fiber is fabricated. Twavelength at which the GDD is zero can thereforeshifted by proper tailoring of the refractive index profiacross the fiber. Such fibers are called dispersion-shiftebers, since the minimum of the GDD is shifted away fro1330 nm, which is the minimum for silica. The cross sectof a fiber whose dispersion has been shifted to locate a mmum for wavelengths near 1550 nm~a standard telecommunications wavelength! is shown in Fig. 4~a!. The core of thefiber is doped with germanium by diffusion. This leads toradial gradient of refractive index that modifies the wavguide dispersion according to the size of the gradient andoverall Ge concentration. The GDD of this fiber is shownFig. 4~b!, and is negative over the entire range of telecomunications wavelengths.

C. Geometrical dispersion

The bending of light at the interface of two dielectricsby diffraction at a periodic interface leads to a very powermethod for engineering GDD. The basic idea is illustratedFig. 5, which shows a pair of prisms arranged in oppositat minimum deviation, with parallel faces. A pulse of ligincident from the left is angularly dispersed by refractithrough the prism. The angle of deviation between the in


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and output rays is set by the refractive index of the primaterial. Since the material has inherent dispersion,angle of deviation depends on the input wavelength. Taction of the second prism is to exactly cancel the angudispersion of the first, so that the rays for different wavlengths emerge mutually parallel to the input ray. They ahowever, laterally displaced according to wavelength, whis known asspatial chirp.

Consider the time taken for wave packets centeredseveral different frequencies within the original pulse sptrum to reach a plane perpendicular to the outgoing rays.clear that the shorter wavelengths traverse a longer geomcal path than the longer wavelengths. The longer walengths, however, experience a longer path in the prismsthe shorter wavelengths, and a straightforward calculationthe total phase accumulation shows that the group delaybe longer for longer wavelength wave packets, providedseparation of the prisms is large enough. The symmetrythe system can be used to eliminate the spatial chirp, whallows the use of a spectral phase function. If a mirrorplaced at the location of the reference plane, then eachretraces its path, and the outgoing beam has every rayrected antiparallel to the input, and collocated on top ofThis arrangement can provide negative or positive dispsion, which can be adjusted by translating the prisms intoout of the beam to insert more or less positively dispersmaterial.16 It is also possible to adjust the prism separati~while keeping the faces parallel! but this is significantlymore difficult, and seldom used.

Diffraction gratings can also be used to assemble antical system with negative dispersion, in the manner shoin Fig. 6. Again a pair of elements is used, with the grativectors parallel. A pulse incident from the left on the firgrating is diffracted, so that the wave packets making uppulse are angularly dispersed, with the longer wavelengdeviated more than the shorter ones. The energy scattinto one of the first-order beams can be 90% or more ofincident energy if the correct geometry is used. The secgrating recollimates the dispersed beam, and a mirror plaat the right-hand reference plane will retroreflect the differwave packets so that they are again spatially superimposethe output. By inspection one can see that in this system

FIG. 2. ~a! A schematic drawing of a Gires–Tournois interferometer~GTI!,illustrating that interference causes some wavelengths to be ‘‘trappedthe structure longer than others, leading to a variation of spectral phasewavelength. Because the right-hand surface has unit reflectivity, all ofincident power is reflected.~b! The group delay dispersion of a GTI fabricated from a 1-mm-thick piece of BK7 glass, with a highly reflective coatinon one surface. The curves show the group delay both calculated from~16! including material dispersion~solid line! and measured~squares!.


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longer wavelength wave packets traverse a longer paththe shorter wavelength ones. In this case, though, there imaterial dispersion to compensate this negative dispersand this arrangement of gratings always has negative Gin an amount proportional to the grating spacing.

The origin of dispersion in both optical systems is tangular dispersion that arises from refraction or diffractioThe amount of dispersion is easily calculated for systemthis class, by calculating the phase accumulated betweeninput and output reference planes, as shown in Fig. 7beam with input wave vectork in in the directionl is scatteredby element 1 into a directionkout. The beam passes betweethe first and second elements and is scattered back intoriginal direction. The input and output reference planestaken perpendicular to the input wave vector at its interstion with the first and second elements. The phase acculated by the scattered beam relative to the~unscattered! ref-erence is simply

f~v!5kout~v!"l. ~18!

For propagation in free space between the two elementsmay takeukoutu5v/c, so the phase transfer function becom

f~v!5v

cu lucos~g2a~v!!5

v

c

D

cos~g!cos~g2a~v!!, ~19!

whereg is the angle between the incident wave vector athe normal to element 1, anda the angle of the outgoingwave vector. The latter is a function of frequency, whofunctional form depends on the details of the scattermechanism.D is the spacing between the scattering elemealong a direction parallel to their normal.

To be concrete, here we treat the case when bothments are diffraction gratings. In that case the input beamdiffracted into a direction corresponding to a wave veckout5k in1mk, wherek is the grating wave vector, andm isthe order of diffraction.k lies in the plane of the grating anhas magnitudeuku52p/d, whered is the groove spacing o

FIG. 3. ~a! A schematic drawing of a fiber waveguide structure, consistof a central core of refractive indexn, and an outer cladding with refractivindex nc smaller thann. Propagating fields are concentrated in the highindex material. The dispersion relation of such a structure is shown in~b!,and consists of a component arising purely from material dispersion acomponent from waveguiding. The latter comes about because of the dent effective paths that rays of different frequencies take in propagathrough the structure, an example of which is shown by the dashed lin~a!.


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the grating. Straightforward algebra then gives the gratequation relating the input wave frequency and the diffracanglea:

m2pc

v5ml5d@sin~a~v!!2sing#. ~20!

Together with Eq.~19!, this allows analytic expressionfor the group delay, GDD, and higher-order spectral phasbe calculated. The expressions for the quadratic~GDD!, cu-bic, and quartic spectral phases are~for single pass!

f~2!~v!524p2cD

v3d2 cos3~a~v!!5

2l3D

2pc2d2 cos3~a!, ~21!

f~3!~v!53~4p2!cD

v4d2 cos3~a~v!! S 112pc sin~a~v!!

vd cos2~a~v!! D5

3l4D

4p2c3d2 cos3~a! S 11l sin~a!

d cos2~a! D , ~22!

f~4!~v!523~4p2!cD

v5d2 cos3~a~v!! H 4182pc sin~a~v!!

vd cos2~a~v!!

1~2pc!2

v2d2 @11tan2~a~v!!~615 tan2~a~v!!!#J5

23l5D

8p3c4d2 cos3~a! H 418l sin~a!

d cos2~a!

r

ar-gin

FIG. 4. Chromatic dispersion characteristics of a dispersion-compensafiber ~DCF!. ~a! Cross-section of the fiber refractive index profile, for a fibwith cutoff wavelength~minimum supported wavelength! of 1.55 mm. Aradial gradient in the germanium doping of the core changes the refracindex by about 1% across the core.~b! The dispersion functionb of the fibershown in~a!.

FIG. 5. Arrangement of prisms providing adjustable group-delay dispersThe dispersion of the refractive index of the prism material leads to a gmetrical dispersion of broadband input light. In the optical region ofspectrum, longer wavelength radiation~dashed line! travels a shorter path inair than shorter wavelength radiation~solid line!. In the material, however, ittravels a longer path, so that by adjusting the separation of the prismsuch a way that these effects compensate, the total dispersion can beeither positive or negative values.


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It is usually only necessary to go to higher orders, oruse the complete~i.e., nonpolynomic! expression for thespectral phase when the transform-limited pulse duratiobelow about 20 fs.

Note that the perpendicular distance between the gings, D, appears in each of these expressions. For gratspaced as in Fig. 7,D is a positive number, and this arrangment therefore gives negative dispersion. It was realizedMartinez,17 however, that ifD could be made negative, thea grating arrangement could be used to generate positivepersion. He showed that this could be achieved in a raingenious manner by inserting an optical system betweentwo gratings in such a way as to form a real image of the figrating, with unit magnification, behind the second gratinThe distance between the second grating and the imagthe first then plays the role ofD in the above-mentionedexpressions. A simple arrangement for accomplishingimaging is the canonical noninverting telescope formedtwo convex lenses of equal focal length, spaced by twtheir focal length, as shown in Fig. 8. It is vital that thimaging system does not introduce any field-dependentvature to the wave front, so that a single lens imaging stem, producing a magnification of21, is not acceptable. Figure 8 shows three systems of this sort that have negazero, and positive GDD, adjusted by displacement ofgratings with respect to the object and image planes ofimaging system.

III. MEASUREMENT OF DISPERSION

A. Introduction

The design of dispersive systems calls for ways in whdispersion may be measured. Since dispersion is geneconsidered a property of optical systems with time-shinvariant response functions~the output pulses are the samno matter at what time they enter the system! certain impor-

FIG. 6. Arrangement of diffraction gratings providing adjustable negagroup-delay dispersion. The geometrical dispersion of different wavelenleads to shorter paths through the gratings pair for shorter wavelengtdiation ~solid lines!, thus to negative GDD. The amount of negative dispsion is set by the grating separation. For fixed separation, higher dispeis available from successively higher orders of diffraction, but at the cosdecreasing diffraction efficiency. Typically reflection gratings are used cto the Littrow configuration~input and first-order output angles are the sawith respect to the grating normal! to obtain the highest reflection coefficients of about 90%.


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eye

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tant relationships hold. For example, the real part oftransfer functionS(v) is related to the imaginary part byHilbert transform:

Re@S~v!#51

pPE

2`

`

dv8ImbS~v8!c

v2v8. ~24!

Since it is not simple to conceptualize a method for msuring one or other of these components, determiningpartner by means of this transform appears moot. Noneless, there are certain classes of systems for which an angous relationship between the modulus and argument oftransfer function holds.

It is straightforward to measure the absorption coecient spectrum of an optical system. This is defineda(v)52Re@ln$S(v)%#. For materials this provides a way iwhich the spectral phase transfer function can be determfrom measured data—a sort of generalized Kramers–Krorelationship.

However, for certain other systems, no such relationsexists. If the complex transfer functionSL(p) ~the Laplacetransform of the response function! has zeroes at some valuof p, then the transform cannot be performed, unless thsingularities in the generalized absorption coefficienta(p)are subtracted from the function. The elimination of theso-called Blaschke factors allows the reconstruction ofphase transfer function from a measurement of the amplittransfer function.18 However, it is not usually possible toidentify the location of the singular points from a measument of absorption: Their positions need to be known,least approximately, beforehand. Generally the applicabof the generalized Kramer–Kronig transform is predicaon a more or less detailed knowledge of the system tranfunction from analysis rather than from measurement.19 But

ehsra--ionfe

FIG. 7. Schematic of optical paths through a transmission grating pair uto derive the spectral phase transfer function of the device.D is the separa-tion of the gratings measured parallel to their surface normals, andd thespacing of the grating rulings.l is the input path vector, and the zero-orddiffraction, andk the wave vector of the first-order diffracted light.a andgare the diffracted and incident ray angles with respect to the grating nor


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such arguments are specious if there exists a way to meathe phase transfer function directly.20

Another reason for wanting direct measurement is tthe structure of the phase function may depend on renances that are located at frequencies quite far from the stral region of interest. Performing the inverse transfotherefore requires quite accurate measurements of thesorption over an extended range of wavelengths, somwhich might be difficult to access. Direct measurementsthe phase transfer, on the other hand, require only the slight sources with which the optical systems are to be us

B. Measurement of dispersion of an optical element

The standard way in which dispersion is measured isinterferometry. Experimentally, the requirement is onlyhave a broadband source of arbitrary coherence, but usicoherent source, such as a laser, usually simplifies meaments for highly scattering or weakly transmissive systeIn the context of ultrafast optics, the earliest attempt to chacterize relevant optical systems was the compensated Melson interferometer of Knoxet al.21 shown in Fig. 9~a!. Inthis device, the system whose dispersion is to be measurplaced in one arm of the interferometer. A light source,tered by a passband spectral filter whose transmission ris centered atvs , is used to illuminate the interferometeand an integrating square-law detector monitors the powethe output port as a function of the delay between thearms of the interferometer. The delay can be adjustedmoving one of the mirrors~M2 in Fig. 9!. The detected signaI (t) is

I ~t!5E dtuEi~ t1t!1Ef~ t !u2

52« in12 ReF E dvuEi~v!u2F~v!S~v!eivtG , ~25!

where « in is the energy of one of the input pulses auEi(v)u2 is the spectrum of the input radiation~defined ac-cording to the Wiener–Khintchine prescription for partiacoherent light sources!, andF(v) the transfer function of thespectral filter. The only contributions to the signal are thofrom a narrow frequency range centered near the filter pband atvs . This frequency is, in effect, a reference frquency for the optical system. If the filter is sufficiently narowband, the spectral phase function of the optical syscan be truncated at the first-order term, the group delavs . ThenI (t) becomes

I ~t!52« in$11F~t2f~1!~vs!!cos~f~v!

