energy-ﬂux characterization of conical and space-time

PHYSICAL REVIEW A 81, 023810 (2010)

Energy-flux characterization of conical and space-time coupled wave packets

A. Lotti,1,2,3,* A. Couairon,2,3 D. Faccio,1 and P. Di Trapani1,3

1CNISM and Department of Physics and Mathematics, Universita dell’Insubria, I-22100 Como, Italy2Centre de Physique Theorique, CNRS, Ecole Polytechnique, F-91128 Palaiseau, France

3VINO-Virtual Institute of Nonlinear Optics, Via Cantoni 1, I-22100 Como, Italy(Received 9 November 2009; published 11 February 2010)

We introduce the concept of energy density flux as a characterization tool for the propagation of ultrashortlaser pulses with spatiotemporal coupling. In contrast with calculations for the Poynting vector, those for energydensity flux are derived in the local frame moving at the velocity of the envelope of the wave packet underexamination and do not need knowledge of the magnetic field. We show that the energy flux defined from aparaxial propagation equation follows specific geometrical connections with the phase front of the optical wavepacket, which demonstrates that the knowledge of the phase fronts amounts to the measurement of the energyflux. We perform a detailed numerical study of the energy density flux in the particular case of conical waves,with special attention paid to stationary-envelope conical waves (X or O waves). A full characterization of linearconical waves is given in terms of their energy flux. We extend the definition of this concept to the case ofnonlinear propagation in Kerr media with nonlinear losses.

DOI: 10.1103/PhysRevA.81.023810 PACS number(s): 42.25.Bs, 03.50.De, 42.65.Re, 42.65.Sf

I. INTRODUCTION

Femtosecond laser pulses are nowadays used for manyapplications that commonly require beam and pulse shapingwhen specific pulse temporal profiles, spectra, or beam profileshave to be maintained over a given propagation distanceor available on a target. For instance, the generation ofsecondary radiation from the UV to the THz range can beachieved by the interaction of moderately intense pulses witha gas over long distances [1,2]. The fast ignition scheme inlaser fusion needs an intense laser pulse to interact with adense plasma over a localized area [3]. Localized energydeposition on a target is required in applications rangingfrom material engineering with lasers to laser surgery [4,5].Particle-manipulation scenarios were proposed with weaklylocalized waves [6,7]. In these applications, wave localizationand stationarity of the interaction over a given propagationdistance are crucial features.

Designing wave packets with the required features necessi-tates excellent control of the phase and intensity distribution ofthe laser pulses. Several pulse- and beam-shaping techniqueshave been developed to spatially, temporally, or spectrallytailor specific features of Gaussian laser beams or pulses.However, complicated wave packets for which space andtime are strongly coupled cannot be routinely generated inthe laboratory by simply combining optical elements actinguniquely on the beam or pulse shape. An example of suchpulses is the family of conical waves recently reviewed byMalaguti and Trillo [8]. These waves are of potential interestin all applications where a stationary propagation of an intensecore is required over distances larger than those achievablewith conventional pulses. Conical waves indeed exhibit thepeculiar property of Bessel beams that the energy flux isnot directed along the propagation axis as in Gaussian-likebeams but arrives laterally from a cone-shaped surface [9].This leads to the appearance of a very intense and localized

*[email protected]

interference peak which propagates without spreading over adistance w0/ tan θ that depends on the cone angle θ and thewidth w0 of the input beam. For Bessel beams, slowly decayingtails in the form of concentric rings contain a major part ofthe beam power and support the long propagation distancefeaturing conical waves. Conical waves can be viewed as thepolychromatic generalization of Bessel beams in the sense thatthey are free of diffraction and dispersion over long distances.Several types of conical waves are well identified from thetheoretical point of view [9–12]. However, their generationand even their characterization raise issues owing to theirspace-time coupling and weak localization.