1f~1!~vs!v02vst!%, ~26!

whereF(t) is the inverse Fourier transform ofF(v). It iseasy to see that the correlation function taken with the ocal system in place is simply a delay-shifted replica ofautocorrelation function without the system, the delay besimply the group delayf (1)(vs) at the reference frequencvs , as shown in Fig. 9~b!. Thus a numerical cross correla


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tion of I (t) measured with and without the test system yiethe group delay at the filter pass frequency with reasonaaccuracy.

It was shown by Naganumaet al.22 that a tunable spectral filter was not needed, and that without it the complespectral phase function could be determined. The apparis shown in Fig. 10~a!. The detected correlation functionI (t)is given by Eq.~25! with F(v) equal to a constant.

A sketch of this function when the input spectrumquite broad is shown in Fig. 10~b!. The spectral phase function is extracted from this data in a direct way. The Fourtransform of the data with respect to the delayt is taken, toyield

I ~V!5FT$I ~t!;t→V%

52d~V!1uEi~V!u2S~V!1uEi~2V!u2S* ~2V!.

~27!

I (V) consists of a large peak nearV50, and two satellitepeaks nearV56v0 , wherev0 is the mean frequency of thinput light. The argument of the term representing the penearV52v0 is the spectral phase transfer function of toptical system atV.

Increased accuracy can be obtained using specinterferometry.23 The apparatus shown in Fig. 11 illustratesthird Michelson interferometer arrangement, in which a tuable spectral filter is placed before the detector. As the pband of this filter is tuned, the detector sees the power inoutput arm modulate, as the spectral interferogram is mapout. This interferogram contains the same information asprevious correlation functions. Returning to Eq.~25!, let the

FIG. 8. Grating configurations for delay lines with~a! negative,~b! zero,and ~c! positive group delay dispersion. The two-lens unit-magnificatimaging system images the first grating~G1! onto G18. @The exact unitmagnification conjugate planes are connected by the dashed lines in~a!.#The location of G1’ relative to the second grating G2 determines the sigthe GDD. In~a! G1’ is a real image so that the effective grating separatD is positive, leading to negative GDD, as in the arrangement of Fig. 6~b! the grating G1’ is still real and coincides exactly with G2, so thatD iszero, and so is the GDD. In~c! G1’ is a virtual image of the first grating, andD is negative, leading to positive GDD. In this latter configuration the sptral phase function is exactly the opposite of the one obtained in configtion ~a! ~and in Fig. 6!. The combination of configurations Fig. 8~c! and Fig.6, are therefore often used as matched pulse stretchers and compress


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spectral filter transfer functionF(v)5Td(v2vs) be a realfunction of the tunable frequencyvs . Considered now as afunction of vs , for fixed t, the measured signal is an inteferogram of the form

I ~vs ;t!52TuEi~vs!u212uEi~vs!u2T cos~f~vs!2vst!.~28!

Now, asvs is varied,t can also be modified so as tkeep the argument of the cosine constant. In this wayinterference maximum defined by the contour can be demined. If t is known to sufficient accuracy, then the spectphase function can be mapped out. In fact it is only necsary to know the relative changes int accurately, providedthe constant spectral phase term is not needed.

Rather than measuring the argument of the cosinerectly, in practice the detector signal is fed back to mirrorM2

of the interferometer, which is moved so that the interferoeter is always locked on to the peak of a fringe in the specinterferogram. The displacement of the mirror is measuusing a separate single-frequency laser~of wavelengthl! toan accuracy of better thanl/50. The displacement is equal tthe group delay at the filter frequency. The increased coplexity of this apparatus nevertheless leads to increasedcuracy in the spectral phase function: the group delay camapped as a function of frequency with an accuracy of ab0.2 fs.

An arrangement that avoids scanning of any kind, eitdelay or spectrometer tuning, is shown in Fig. 12~a!.24 Arange of delays is provided by tilting the mirror in the refeence arm of the interferometer, into which a quasi-odimensional~broadband! light source is collimated. At anygiven frequency, tilt fringes are seen at the output ofinterferometer, in a direction parallel to the tilt of the refeence mirror. A slit is placed at the output of the interferoeter, with a grating behind it, arranged with grooves parato the slit direction. This causes the interferograms for ewavelength to be laterally separated at the image planelens placed beyond the grating. A two-dimensional charcoupled device camera is placed at the focal plane ofimaging lens, and the interferograms recorded. Examplethese are shown in Fig. 12~b!. The horizontal axis~the direc-

FIG. 9. ~a! Apparatus for measuring the dispersion of an optical sysfrom an interferometric correlation function. The correlation function of tinput optical field, centered at frequencyvs , is measured by scanning thdelay of one arm of the interferometer through a distance greater thancorrelation length of the input radiation.~b! With no optical system presentso that the two arms are balanced, the peak of the correlation functionS(t)is located at delayt50. When a dispersive system is placed in one armthe interferometer, the peak of the field correlation function is shifted frt50 by the group delay of light through the system at the filtered walength.


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tion of the grating vector! represents wavelength. The vertcal axis is the tilt interferogram for each wavelength, so tthis axis represents phase—a 2p shift between adjacenfringes.

The resulting two-dimensional interference pattern hfringes that follow contours of equal phase. As a result iteasy to read the spectral phase directly from the pattern—shape of each fringe tells the form of the phase polynomThe quadratic phase appears as parabolic fringes, cubs-shaped patterns, and so on.

Currently, perhaps the most widely used techniquethe measurement of linear transfer functions is the methoFourier transform spectral interferometry. PioneeredFroehly,25 this technique has found wide specific applictions in the field of ultrafast optics in the last few yearhowever, as for the other interferometric techniques, itsplication to the measurement of dispersion26–28only dependson the spectral density of the available light source. In tmethod, the output of the interferometer can be frequeresolved for a fixed delayt in a single shot using a spectrometer. The apparatus is identical to that shown in Fig.except that the delay is set at a value that is longer thancorrelation time of the input light.29 The spectrally resolvedinterferogram is

I ~vs ;t!52TuEi~vs!u212uEi~vs!u2T cos~f~vs!2vst!.~29!

Extraction of the spectral phase transfer functionstraightforward. The Fourier transform of this interferograwith respect to the frequency is composed of three peakst50, one finds the Fourier transform of the noninterferiterms while att5t and t52t stand the Fourier transformof uEi(vs)u2T exp@i(f(vs)2vst)# and uEi(vs)u2T exp@2i(f(vs)2vst)#. A large enough value oft allows the fil-tering of one of these interferometric components, whiwhen Fourier transformed back to the frequency domagivesf(vs)2vst as its argument. This allows the complemeasurement of the transfer function of the optical elemeand therefore of its dispersion, using a single delay. Itexperimentally much simpler than the previously describmethods; when only orders higher than the quadratic spephase~or GDD! are needed, there is no need to stabilizedelay or even to know its value, as any linear term cansubtracted from the retrieved spectral phase. Because o

he

f

-

FIG. 10. Apparatus for measuring the dispersion of an optical system uFourier-transform interferometry. A broadband source generates ‘‘whlight’’ interference fringes as the delay mismatch is varied. The delaywhich these fringes have greatest modulation and the symmetry offringes with respect to delay are changed by the dispersive optical systethe test arm of the interferometer.


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large value of the delayt, the removal of the linear termvstfrom the measured spectral phase difference is very sensto the calibration of the spectrometer, but efficient techniqare available to counteract, and even make use of,sensitivity.30

Other methods of dispersion measurement involve ustunable laser sources or ultrashort pulses. The former meis often used to measure the dispersion of fibers at telecmunications wavelengths, and involves measuring the rtive phase of amplitude modulation as a function of the ting of the laser. The latter method uses the time of flightthe pulse, measured either using a fast detector~for largedispersions! or an intensity autocorrelator~for smaller dis-persions!. It is therefore not possible to extract the entspectral phase function from this method, unless the lasource is tunable, for similar reasons as in the Knox meth

C. Measurement of dispersion of a laser resonator

The intracavity dispersion of a laser is important in dtermining the duration of pulses that the laser can generThe reason for this is that the coherence of the oscillamodes is established by periodic loss or phase modulaUnless the periodicity of the modulation equals the invemode spacing, the number of modes that can be lockedbe small. Dispersion causes a nonuniform mode spacthereby restricting the bandwidth of the laser and the puduration.

One can imagine that it is difficult to estimate the intrcavity dispersion by adding the dispersion of each ofoptical elements therein, since the position of the laser stial mode in the entrance pupil of each may vary with tdetails of the cavity alignment. What is needed is a waymeasure the dispersion of the cavity elements in the confiration that they are in when the laser is operating. Two teniques have been developed with this in mind.

The first of these uses a variant of the interferomeschemes outlined previously.31 The major modification is

FIG. 11. Apparatus for measuring the dispersion of an optical system bon a direct measurement of the spectral phase difference between tharms of the interferometer using phase-locked interferometry. The inteometer is kept locked to a fringe maximum as the spectrometer is tuneddelay required to accomplish this is measured accurately using a monomatic reference laser. The group delay dispersion is related to the meadelay as a function of frequency by a simple derivative, as in Eq.~28!.


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that the light source is placed inside the laser cavity,output of which is directed to a balanced Michelson interfometer, as shown in Fig. 13~a!. The light source is often thegain medium of the laser itself, pumped to just below threold. The light leaving the cavity consists of periodic noibursts, separated by the round-trip time of the cavity. Eburst has a slightly different temporal shape than thepreceding it, because it has taken one extra roundthrough the dispersive cavity than the previous burst. Ifarms of the Michelson are therefore adjusted to providdelay that is equal tom round trips of the laser cavity, it ispossible to measure the first-order correlation betweenburst and the same burst afterm round trips, in order to gettheir spectral phase difference, and therefore the disperof the cavity. This is done by scanning the reference armin the Naganuma scheme. In Fig. 13~b!, the first-order cor-relation of a burst with itselfS0(t) and with the next pulse~i.e., m51! S1(t) are plotted. As these two functions adifferent, one can predict that significant dispersion has bintroduced by one round trip.

A second method, illustrated in Fig. 14~a!, actually mea-sures the intracavity dispersion when the laser is operatin32

The output of the laser, when mode locked, is a trainpulses separated by a repetition period that is the caround-trip time at the mean wavelength of the pulse tr

edtwor-hero-red

FIG. 12. Apparatus for measuring the dispersion of an optical system uspectral interferometry~a!. Spectral interference between the light passithrough the dispersive test system and the reference arm of the interfeeter is measured using a grating and a charge coupled device~CCD! camera.The orthogonal dimension of the CCD camera is used to record tilt frinat each wavelength simultaneously. The spectral phase transfer functiobe reconstructed by determining the local slope of the two-dimensiospectral-spatial interference fringes. Examples of recorded fringes aresented in~b!. Reprinted with permission from Ref. 24.


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spectrum. This is determined both by the physical lengththe cavity and the dispersion of the intracavity elementeach adds a certain group delay to the pulse as it bouback and forth in the resonator. Thus a measurement ofrepetition rate of the laser as a function of frequency msures the change in group delay as a function of frequewhich is just the GDD@Fig. 14~b!#. The scheme devised bKnox does exactly this. The mode-locked pulse train is indent on a fast photodetector, whose output is sent to aquency counter. The repetition frequency is recorded afunction of the tuning of the laser, and the GDD map builtdirectly. This method requires that the laser be tunable wremaining mode locked. This implies that the entire bawidth of the gain medium not be mode locked while tmeasurement is made, since tuning is ipso facto impossin such a circumstance.

A remarkable result of measurements of cavity dispsion using the Knox method is that the measured GDDvery close to that of the constituent bulk elements incavity. This is surprising for the following reason: The cavmode spacing in frequency is not constant when the cacontains dispersive optical elements. The superposition pciple allows that when a coherent superposition of suchequally spaced modes is effected, the result will not bsteady train of identical pulses, but rather a modulated trSuch an output is not seen from a well-mode-locked lafor good reason: The process of mode locking is both nstationary and frequently nonlinear. The overall effectthese processes is to force the modes to be equally spacewas experimentally checked.33 It is not obvious that such aneffect will yield a local mode spacing that is so directly rlated to the local dispersion. Nonetheless, extensive expmental studies have shown that this appears to be the cTherefore it would seem that it is only necessary to measthe dispersion of the intracavity elements individually in islation, and to simply combine them to find the dispersionthe entire resonator, and this is the method used most oftepresent.

IV. DISPERSION IN ULTRAFAST LINEAR OPTICS

A. General considerations

In order to make effective use of short optical pulses inecessary to be able to deliver them undistorted to an expment. For pulses with durations of less than about 50 fs, cmust be taken to ensure that there is no detrimental efdue to uncompensated dispersion between the laser antarget.

The most common form of dispersion between the laand the experiment is that of materials. Since this is moften positive, a system with negative dispersion is neefor compensation. The most useful systems for this purpare the double-pass grating or prism delay lines, eachwhich provides adjustable GDD~with associated variationof higher-order spectral phase terms!. A diagram of such asetup is shown in Fig. 15. The output of a laser system whproduces, say, transform-limited, 24 fs Gaussian pulsedirected toward a target experiment by an optical syswhich contains 4 cm of BK7 glass. Without compensatio


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the material dispersion in this glass would cause the pulsetarget to be stretched to roughly 60 fs. In order to optimthe pulse width at the target, a prism delay line is insertedthe output of the laser system. This delay line introducnegative dispersion, which cancels the dispersion introduby the optical system. The method of adjustment is prmatic. A device for measuring the pulse duration is set upclose to the experimental location as possible, and the pduration is monitored as a function of the setting of the deline. The prism or grating separation is adjusted until tpulse duration is minimized.

B. Dispersion in the measurement device

Several issues are important in an operation of this kiThe measuring instrument should not introduce any dispsion itself, and should be able to provide a reliable valuethe pulse-quality measurement~in this example the duration!as rapidly as possible. For pulses from a mode-locked laoscillator, which have a repetition rate of several tensMHz, this is easy to do, especially if the pulse durationsabove 50 fs. In this regime the dispersion of most standmeasurement instruments is sufficiently small that the msured pulse duration is quite accurate, and the update rathe measurement is high enough that real-time feedbacpossible. In fact, it is possible to automate this procedentirely, so that adaptive control of the dispersion is possi

For pulses with durations of 20 fs or less, the specbandwidth is large enough that care must be taken to remall dispersion in the measuring device, otherwise the resing measurement will be inaccurate. All instruments fcharacterizing ultrashort optical pulses make use of two rlicas of the input pulse that are mixed with each other or wanother field in a nonlinear mixing process.

An example of such a technique is the intensity autocrelator, in which two replicas of the input are mixed inmaterial with a second-order nonlinear susceptibility in ordto generate radiation at the second-harmonic frequency.second-harmonic power is larger when the two pulses olap in time, due to an interference effect, compared to whthey do not. Therefore the instrument maps out the intenautocorrelation function of the input pulse as the delaytween the replicas is varied.

FIG. 13. Measurement of the intracavity dispersion of a laser cavitysource of broadband radiation is placed in the laser cavity~a!. The sourcedoes not provide sufficient gain to reach threshold, but excites sevmodes below threshold. The cross-correlation function of the light leakfrom the cavity on successive roundtrips is measured. On each succeround trip the light accumulates a spectral phase reflecting the total dission of all the cavity elements, so that the spectral phase transfer functiothe cavity can be extracted from the interferogram using the proceddescribed in the text. Examples of the cross correlation of the pulseitself S0(t) and after one round tripS1(t) are plotted in~b!.


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A more sophisticated variant of this is the techniquefrequency-resolved optical gating~FROG!,10,34 in which theoptical spectrum of the second-harmonic radiation is msured at each delay setting. The complete spectral ampliand phase of the input pulse can be obtained from sucmeasurement using numerical algorithms for iterative decvolution.

Another technique, spectral phase interferometry forrect electric field reconstruction~SPIDER!,11 makes use ofthe two replicas to generate a spectral interferogram frwhich the spectral phase of the input pulse can be extradirectly.

In all cases, it is important to ensure that the two pulare as close to being identical as possible, since this allthe compensation of small spectral phase accumulated ininstrument.

Replicas of the input pulse are usually prepared usinmodified Michelson interferometer. Minimal spectral phacan be added to each pulse if the beamsplitter is made asas possible. It is possible to use pellicles for this applicatialthough it is usually the case that small amounts of surfcurvature cause a large mismatch of the beams from thearms of the Michelson at the focus of the imaging lens. Tmeans that the mixing signal may be affected not only bypulse temporal shape, but also its spatial shape. Usinthicker beamsplitter avoids this problem but adds a larspectral phase onto the two pulses. If this is added symmcally, however, it can be compensated in certain measment schemes. The reason for this is that the pulses intwo arms of the Michelson interferometer remain replicaseach other, albeit not of the input pulse. Therefore anypersion in the instrument can be subtracted from the msured spectral phase function of the pulse, if the instrumecapable of extracting this function~which is not the case othe autocorrelator!. For reasons of accuracy, it is still desiable to make instrumental dispersion as small as possiblthe dispersion is large, then the measured spectral phasebe large, even if the input pulse has relatively little variatiin phase across its spectrum. In this case a precise meament of the input phase requires the subtraction of two lanumbers, so that the relative accuracy of the input phasdramatically reduced.

FIG. 14. Measurement of the intracavity dispersion of a mode-locked launder operating conditions.~a! The repetition rate of the laser is measureda function of the mean wavelength of the output spectrum. The spectruthe laser is tuned using an intracavity element. The time between pulssimply related to the group delay of one round trip of the cavity, and canread directly off the graph, as shown in~b!. Reprinted with permission fromRef. 32.


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Designs for low dispersion autocorrelators35

FROGs36–38 and SPIDERs39 have been developed, and usto measure pulses with durations in the regime of two optcycles. In these designs, care was taken to keep thepulses as replicas of one another, since it is only in tsituation that any instrumental phase can be removed athe measurement.

These methods have been applied to the characterizaof ultrashort pulse shapes at the location of the experimA particularly interesting example of this is the measuremof a femtosecond optical pulse at the focus of a microscused for confocal imaging applications.40 In this case thecomplete dispersion of the imaging system was quite coplicated, so a direct measurement of the pulse was imporin order to establish the temporal resolution of the expemental apparatus. In the measurement, which was done uthe FROG technique, the sample at the focus of the micscope objective was replaced with a nonlinear crystal, sothe pulse shape at precisely the right location could be demined.

C. Optimization of a short optical pulse

Another situation in which such methods are useful isthe characterization of pulse propagation. There are bothear and nonlinear aspects to this problem, and it is impsible to discuss them in their entirety here. A single examwill suffice to make the point. In telecommunications sytems using time-division multiplexing of the data, the bit rais limited by the dispersive properties of the optical fibalong which the pulses propagate. If the dispersion islarge, the pulses will lengthen in time, and eventually ovlap one another, making it impossible to extract the bit ptern from the measured output power. One method to ovcome the positive dispersion of common single-mode fibat telecommunications wavelengths~in the near infrared! isto compensate it using a piece of fiber engineered~using thetechniques described in Sec. II! to have negative GDD at theappropriate wavelength. The concatenation of sectionsthis fiber with regular fiber leads to a net zero dispersacross the entire link~of course, if the pulse intensity is largenough to give rise to nonlinear effects, then such cancetion is not possible!. An experiment by Chang andWeiner41,42 demonstrates that such compensation is possin the linear regime. A diagram of their setup is shownFig. 16~a!. The stretched output of a 62 fs fiber laser is ijected into a 2 kmfiber. Without compensation, the dispesion of this fiber would cause the pulse duration to incre

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FIG. 15. The dispersion of an optical system is compensated using a dopass prism pair so that the laser pulse width is minimized in the target pl


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from about 250 fs to over 10 ps. By inserting a 0.5 kdispersion-compensating fiber, the output pulse width ofoverall system could be reduced to about 500 fs. Fig16~b! shows a comparison of the input and output pulsesthis system. The fact that the output pulse is not complerecompressed to its original duration is indicative of the fthat the two fibers have different amounts of cubic specphase. Other techniques for compensating fiber disperhave also been demonstrated.43,44

Once the pulse duration or shape at the target is knothen its shape can be altered by introducing the appropamount of dispersion or attenuation. Many techniques hbeen developed for these pulse-shaping applications.45,46 Acommon and practical way to do this is to make use oFourier-plane pulse shaper. This device uses a dispersivelay line of the sort shown in Fig. 8, set for net zero dispers~i.e., with the two images of the grating superimposed onanother!. At the mutual focal plane of the two lenses in thdevice, spectral components are spatially dispersedfocused.47 Spatial phase modulation in this plane then trafers into spectral phase modulation. Such an arrangementhen be considered as a programmable device inducingarbitrary dispersion. To date, the most frequently usedvices for adjusting the group delay in this manner have ba multipixel liquid-crystal array46 and a broadband acoustooptic modulator,48 but alternative solutions have also beused.49–51 Details of the operation of these devices areyond the scope of this article. For our purposes it sufficenote that the spatial phase at any given location in the implane of the delay line can be changed by means of a ucontrolled signal. Driving this signal then allows the quasibitrary control of the induced spectral phase.

Another technique, which does not rely on a spamodulation at the Fourier plane of a zero dispersion liuses a collinear interaction between the optical pulse anacoustic wave in a birefringent acousto-optic crysta52

Proper phase matching allows the diffraction of speccomponents of the input pulse polarized along one ofaxes into the output pulse polarized along the other axisthe optical indices along the two axes are different, the grdelay for the diffracted frequency propagating in the crysdepends on the position where diffraction occurs. Propedriving the acoustic wave then allows one to shape the grdelay in the output pulse.

The addition of a specific spectral phase to eachquency component of the light individually means thpulses of nearly arbitrary shapes can be generated. Theticular spectral phase that must be applied to the input pis derived from the measured pulse shape~or other quantity!by defining an error signal using either local- or quasiglooptimization algorithms.

Such schemes are most useful to the control of coheoptical processes, where not only the temporal intenshapes of the pulse, but also its phase are important.amples of optimization applications of this type includcompression of ultrashort pulses,53–55 maximization of two-photon absorption56 or particular channels of photodissociation,57,58 strong-response excitation of a molecu


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electronic–vibrational transition,59 and weak-field control ofthe quantum state of a Rydberg electron.60

D. Focusing of a short optical pulse

If the full variation of the field with transverse coordnates is accounted for, another important effect of dispersis apparent. When a short pulse is focused by a lens, portof the wave front at the edges of the lens aperture experiea different thickness of material than the portion at the cenof the aperture. It is this variation in thickness that gives rto the lenticular shape of the element and hence to its focing power.~Although the full thickness of the material is noneeded, hence the ability of Fresnel lenses to focus light,the most convenient way to make a lens.! A corollary to thisis that the portions of the beam at the edges of the aperalso see less dispersion, so that they have a shorter gdelay. This means that they arrive at the focus of the lbefore the center of the beam, so that the ‘‘group front’’the beam~the surface of constant group delay! will no longerbe a spherical wave centered on the focus. In other wordone takes into account all the converging rays, the appaduration of the pulse at this point can be dramatically larthan the effective duration of each ray.

This phenomenon was first pointed out by Bor, usisimple ray tracing calculations.61 The radius-dependent paof the propagation time of a ray from a plane before any kof focusing system to the focal pointF ~Fig. 17! can becalculated as

DT~r !52r 2

2c f2 ld f

dl, ~30!

wherer is the input radius of the ray,f is the focal length atthe mean wavelengthl of the short pulse, andd f /dl is thevariation of the focal length with the wavelength.

For a singlet lens made of a material of indexn(l), onehasd f /dl5@ f /(n21)#(dn/dl). For example, a BK 7 lensof focal length 10 cm will induce a delay difference equal50 fs between the paraxial ray and a ray going throughlens at 1 cm from the axis. This is far from negligiblepulses of shorter durations are now widely used in expments.

For an achromat optimized for the mean wavelengththe short pulse, the variation of the focal length with twavelength is zero, so the group delay is, in this approximtion, independent of the radius. This means that all pulfrom any point of the aperture will arrive at the same timethe focus. Unfortunately, most doublets are optimized invisible, and have a nonzero value ofd f /dl at 800 nm. Forexample, the Melles Griot doublet 01LAO123 is optimizefor minimal chromatic aberrations in the visible. It yieldsfocal length of 100 mm and a value ofd f /dl equal to zero at546.1 nm, but this quantity is roughly equal to 1 mm/mm at800 nm. The inferred delay difference on the two rays cosidered previously is 20 fs, i.e., far from negligible.

The group velocity dispersion also becomes a functof the radius because the rays travel through different thnesses of a dispersive material~the lens itself!. This effect,


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proportional tod2n/dl2 for a singlet lens, tends to modifthe temporal shape of the pulse traveling on each ray.usually smaller than the previous effect.

These results obtained from geometrical optics hbeen confirmed using simulations based on a wave opdescription of the problem,62 including the effect of sphericaaberration.63 The manipulation of a short pulse with convetional optics is thus not an easy task, and other techniquecounteract the effects of dispersion have been studied.64,65

The distortion of the group front of a short pulse uppropagation through a lens has been measured by Radzeet al. using an interferometric technique.66 Their method canbe thought of as an extension of the method of Liet al.67 formeasuring GDD, except now in a system where the dispsion varies within the system exit pupil. Therefore measurthe location within the pupil of fringes arising from interfeence between the focused and reference pulses as a funof the delay between the two, as in Fig. 12, allows onemap out the group delay across the pupil. In all casesduration of the pulse at the target is increased, and the pintensity is reduced over that expected from a simple calation.

V. DISPERSION IN MODE-LOCKED LASERS

A. Introduction

The most common method of generating ultrashort ocal pulses is from a cw mode-locked laser. Figure 18 sho

FIG. 16. ~a! Fiber laser system using a concatenation of fibers with oppoquadratic spectral phase.~b! Comparison of input and output pulses for thfiber laser system in~a!. Reprinted with permission from Ref. 41.


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a common layout for this class of lasers. They consist ocontinuously pumped gain medium which has a very lagain bandwidth. The laser also has a mechanism for molating the intracavity loss or phase. This mechanism mayeither active~controlled by an external signal! or passive~controlled by the intracavity intensity itself!. In either case,its function is to cause the phases of the cavity modesare above threshold to be locked, that is, to have the sphase~or for adjacent modes to differ by the same constphase!. In the case of homogeneously broadened gain methe mode-locking mechanism must also couple powertween the modes.

When the modes are completely phase locked, the fiinside the laser cavity consists of a single pulse, whoseration is equal to the inverse of the total mode-locked bawidth ~this is nominally equal to the bandwidth of the gamedium!. The output of the laser is a train of short pulsespaced in time at the round-trip time of the cavity. A pulseadded to the train every time the intracavity pulse is partiatransmitted through the output coupler.

Dispersion plays an important role in such lasers becait causes the mode spacing to become uneven in frequenthat is the mode spacing at line center~the frequency atwhich the gain is maximum! is different from that away fromline center. The frequencies at which the cavity suppostable Gaussian modes are given by the solutions to

f~vn!52np, ~31!

wheref~v! is the spectral phase accumulated by a wavefrequencyv during a single round trip of the cavity. Therare several sources of phase accumulation. That due toterial propagation is simply

f~v!5v n~v!L

c, ~32!

whereL is the physical path length of material traversed, an(v) the refractive index at frequencyv. L may also varywith frequency because of geometrical dispersion, asscribed in Sec. II. A second source of dispersion is duespectrally dependent mirror reflectivities. Mirrors are typcally chosen to have very flat phase transfer functions, sas is the case with metallic mirrors in the visible and neinfrared, or are engineered to have very specific phase trfer functions, using sophisticated thin-film coating methothat produce dispersion via interference. Another phaseset arises from the Gouy phase shift associated with a fo

If the modes are equally spaced in frequency, that is,solutions$vn% to Eq. ~31! satisfyvn2vn115Dv for all nsuch thatvn is within the operating spectrum of the lasethen the repetition period for the laser will beTrep

52p/Dv. If the spacing is not equal, however, it will bimpossible for a periodic modulation of the cavity round-trloss or phase to effectively lock the phases of all of tmodes across the gain bandwidth.

It is clear from Eq.~31! that the uniformity of the modespacing depends on whetherf~v! is a linear function of fre-quency. Any deviations from linearity cause the mode sping to become unequal. In the limits where the spectral ph

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is a slow function of frequency, and the mode spacing is vsmall compared to the bandwidth of the mode-locked putrain, the mode spacing is approximately

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wheref (1)(vn) is the group delay at mode frequencyvn .As a consequence of this fact, a large component ofdesign of mode-locked oscillators is to find configurationsthe cavity in which the dispersion is very close to lineacross the widest possible range of frequencies.

The first consideration is usually to make the gain mdium as short as possible, since this is the main sourcpositive dispersion in the laser~because of the nonresonainteraction of the laser light with the host medium of tactive ions!. The limit to the thinness is usually set by thdoping level of active ions that can be sustained withcompromising either the gain lifetime or the stimulated emsion cross section through quenching of the excited spopulation via nonradiative transitions.

The next step is to arrange for sufficient negative dispsion, usually in a way that can be adjustedin situ. Whetherone chooses the dispersion to exactly cancel that of themedium or not depends on the details of the mode-lockmechanism.

The shortest pulses are generated using passive mlocking. That is, it is self-action of the pulse inside the lascavity which causes the time-dependent amplitude and pmodulation that are necessary to lock the mode phases.most effective of such mechanisms is self-amplitude molation. This is often implemented using a nonresonant nlinearity ~in which the phase accumulated by the pulsepends on its intensity! together with a linear mechanism thconverts phase to amplitude, such as polarization rotatiospatial filtering.

B. Kerr lens mode locking

An important example is provided by the Kerr-lenmode-locking mechanism that is the basis of sustaimode-locked operation of cw Ti:sapphire lasers.68,69 In thistechnique, the amplitude of a short pulse is modulatedsuch a way that the intense pulses experience less lossthe weaker pulses. The passage of the pulse through the

FIG. 17. Focusing of a plane wave. The propagation time between a rence plane before the focusing element and the focus is a function oradius, leading to different curvatures for the group front and the wave fr


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linear medium~in the case of the Ti:sapphire laser thisusually the sapphire host itself! causes the beam to selfocus. The pulse will pass through a subsequent aperonly if it was intense enough to be focused to a size smathan the aperture. If not, it will experience loss. Since tpulse energy is fixed by the saturation intensity of the gmedium and the repetition rate, then shorter pulses expence lower losses. The laser then finds stable operationthe shortest pulses that can be sustained due to otherstraints, such as dispersion.

In the case of Kerr-lens mode locking, the nonlinear seaction also causes a intensity-~and therefore time-! depen-dent modulation of the pulse phase. When averaged ovepulse train, this is equivalent to the intermodal transferpower, a necessary condition for mode locking. Nonethelethe particular way in which the time dependence of the phoccurs~rising intensity shifts frequencies to the red! meansthat the dispersion must be adjusted in the right way to prerly compress the pulse. Although this is not possibleactly, because the frequency shifts depend on the detaithe pulse intensity, it is generally the case that the phasenear the peak of the pulse is a linear function of time. Trequires linear negative group delay dispersion, so thatblueshifted frequencies and redshifted frequencies are rechronized, leading to a shorter duration pulse. In some cait is possible to operate the laser in a solitonlike regime,which the combination of the nonlinear temporal phase sand quadratic spectral phase interact to form solitons.

Thus the usual situation in a Ti:sapphire laser is thatintracavity dispersion is adjusted to be somewhat negatbut still very close to linear. In order to do this, geometricdispersion is employed to provide adjustable negative lindispersion, and interferometric or material dispersion is uto flatten the remaining higher-order phase. Since both stems are used inside the laser, it is vital that they havetremely low loss. For this reason a two-prism delay lineused to provide the adjustable negative dispersion.prisms are cut so that the light is incident at Brewster’s anat the front surface, and they are used at minimum disperso that the incident and refracted angles are nearly the sfor all wavelengths. This means that the insertion loss canvery small, only a few percent. However, it is the case tmaterials with low dispersion will give low angular dispesion. Therefore the separation of prisms must be largerlower dispersive materials in order to obtain the same qdratic spectral phase. This means that the laser cavitybecome unwieldy if the GDD of the gain medium is tolarge. Alternatively one can use highly dispersive mate

r-het.

FIG. 18. Common layout for a cw mode-locked laser. ML: mode-lockDDL: dispersive delay line, OC: output coupler, HR: high reflector.


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for the prisms, and reduce the cavity to a more practlength. The cost of this is that the prism delay line introduchigher-order spectral phase which must also be compensalong with higher-order material dispersion. Nonetheless,signs of prism delay lines with reduced higher-order dispsion for a given quadratic dispersion have been develoand used successfully in mode-locked lasers.70 As an ex-ample, Lemoff and Barty71 showed that careful selection oprism material allows one to compensate both second-third-order dispersion in the laser cavity. The prism sepation is chosen to balance the second-order dispersion olaser rod. Then, as illustrated in Fig. 19 for a 1 cmTi:sap-phire rod, given a particular center wavelength, the optimprism material is chosen to compensate the third-orderpersion of the rod.

This strategy has been a successful one for the minzation of the pulse durations from Ti:sapphire lasers. It wevident from work with mode-locked dye lasers that copensation of the quadratic spectral phase~the linear GDD!was insufficient to generate pulses with gain bandwidth lited durations.72 The strategy adopted in Ti:sapphire lasehas been to make the material dispersion as small assible, by making the gain medium as short as possible.71 Thismeans that the prism delay line need only compensasmall amount of quadratic dispersion, which can be dowithout introducing significant higher-order dispersion.

Another approach has been to design thin film coatito provide both a high reflectivity across a broad bandwiand a specific phase transfer function.73 This process is rathecomplicated and requires mirrors with several tens of layUsually there is a residual fourth-, fifth-, or sixth-order phathat leads to a small periodic modulation of the phase acthe pulse spectrum. That remaining can be compensateding so-called ‘‘chirped’’ mirrors.74,75 Figure 20~a! illustratesthe concept of a chirped mirror. They are carefully fabricaso that the effective penetration depth for each wavelencan be controlled. This allows one to tailor the dispersintroduced by each mirror. Figure 20~b! shows the standingwave electric field patterns in a double-chirped mirror afunction of wavelength. The use of these mirrors not oleads to the simplest laser cavity with the fewest opticalements, but also allows one to produce pulses in the reof 6 fs.76,77 Measurements of the amplitude and phasethese pulses from a variety of laser systems have shownthe durations appear to be limited by the residual highorder phase in the laser cavity~that is, spectral phase function is not constant across the pulse spectrum, which iscriterion for generating the pulse with the shortest rms dution from any given spectral amplitude!.

C. Fiber lasers

The design of short-pulse fiber lasers is somewhatferent. The cavity consists of several lengths of fiber cnected together, with a mode locker inserted betweenpair of sections. In this case, the transverse mode size infiber is fixed, so that Kerr-lens mode locking cannot be usInstead, a nonlinear rotation of the laser polarizationimplemented using the same physical mechanism—the K


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effect—but now to induce a cross-phase modulation betwthe two polarization modes of the fiber. The polarizationthe light in the fiber is therefore rotated in a way that dpends on its intensity, so that a polarizing element will tramit a shorter pulse than the input pulse.

A further consideration for fiber lasers, however, is ththe pulse inside the laser cavity cannot be too short, othwise the power will be limited by uncompensable timdependent nonlinear phase shifts, or by optical damage. Nthat the mode size in the fiber is a few microns, and the filengths much longer than the material lengths used in blasers. This means that pulse trains with even modest aage intensities can have peak intensities that will induce seral radians or more of phase shift. On the other hand,intracavity laser intensity should be comparable to the saration intensity of the gain medium if the laser is to operaefficiently. The solution to this dilemma is to arrange tcavity dispersion so that the pulse inside the cavitychirped, and consequently stretched in time.78 If the durationis long enough, then the time-averaged laser intensity is laenough to saturate the gain medium without inducing nlinear spectral phase shifts that cannot be compensatedside the laser. A short length of fiber with the opposite sof dispersion to that used inside the laser can be madcompress the pulse outside the laser. Figure 21 showmode-locked fiber laser which uses this technique to prodsubpicosecond pulses.

A second mode of operation is also possible in succonfiguration. Whereas in the mode of operation justscribed amplitude modulation by nonlinear polarization rotion plays the dominant role in mode locking, it is possiblecarefully balance the nonlinear phase shifts accumulatedthe stretched pulse by negative dispersion~in passive sec-tions of the fiber! in such a way as to generate solitons. Tamplitude modulation is then relegated to a secondary rolstabilizing the soliton propagation against the growth of dpersive waves from amplified spontaneous emission fromgain section of the fiber laser. In these lasers, the residhigher order spectral phase limits the pulse duration, anwas found that the minimum duration is not obtained for tbroadest spectrum.79,80

Dispersive-wave fiber lasers typically generate pulsesthe 100 fs region, at wavelengths of 1.5mm, comparable tomodern telecommunications fibers. On the other hand, stonic lasers produce pulses in the picosecond regime, at slar wavelengths.

Soliton lasers can be operated without amplitude stabzation. In this case they may be unstable with respect togrowth of dispersive waves. This instability can be usedmeasure the dispersion of the fiber cavity while the laser ioperation. When a soliton is scattered, it may shed energa dispersive wave. If the scattering is periodic, as it is insa fiber laser cavity, then the dispersive waves from eperiod interfere with each other. For wavelengths whereround-trip phase accumulation is 2p, then dispersive scattered waves on successive round trips interfere consttively. Frequencies for which the phase accumulation is jp will interfere destructively, so that energy scattered frothe soliton on one round trip will be rescattered back into


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soliton on the next. For this reason, as shown in Fig. 22,spectrum of the output pulse train consists of a central brpeak, corresponding to a train of solitons, on which is supimposed a set of peaks whose locations correspond tofrequencies that satisfy the cavity mode condition Eq.~31!.These are the so-called Kelly sidebands of the soliton,their spacing provides a map of the cavity dispersion.81

Finally, the simple pictures of the role of dispersionultrafast mode-locked lasers presented here is not the mcomplete. It turns out to be possible to operate these lastably in regions where the linear dispersion of the cavitynot zero, and the sign of the linear dispersion is positivethat solitonlike pulse formation is not operative. In thecases there is a trade-off between the nonlinear ampliand phase modulations and the intracavity dispersiondefies a simple explanation. However, in all cases the pu

FIG. 19. Cubic spectral phase vs wavelength for a 1 cm Ti:sapphire rod andfor prism pairs made of various materials. By selecting the correct prmaterial, one can compensate the third-order dispersion of the Ti:saprod at a given center wavelength. Reprinted with permission from Ref.


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in these nonsolitonic regimes are longer than in the cawhere some positive nonlinear chirp is compensatednegative group-delay dispersion, so that practically speaksuch lasers are a curiosity.

VI. DISPERSION IN ULTRAFAST AMPLIFICATION

One of the major features of ultrashort optical pulsesthat they have very high peak powers even for a smallergy. For example, a 10 fs pulse from a 100 MHz modlocked oscillator with 10 mW average power, has a pepower of 10 kW. This itself is sufficient peak power to ufor nonlinear optics in certain applications. It is possibhowever, to generate femtosecond pulses with peak powin the Terawatt regime in rather compact systems, and, ggood spatial beam quality, to focus them to intensities inneighborhood of 1019W/cm2 or greater. This makes possibphysics in a regime where the Coulombic binding forcethe valence electron in an atom is smaller than the elecfield of the laser. This highly nonperturbative regime of nolinear optics has blossomed in the past few years becausthe development of high energy, ultrashort pulses frsmall-scale laser systems.

The key development in ultrashort pulse amplifiers isidea of chirped pulse amplification~CPA!.82,83 Illustratedschematically in Fig. 23, CPA requires proper dispersmanagement in order to produce ultrashort, high enepulses.84 In this technique, an ultrashort pulse is first temprally stretched by sending it through a dispersive delay lwith very large ~typically positive! GDD. This procedurelowers the peak intensity so that the pulse can then beplified by several passes through a broadband gain medAfter amplification, the final stage of CPA is to recomprethe pulse in a dispersive delay line with the opposite s

ire.

n

-d

FIG. 20. Double chirped mirror opti-mized for broadband high reflectivityand engineered group delay betwee650 and 1050 nm.~a! Principle of op-eration. ~b! Standing-wave electricfield patterns in a double chirped mirror structure vs wavelength. Reprintewith permission from Ref. 4.


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~typically negative! GDD to that of the stretcher. This technique is used to avoid uncompensable or uncontrollable nlinear effects in the amplifier gain medium. At lower intesities, these nonlinear effects can appear as self-pmodulation, and at higher powers, as multiphoton ionizatof the gain medium itself, which can lead to avalancbreakdown of the amplifier or other optical material and thto permanent optical damage.

An optical amplifier operates most efficiently when tinput fluence is comparable to the saturation fluence,Fs . InTi:sapphire the small stimulated emission cross sectiosmeans thatFs(5\v/s) is rather large—about 1 J/cm2. Ifone wished to amplify directly a 100 fs duration pulse tomJ energy with high efficiency, the pulse at the output ofamplifier would have a peak intensity of more th1015W/cm2, well above the damage threshold of commoptical materials and coatings. However, stretching the pby a factor of 103 or so will reduce the peak intensity to thpoint where optical damage is quite improbable. The amfier will still operate in the saturated regime provided tduration of the stretched pulse remains much shorter thanspontaneous lifetime of the gain. For Ti:sapphire this mea duration of less than 3ms. A larger stretch factor allowsone to operate closer to saturation while avoiding uncontlable nonlinear effects. Recent works85,86have shown that upto 90% of the theoretical maximum quantum efficiency foTi:sapphire amplifier can be achieved by using stretch ftors of 105.

Under these conditions, the effects of nonlinear seaction are also negligible. A common rule of thumb in higpower laser amplifier design is that the peak nonlinear phshift should be less than 1 rad. Although nonlinear phshifts may be treated like any other phase as far as dispecompensation goes,87 they often lead to a complicated spetral phase function for the output field which is usually ncompletely compensable by the sort of dispersive syst

FIG. 21. Chirped pulse mode-locked fiber laser. Output pulses are cpressed using a fiber whose dispersion is opposite in sign to that of theReprinted with permission from Ref. 78.


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ts

one can easily construct. As a consequence, the usual destrategy is to ensure that they are as small as possible.

The most common effect that causes nonlinear phshifts is self-phase modulation. Every optical material psesses a nonlinear refractive index, labeledn2 , that producesa phase accumulation on a beam of intensity,I, passingthrough it. The nonlinear phase shiftfNL(t) is

fNL~ t !52pv

cn2E

0

L

dz I~z,t !, ~34!

where L is the length of the crystal, andn2 for sapphireequals 2.5310216cm2/W. With an intensity of 1010W/cm2,corresponding to the peak of the stretched pulse,fNL(t)<0.2 rad. It is also possible to negate the effects of smamounts of self-phase modulation by introducing nonlinmedia with ann2 of opposite sign to that of the amplifiemedia.88

The key attributes of the stretcher and compressor uin any CPA system are:

~i! ~output system! high-power and high-energy handlincapability,

~ii ! adjustable spectral phase to some reasonable oover some reasonable range.

These attributes affect not only the specifics of thesign, but also the physical size of the systems. The mdispersion required and the larger pulse energies to benipulated, the larger the optical elements must be.

As the labels below the diagram in Fig. 23 indicate, tideal situation is that the compressor have exactly the opsite second-order dispersion as that of the combination ofstretcher and amplifier. The stretcher is usually chosenhave positive dispersion so that there is no danger thatpulse be compressed in the amplifier, which itself has pdominantly positive dispersion due to the host lattice. Inder to achieve the required stretch factors, GDDs on theder of 104– 106 fs2 are required.

The largest dispersions for the shortest total path lenare found in angularly dispersive systems. In particular,double-grating arrangement analyzed by Treacy in 19689

and discussed in Sec. II C, provides a prototype for the copressor portion of a CPA system. The throughput of the copressor is limited by the number of interfaces that the putraverses. For example, since there are four reflections off

-er.

FIG. 22. Kelly sidebands of a soliton. Linear~black! and logarithmic~gray!plots of the spectrum of a soliton laser operated without amplitude stazation. Reprinted with permission from Ref. 81.


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gratings, the typical transmission of this system is ab50%–70%. An important issue with this system is determing the size of the optics, in particular the gratings. The sof the first grating depends on the output pulse peak insity, which can damage the optic severely via multiphoionization. The spatial extent of the dispersed amplifipulse’s spectrum determines the size of the second graThe size of the dispersed spectrum on the second gratinproportional to the angular dispersion of the first grating]a/]v, the pulse spectral bandwidthDv, the perpendicular distance between the gratingsD, and the angle of diffractionfrom the gratinga~v!,

S}]a

]vDv

D

cos~a~v!!. ~35!

Using Eqs.~20! and ~21!, Eq. ~35! can be written as

S}f~2!Dvd cos~a!

l, ~36!

whered is groove spacing of the first grating.As discussed previously one of the important parame

in the design of a CPA system is the duration of the amfied stretched pulse. For large dispersions, using Eq.~7!, theduration of the stretched pulse can be related to the origpulse duration bytstr}f (2)/torig , where for a transform-limited pulse,torig}1/Dv. Thus, for a given stretched pulsduration and central wavelength, the size of the second ging is proportional to the grating grove separation anddiffraction angle,

S}tstr

d cos~a!

l. ~37!

Note that this expression does not include the finite sizethe input beam.

The corresponding stretcher, matched to all ordersdispersion~in the limit of paraxial imaging! to the Treacycompressor, is the design of Martinez.17 The throughput ofthis system is typically reduced because of the larger numof optical elements, so its positioning before the amplifieconvenient in this regard also. The presence of monocmatic aberrations in the imaging system can cause amatch between the cubic and quartic spectral phase ofstretcher and compressor, which limits the fidelity of purecompression.

However, if one includes the dispersion introducedthe amplification process, it is then not necessarily desirafor the stretcher and compressor to be completely matcheall orders of dispersion. A typical 20-pass regenerative aplifier contains a total of about 1 m of optical material, andintroduces GDD of about 39 000 fs2, cubic phase of 38 000

FIG. 23. Schematic representation of chirped pulse amplification.


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fs3, and quartic phase of approximately222 000 fs4. A mul-tipass amplifier, having fewer passes and less optical mrial, introduces about half the spectral phase of a regeneraamplifier. In order to compensate for the additional quadraphase, one can mismatch the stretcher and compressosimply changing the distance between the two compresgratings@D in Eq. ~19!#. Figures 24~a! and 24~b! show simu-lations of a 50 fs Gaussian pulse that has been amplifiedregenerative amplifier using CPA. In this example, the copressor grating separation was adjusted so that thesecond-order dispersion of the entire CPA system is zFrom Fig. 24, it is clear that merely adjusting the compresgrating separation is not sufficient to achieve a short amfied pulse. This is because a change in grating separationintroduces additional third- and fourth-order spectral phaThe sign of the additional cubic is the same as that ofthird-order phase introduced by the amplifier. In additiobecause optical materials typically have very low fourorder dispersion, amplifiers contribute only very smamounts of quartic phase. Therefore, the central problemthis: If one adjusts the compressor to compensate foradditional GDD of the amplifier, then excess higher ordphase is introduced which severely limits the fidelity of trecompressed, amplified pulse.

In many cases,90 by adjusting the compressor gratinincidence angle and separation, one can minimize bothsecond- and third-order dispersion of the CPA system. Tresulting amplified pulse~for a 50 fs Gaussian input pulse! isshown in Figs. 24~c! and 24~d!. Although the recompressionis much improved, the amplified pulse is still not transforlimited. This is due to uncompensated higher order specphase. For desired amplified pulse widths below 50 fs,must also completely compensate fourth-order dispersion

There have been many other solutions to the problemhigher order phase compensation in CPA systems. Oneuse a separate optical system to compensate the higher-terms. For example, the cubic dispersion for prism pairs isthe opposite sign to that of gratings pairs, even when bhave the same sign of GDD, so that these elements togecan be used to adjust both second and third-order dispersThis idea was first applied to pulse compression in genetion of 6 fs pulses from a dye laser system.72 This concepthas been extended by Kane and Squier90 to modern CPAsystems by using a monolithic grating and prism combition ~with the gratings written on the surface of the prism!to compensate cubic phase. In fact, a compressor of thissign allows one to use a much simpler stretcher—a piecoptical fiber— and still obtain good recompression for pulson the order of 100 fs.

A different approach to this problem is to use thstretcher to introduce the compensating higher order phterms, rather than the compressor. This is somewhat eabecause the larger number of elements in the system omore degrees of freedom. As an example, Whiteet al.91 usedan air spaced doublet for the imaging lens. This allowedlateral and longitudinal positions of the lens elements toadjusted to accommodate various amounts of cubic and qtic phase. They therefore modify the spectral phase by deerately introducing several monochromatic aberrations


ofgs.

tortitionhase,


FIG. 24. A 50 fs pulse~dashed! is amplified in a CPA system to a high energy pulse~solid line!. The pulse from the oscillator is first stretched by a pair1200 groove/mm gratings, then amplified in a regenerative amplifier~12 round trips!, and finally compressed by another pair of 1200 groove/mm gratinWhen the grating separation in the compressor is adjusted to compensate only for the additional quadratic phase introduced by the amplifier, disons dueto the remaining third-order dispersion are significant, as shown on a linear@~a!# and logarithmic@~b!# scale. When the grating incidence angle and separaare adjusted to minimize the total quadratic and cubic phase of the CPA system, the remaining pulse distortions are due to residual higher order pas canbe seen on a linear@~c!# and logarithmic@~d!# scale.

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to decentering and defocus~such as on-axis astigmatism!.Additional methods for higher order phase compensa

include placing an additional optical element betweentwo gratings of the compressor, which has been showncompensate primarily cubic phase,92 and using gratings withspatially nonuniform groove spacing which could, in specconfigurations, be used to induce purely linear group dedispersion of positive or negative sign.93,94

The most comprehensive approach to the designstretchers and compressors for ultrashort pulse CPA sysis that of Lemoff and Barty.95 They deliberately set out toconfigure the grating angles, separations, and the aberrain the stretcher in just the right way to provide exactly tcorrect amounts of cubic and quartic phases. The penalan added complexity in alignment because of the grenumber of optical elements, but this technique has been uon a CPA system producing sub-20 fs, 100 TW pulses.85

In previous examples, the fact that aberrationspresent in the imaging system of the stretcher was usecompensate higher order dispersion. However there arevantages to aberration-free stretcher designs. It is wknown that imaging systems whose surfaces have a comcenter of curvature have no monochromatic aberrations toorders. An important system of this sort for these purposethe Offner triplet stretcher.96 This is an all-reflective designin which the imaging system consists of a convex andconcave mirror used in an off-axis configuration with objeand image planes mutually parallel and both containingcenter of curvature of the mirrors, as shown in Fig. 25. Tcomplete absence of aberrations in this design allowedet al. to stretch a 30 fs pulse by a factor of 104 and to re-compress it completely.97 An important practical feature othis design is that it is easy to align, and rather tolerantslight misalignments. This design has been used to prod


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30 TW, sub-30 fs pulses from a multipass amplifier CPsystem.98

While the fact that the Offner triplet stretcher is aberrtion free means that one can stretch and compress a pwith excellent fidelity, it also means that one cannot useaberrations of a standard stretcher to introduce fourth-odispersion. If the amplifier of a CPA system requires a lotmaterial or additional passes through the gain medium~as ina regenerative amplifier!, then when one adjusts the compressor to compensate for the extra quadratic and cphase, there is no way to compensate the additional fouorder dispersion. A solution to this problem, introducedSquier and Kane,99 is to use a compressor whose gratinhave a different groove spacing than that of the Offner tripstretcher. In fact, it works out that one can use a higgroove density grating in the compressor, leading to bediffraction efficiency, and still compensate up to fourth-ordphase. This technique has been used to produce 20 mJ,pulses at a 1 kHz repetition rate using a CPA system withregenerative preamplifier.100 Because of their simplicity andflexibility, these solutions have been implemented in macommercial CPA systems.

As these systems have a limited number of parameteris not possible to exactly get a flat spectral phase overcomplete bandwidth of the output pulse from a CPA systewhich would lead to the optimal pulse in terms of duratioFor example, the exact cancellation of the second- and thorder dispersion for the output pulse might leave enoufourth-order phase to deteriorate the quality of the pulThis then calls for a global theoretical optimization of thparameters of the system~e.g., parameters of the stretchand compressor!, taking as a target a merit function for thoutput pulse~e.g., its duration!.99,101Also, this procedure al-lows one to quantify the sensitivity of the optimal set


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parameters regarding slight misalignments, which is anportant practical point for the experimental implementatio

It must be noted that in all these systems, the surfquality of the optical elements plays a key role: Wherethe optical spectrum is spatially dispersed, any spatial phfeature will induce a spectral phase modulation. This effcan easily be experimentally demonstrated, e.g., when msuring the dispersion of a stretcher set in a zero disperposition. These high order or periodiclike induced featuusually cannot be compensated by the plain action ofcompressor. They have been shown as detrimental topulse quality at the output of CPA systems.102

Also, the use of pulse shaping devices, which are tgreat extent capable of an arbitrary control of the specphase, on CPA systems allows one to compensate thesidual spectral phase of the output pulse. These devicesplaced before the amplifier in order to prevent damage tooptical modulator~for high-resolution control of the spectraphase, the spectral components need to be focused on tmodulator!. This technique should be able to correct uncopensated spectral phase, either originating from theoremismatches in the complete CPA system or from experimtal mismatches~quality of the optical elements or day-to-damisalignments!.

The ability to stretch a short pulse, amplify it withoudetrimental nonlinear effects, and recompress the highergy pulse to a short high-power pulse with a good contis now routine using the optical systems presented prously. Increasing the number of parameters of one of thsystems, or the number of independent systems, allowbetter control of the pulse quality at the output of the CPsystem. But this also usually brings an increased compleor an overall low energy transmission. In this sense, a pshaper is perhaps the most radical solution, because alarge number of parameters can be independently set. Wever the diagnostic used at the end of the system, thesesatile solutions for spectral phase control, along with goquality conventional stretchers and compressors, seem ta reasonable direction for the control of dispersion in futCPA sources.

VII. DISPERSION IN ULTRAFAST NONLINEAROPTICS

A. Interaction of short pulses

The role of dispersion in nonlinear optics takes seveforms, mainly relating to over what durations and palengths the nonlinear interaction between pulses canmaintained. Consider, for example, the frequency conversof an intense optical pulse by its interaction with a weapulse of different frequency in a nonlinear medium,shown in Fig. 26. The efficiency of the transfer of enerfrom the stronger to the weaker pulse depends on a lanumber of parameters: the length of the nonlinear mediwhether the process involves phase matching, the diffegroup velocities of the two pulses; and the degree to wheach pulse is temporally broadened due to quadratic spephase accumulation as it propagates through the medium


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These parameters affect the nonlinear interaction infollowing ways. The most efficient interaction occurs withe longest interaction path length, so the medium itsshould be as long as possible. On the other hand, the auseable length of the medium is set by the other parameAssuming that phase matching can be achieved~if it is nec-essary!, then a difference in the group velocity of the twpulses causes them to walk away from one another, sothey no longer interact.

The time taken for a pulse propagating at the velocityv i

to traverse a distanceLe is

t i5Le

v i; i P~1,2!. ~38!

The difference in traversal times istm , where

tm5LeU 1

v12

1

v2U. ~39!

If this time tm is less than the input pulse durationt,then the pulses will still overlap at the output of the mediuSo the effective interaction length is then approximately

Le5tS v2

Dv D , ~40!

where Dv is the difference in group velocity andv is thegeometrical mean group velocity of the two waves. Thetual interaction length is thereforeLact5min(Le,L), whereLis the physical length of the medium. Group velocity dispsion ~or GDD! affects the interaction in another way. Nonzero GDD causes the pulse to be temporally stretchTherefore its peak intensity will decrease, and the nonlininteraction will become much less efficient. Moreover, tpresence of dispersion will cause the duration of the outpulse to be longer than that of the input, in general, for breasons pertaining to the nonlinear interaction, as discushere, and the linear propagation. It is therefore criticalunderstand the compromise between these various paeters that leads to the optimal nonlinear interaction.

FIG. 25. All reflective Offner triplet stretcher.


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The field of ultrafast nonlinear optics is too large toeffectively surveyed here, even if that were warranted bysubject matter. Therefore we shall choose a few exampleillustrate the ways in which the important parameters affthe nonlinear mixing of ultrashort optical pulses.

B. Phase-matched processes

All parametric processes require a phase-matched inaction. A simplified description of this will suffice for oupurposes, The reader is referred to one of the many textboon nonlinear optics for a more rigorous description. Moover we restrict ourselves to the two simplest situations,generate and nondegenerate three-wave mixing.

The nonlinear polarization source term generated bywaves of frequencyv1 and v2 at positionz in a mediumwith a x (2) nonlinearity is~for sum frequency generation, ithe weakly interacting limit!

P~v,z!5x~2!~v5v11v2 ;v1 ,v2!E~v1 ;0!

3eik~v1!zE~v2,0!eik~v2!z, ~41!

whereE(v,0) is the field at the input face of the medium,zis the propagation distance,k(v) is the wave number of thefield at frequencyv, The effectiveness of the nonlinear interaction is predicated on this term having the same stiotemporal behavior as a field at the same frequency progating through the medium. It is therefore common to facout the propagation constant of the generated wave and wwith the envelope of the field.

The field generated by the interaction of two short pulis found by integrating this source polarization over tlength of the nonlinear medium, and all input frequencies

E~v,L !5E dv1 x~2!~v;v1 ,v2v1!E~v1,0!E~v2v1,0!

3E0

L

dz ei @k~v1!1k~v2v1!2k~v!#z, ~42!

where we have taken into account both the conservatioenergy, so thatv11v25v, and the propagation constantthe electric field at frequencyv. The integral overz on theright-hand side of Eq.~42! is called the phase-matching termand contains information on the propagation parameterscussed previously.

The phase-matching term has the form

A~L !5E0

L

dzeiDkz5LeiDkL/2sin~DkL/2!

DkL/2, ~43!

where

Dk~v,v1!5k~v1!1k~v2v1!2k~v! ~44!

is called the wave vector mismatch.The maximum of the phase-matching function occ

when its argument is zero, so that the efficiency of the nlinear process is highest when the wave vector mismatczero over the entire range of frequencies contained ininput pulse.


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In order to understand the effects of dispersion onnonlinear process, it is convenient to expand the wave vemismatch about points near the center of the input and ouspectra, sayv i and vo . Let the expansion frequency varables beD5v12v i and d5v2vo , respectively. Then,taking into account energy conservation, the expansionDk to second order in frequency mismatches is

Dk~v,v1!5@k~v i !1k~vo2v1!2k~vo!#

1F ]k

]vUvo2v i

2]k

]vUvo

Gd1F ]k

]vUv1

2]k

]vUv22v1

G3D1

1

2 F ]2k

]v2Uvo2v i

~d2D!2

1]2k

]v2Uv i

D22]2k

]v2Uvo

d2G , ~45!

where ]k/]vuvois proportional to the group delay of th

radiation at frequencyvo and]2k/]v2uvois the group delay

dispersion.It is now easy to see the role of dispersion in determ

ing the efficiency of the nonlinear process. If the first termsquare brackets on the right-hand side of Eq.~45! is set tozero, then the process is said to be phase matched fofrequenciesv i andvo . The phase mismatch is still nonzerwhen this is done, however, because in general the grdelay and GDD mismatch terms~the second and third lineon the right-hand side of Eq.~45!, respectively! are nonzero.It is clear that the process will be more efficient provided tgroup velocities of all three waves are the same. In the cwhere the group velocity of the input waves at the frequcies v i and vo2v i are the same, the second term in tsecond line of Eq.~45! vanishes. This means the process wbe phase matched over a broad range of frequencies.

Unfortunately, there is no such simple cancellationthe GDD terms. It is straightforward to show that the onway for the third line in Eq.~45! to vanish is for the groupdelay dispersions of all three frequencies to be zero. Timplies that both input pulse and generated pulses dobroaden temporally as they propagate through the mediand their peak intensity does not therefore diminish.

A common method of adjusting the wave numbers ofinteracting waves is to use a birefringent crystal, alongaxes of which the refractive index differs. An example of this shown schematically in Fig. 27. Figure 27~a! indicates thesymmetry axisc of a uniaxial crystal, and the direction of thwave vector of the lightk(v i). The axes orthogonal to thwave vector are the two different polarizations that expe

FIG. 26. Nonlinear interactions of short pulses with different group veloties.


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ence the largest index difference. The polarization lyingthe plane containing the vectorc and k(v i) is called theextraordinary wave and the polarization orthogonal to tplane is the ordinary wave. The refractive index of the etraordinary wave can be varied by changing the angles be-tween thec axis and the wave vector. The dependence ofrefractive index on frequency in this configuration is shoin Fig. 27~b!. The two curves represent the ordinary aextraordinary wave refractive indices, the latter curve gosmoothly into the former as the angles is changed. Thedashed lines represent the phase-matched conditionsecond-harmonic generation in a type-I geometry. In tcasevo52v i , and it can be seen from Eq.~44! that thisimplies k(vo)52k(v i), or n(vo)5n(v i).

The next step is to ensure that the group delays ofinput and generated waves are equal. This can be doncareful choice of crystal parameters.103 Alternatively, thesame effect can be achieved using the angular dispersiothe birefringent crystal to match the two velocities. For eample, the method of Radzewiczet al.104 sets two inputwaves at different angles. The projection of the two wavectors onto a vector that bisects them is adjusted by ching the angle between them. The phase-matching condcan be set by changing the crystal orientation, so that thare two degrees of freedom. These can be used to simneously satisfy the phase- and group-velocity matching cditions.

A second approach to the problem of broadband phmatching is to angularly multiplex the input wave so thdifferent frequencies are incident on the nonlinear crystajust the right angle to satisfy the phase-matching conditiThen each frequency is doubled most efficiently, andresulting second-harmonic radiation can be reassembleda pulse by angularly demultiplexing. The mux–demux steare implemented using two prisms and an imaging larrangement.105

A more complicated situation arises in the case of pametric downconversion, in which a pump pulse of high frquency is converted into two pulses of lower frequencalled the signal and idler waves, as shown in Fig. 28. Tprocess is used to generate ultrashort optical pulses innear and midinfrared region of the spectrum. In this sittion, it is necessary to match all three group velocitiesclosely as possible. For type-I downconversion in the degerate limit ~when the signal and idler have the same fquency, exactly half that of the pump! the bandwidth of theprocess is often very large, since the group velocity mmatch goes almost to zero. In fact, for certain crystpumped near 400 nm, it is possible to get a phase-matcbandwidth of several hundred nanometers,106,107and this hasbeen used to generate tunable pulses of less than 10 fstion from a seeded parametric amplifier.

This process also provides enough gain that it is possto build a femtosecond parametric oscillator. A typical arangement for such a device in shown in Fig. 29. It consof a parametric downconverter pumped by a train of pulfrom a model-locked oscillator. The signal pulses from tdownconverter are resonated in an optical cavity, arrangethat the round-trip time for the signal pulses equals the


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riod of the pump pulses. In this way the energy of the sigpulse builds up and brings the oscillator above threshold.intriguing feature of this synchronous pumping is that tdispersion in the cavity causes tuning of the wavelengththe oscillator output. Recall that the round-trip period of toptical cavity is set by both its physical length and the dpersion of the intracavity material. If the GDD dependswavelength, then the wavelength of the signal~and thereforethe idler, which is not resonant with the cavity! changes sothat the total round-trip timet rt5(2L/c)1f (1)(vsignal)2f (1)(vpump) is equal to the periodT of the pump laserpulse train. In this way, small variations in the pump perican be accommodated, or even used to tune the OPO. Inthis mechanism of dispersive tuning can be developed inmeans to stabilize the lengths of the pump and OPO caviby simply ensuring that the spectrum of the OPO remaina fixed optical frequency.

The output signal pulse duration is usually limited by tGDD at the signal wavelength. This can be compensatethe same way as in lasers, so that such oscillators canerate pulses of 40 fs or less duration in the near infrared,wavelength of 1.3mm or so.108

For generating wavelengths in the midinfrared, at walengths in the range from 2 to 10 microns it is commontake the signal and idler outputs of a parametric downcversion source, and to generate their difference frequeThe optimal method for this depends greatly on the mateand pump wavelengths available. Nonetheless, the aments presented previously hold in this situation also.example, pumping a lithium niobate (LiNbO3) crystal at awavelength of 767 nm to obtain idler wavelengths in the 2mm region limits the effective interaction length to 1 mmso, leading to low output power.108 On the other hand, asilver thio gallate (AgGaS2) crystal pumped at 1–3mm hasall three waves with nearly the same group velocity in tfrequency range 2–12mm, ~with them exactly equal at 9.5mm! so that a long crystal can be used to obtain short puwith high output powers.109 Plots of the relevant group delamismatches for this particular process are shown in Fig.

Up to this point we have discussed only the regimeweak nonlinear interactions, in which the pump pulse isdepleted. In the regime of strong interaction, the situationmuch more complicated, and no simple rules can be givAs an example of the modified role of dispersion in thesituations, we consider the method of Umbrasaset al.,110 inthe generation of second-harmonic radiation of short optpulses, and their simultaneous compression. The experimtal arrangement is shown schematically in Fig. 31~a!. Theinput waves have orthogonal polarization and are incidenthe nonlinear crystal in a collinear fashion, but delayed wrespect to one another. The crystal is configured for typphase matching for frequency doubling. In this case,group velocities of theo- and e-wave pulses are not thsame, and they propagate through the crystal so that thelay between them at the output is exactly the negativewhat it was at the input—the fundamental pulses ha‘‘walked through’’ each other inside the crystal, as shownFig. 31~b!. If, however, the group velocity of the secondharmonic wavelength is close to the mean velocity of the t


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input waves, and the power is sufficient to deplete the fdamental pulses during their interaction, then the genersecond-harmonic pulse will be quite short. This is becaenergy is fed into it only from a small region where the twfundamental pulses overlap, at which point they are depleIn this way pulses of about 0.5 ps have been generatedfrequency doubling pulses of several tens of picosecondduration.

Another way to adjust the GDD to phase match a nlinear interaction is to use waveguide dispersion to balathe dispersion of the material. This approach has beenticularly successful in application to the generation of hiharmonics of infrared pulses. Radiation at frequencies cosponding to the 150th or so harmonic of the input frequecan be generated when atoms are illuminated with an opfield whose strength is comparable to that of the Coulombinding field of the outer-shell electron of an atom.111 In thiscase of extreme nonlinear optics, the efficiency of the pcess depends on both the brevity of the input pulse andthe proper phase matching between the fundamental andmonic radiation. Since the dilute atomic gas in which thprocess takes place has positive dispersion, some negdispersion is required to improve the efficiency. This candone by situating the gas in a hollow-core waveguide,diameter of which is such that the waveguide dispersion ithe correct sign and magnitude to balance that ofmaterial.112 This type of source holds promise for ultrafatime-resolved soft-x-ray spectroscopy.

C. Quasi-phase-matched processes

The previous techniques enhance the efficiency ofnonlinear process by using materials and configurati

FIG. 27. Angle phase matching for second-harmonic generation in alinear type I crystal. A proper choice of the phase-matching angle~a! allowsthe equality of the index of the fundamental along the ordinary axissecond-harmonic frequency along the extraordinary axis, leading to thecellation of phase mismatch~b!.


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where the phase mismatch between the interacting wavezero over a broad range of frequencies, thus maximizingcoherence length for a wide wavelength range. The tenique of ‘‘quasi phase matching’’ relies on the engineeriof the nonlinear medium itself in order to cancel the destrtive interference of the interacting waves.113,114This can beachieved by making the nonlinear susceptibility a functionthe position in the nonlinear material. In bulk material, whthe phase mismatch is equal to zero, the energy in the cverted field grows quadratically with the length of interation. In other cases, the energy increases for values ofL inthe range@0,Lc#, whereLc5p/Dk is the coherence lengthIt reaches a maximum atL5Lc and decreases over the ran@Lc,2Lc#, finally going back to zero because of reconversifrom the second-harmonic field to the fundamental fieConsider a material with a nonlinear susceptibility equalx (2)(v;v1 ,v2v1) for values ofz in the range@0,Lc# andequal to2x (2)(v;v1 ,v2v1) in the range@Lc,2Lc#. Forthis material, the product of the nonlinear susceptibility wthe phase-matching term after a distanceL5Lc1L8 can bewritten as

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FIG. 28. Optical parametric generation of short pulses of different grovelocities.

FIG. 29. Diagram of a synchronous pumped femtosecond optical paramoscillator.


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The change in the sign of the susceptibility in the ran@Lc,2Lc# precludes the backconversion from the secoharmonic to the fundamental field. The efficiency of secoharmonic generation in this range is then the same as inprevious range@0,Lc#. When propagating over multiple periods of such a structure, the field then grows linearly wthe propagation length, and therefore the energy grows qdratically as in the case of a perfectly phase-matched cfiguration ~Fig. 32!. This leads to very efficient frequencmixing arrangements for sum frequency generation and pmetric downconversion. An interesting aspect of quasi phmatching is that as exact phase matching is not requiredpolarization of the interacting waves can be chosen to tadvantage of the highest nonlinear coefficients in the mrial. A complete treatment of such a structure with periodity L shows that the phase-matching factor for the convsion efficiency has the same form as in a bulk material,with a phase mismatch shifted by the value of the wavector of the periodic structureKm52pm/L, chosen closeto Dkbulk :

Dk5Dkbulk2Km . ~48!

FIG. 30. Group delay mismatch between signal and difference wavele~solid curve!, signal and idler~dashed curved!, and idler and differencefrequency~dashed-dotted curve! for difference frequency generation asfunction of the generated wavelength in the case of LiNbO3 pumped at 767nm ~a! and AgGaS2 pumped between 1 and 3mm ~b!. In the latter case, allthree waves travel at nearly the same speed for difference frequencyeration at 9.5mm. Reprinted with permission from Ref. 109.


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There is thus no specific improvement of the specacceptance as far as a periodic structure is concerned. Hever, the possibility to use aperiodic structures enhancesspectral acceptance, which is particularly useful for ultrafoptics. In this case, local sections of the structure canengineered in order to upconvert different portions of tbroad spectrum by the choice of the period. As the upcverted spectral components travel with their own grouplocity, such a structure modifies the group delay in the uconverted pulse. It has been shown that these structureshave the same effect as the compressor at the endchirped pulse amplification system by using a chirped perin the quasi-phase-matched device, keeping in mind thatmean frequency of the compressed pulse is twice the mfrequency of the amplified pulse.115,116Other structures haveproduced trains of upconverted pulses from a single funmental pulse.117

D. Non-phase-matched processes

Nonlinear processes that do not require phase matcor are automatically phase matched are nonetheless affeby dispersion. Examples of such processes are the self-cross-action effects induced by the nonlinear refractive inand gain processes such as stimulated Raman scatteringeffects of dispersion are quite similar to those in the phamatched case—a mismatch of the group velocity betwthe interacting waves reduces the efficiency of the procesdoes nonzero GDD.

For example, it has already been mentioned thatcombination of positive nonlinear refractive index and negtive GDD occurring in the same system can be used to gerate optical solitons. These are optical pulses of particshape, whose energy is specified by the GDD of the systThis phenomenon has been observed in waveguide geetries by many scientists, and is now the basis of high-sp

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FIG. 31. Experimental arrangement for simultaneous high-efficiencyquency doubling and pulse compression in a type II crystal.~a! Relativepositions of fundamental input and output pulses.~b! During the propaga-tion, the fundamental pulses walk through each other because of diffegroup delays along each axis of the birefringent nonlinear crystal.


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time-division multiplexed optical telecommunications tebeds.

A critical issue in such systems is maintaining the bance of self-phase modulation and GDD even in the preseof imperfections in the fibers. If the two effects do not baance exactly, then the soliton will cease to be such, andpulse will gradually disperse. Since all fibers have at lelosses due to Rayleigh scattering in the fiber core, it mighexpected that the range of a soliton communicationswould be restricted. In fact, clever management of dispersallows one to circumvent this problem.118 For example, anarrangement such as that shown in Fig. 33~a!, where theGDD of the fiber is a monotonically decreasing functiondistance, allows a soliton to be sustained even in the pence of losses. This is because the decreasing energy osoliton as it undergoes scattering causes smaller self-pmodulation at the peak of the pulse, and thus lower GDDrequired to maintain the solitonic shape. The balancetween these effects is maintained as the pulse propagatetailoring the dispersion in the correct way to match tlosses. Examples of the output of the fiber~using a 1.5 psinput pulse! for uncompensated and compensated fibersshown are shown in Fig. 33~b!. In the case of the uncompensated fiber, the increased pulse duration indicates thatpulse became dispersed rather than remaining a solitonpropagating a fraction of the distance along the fiber.

A similar argument holds for spatiotemporal solitonsbulk materials. Here there is no material-imposed waveguing, and a spatial soliton is formed by a balance betwself-focusing and diffraction. The coupling of spatial atemporal degrees of freedom via these mechanisms stabthe soliton, leading to a so-called optical bullet that propgates as a localized excitation in space and time. A tempsoliton requires that the dispersion and nonlinear phasebalance one another. This is not possible in a medium wpositive nonlinear refractive index and positive dispersionorder to overcome this effect, Liuet al. used a nonresonancascadedx (2) process to synthesize a nonlinearity of the crect sign.119 This process effectively concatenates an upcversion and a downconversion process in sequence, inrefringent medium far from phase matching. The interfereof the fundamental wave with the light that undergoes cversion to the second harmonic and back again causeintensity-dependent phase shift that mimics a nonlinearfractive index. The effectiveness of this process is limitedthe group velocity mismatch between the fundamentalthe second harmonic. Because the material has positivepersion, it is necessary to ‘‘precompensate’’ by slighnegatively chirping the pulse before entering the nonlinmedium. This is done using angular dispersion, by placthe nonlinear crystal at the focal plane of a dispersive deline.

The interplay between dispersion and nonlinear phshifts in the propagation of ultrashort pulses is currentlysubject of intense study, and a comprehensive survey ispossible here. In many cases simple pictures have beenveloped onlyex post facto, and usually they provide only apartial description of the effect. Typically, a complete nmerical solution of the problem is required. Nonethele


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great progress has been made in several areas, includingeneration of continuum radiation in bulk120,121 andwaveguided geometries,122,123 as well as four-wave mixingusing third-order and cascaded second-ornonlinearities.124

A third example is provided by group velocity matchinfor effective stimulated scattering. The method to be dscribed was originally developed for traveling wave amplers with large bandwidths.

In Raman scattering, an intense pump pulse scattersinto a pulse of lower frequency, the Stokes pulse, leavingmedium in an excited vibrational, rotational, or electronstate. The process can become stimulated, and this hasto the development of Raman amplifiers and frequenshifters for pulses from the nanosecond to the femtosecregime. The Raman cross section is larger the closerpump and Stokes frequencies are to resonance. But thismeans that there is a difference in group delay betweentwo pulses. As a consequence, the effective length ofinteraction region may be shorter than its physical lengBecause the material in which Raman scattering occurusually not birefringent, it is not possible to use angular dpersion to adjust the two group velocities of pump aStokes pulses to be the same. One method that hasadopted to overcome this problem is to configure the pubeam so it has a particular geometry, that of a ‘‘Besbeam.’’

If a planar pump beam is incident on an axicon~a coni-cal refractive element!, then the beam is deviated from itinput direction in such a way to form a region along the aof the axicon in which the intensity is very high. Moreovthe region of high intensity moves along the axis withvelocity that depends on the cone anglew of the focusedpump light. The velocity of the illuminated region along thcone axis is approximatelyc/n cos(w), wheren is the refrac-tive index of the medium. This allows the possibility omatching the speed of the moving pumped region tospeed of the Stokes pulse. Even though the direction ofpump light itself is at an anglew to the axis, phase matchinis automatically satisfied for Stokes scattering, and therethis angle is a free parameter that can be adjusted to mthe speed of the gain region along the axis to the grovelocity at the Stokes wavelength.125

Raman conversion efficiencies of up to 1% for 100pulses in liquids have been observed using this techniq

FIG. 32. Energy of the harmonic field in the case of perfect phase matcin a uniform material~a!, nonphase matched interaction~c!, and quasi-phase-matched interaction when reversing the sign of the nonlinear sutibility every coherence length~b!. Thex axis is in units of coherence lengthfor the non-phase-matched process.


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Normally in liquids the competing process of four-wamixing dominates frequency conversion, but this processquires phase matching, and the situation with Bessel bexcitation precludes such.

VIII. DISCUSSION

It is critical to pay attention to dispersion when dealiwith ultrashort optical pulses, since it can severely compmise the focused intensity and the pulse duration of beama target. Even more important, it is crucial to understand hto manipulate dispersion in order to compensate for therimental effects of too much or too little in an optical systeThis, in turn, demands a set of methods for measuringpersion.

A reasonable rule of thumb, applicable in most caseslinear pulse propagation, and to a lesser extent in weanonlinear situations, is that pulses with bandwidths of lthan 1 nm and durations of greater than 1 ps are not maffected by dispersion~at least in transparent media, far froabsorption resonances!. On the other hand, pulses witgreater than 1 nm of bandwidth or durations of less than 1may be affected quite dramatically by dispersion. The magement of dispersion becomes increasingly more imporas the brevity of the pulses decreases.

The options available for controlling dispersion depecritically on the system in which the dispersion must be maged. The trade-off is usually between a low-loss syswith a small range of dispersion adjustment or a higher lsystem with a much wider range of adjustment. The formstrategy is nearly always adopted with mode-lockoscillator-based experiments, whereas the latter is usedamplified systems. Material dispersion in the visible is

FIG. 33. Engineered dispersion fiber for improvement of the propagadistance of a soliton.~a! Dispersion as a function of the distance in a costant dispersion fiber~CDF! and decreasing dispersion fiber~DDF!. ~b! Out-put duration for a 1.5 ps input soliton propagating in 40 km of CDF~top!and DDF~bottom!. Reprinted with permission from Ref. 118.


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ways positive~except in amplifiers, where it can be weaknegative!. In combination with other elements, angular dpersion may be used to obtain controllable negative or ptive group delay dispersion. Interferometric thin-film coaings can provide dispersion of arbitrary sign, but with velimited adjustability, so they are effective only as one-odevices, for particular system configurations. Nonethelethey are very compact and have low loss, in contrastgrating-based dispersive systems.

Dispersion plays a central role in nonlinear optics. Athough it is difficult to make any completely general stament about how it affects the details of the nonlinear proceit is clear that all wave mixing phenomena involving pulswith disjoint spectra will depend upon whether the pulswalk away from each other, as well as whether their peintensity is reduced upon propagation. The subtle interpof dispersion, diffraction, and nonlinearity will undoubtedprovide a rich field of research for many years to come.

It is safe to predict that dispersion management will bethe forefront of advances in ultrafast science and technolo

ACKNOWLEDGMENTS

The authors are grateful for numerous conversatiwith and suggestions from many persons. Of particular hin the preparation of material for this article were: ChBarty, Eric Buckland, Irl Duling, Turan Erdogan, Chris Iaconis, Jim Kafka, Wayne Knox, Bill White, and Frank WisWe are especially grateful to Chris Barty for helpful comments on the manuscript. The work was supported byNational Science Foundation.

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REVIEW ARTICLE The role of dispersion in ultrafast opticslochi/ultra/Zusatz... · solids, the...

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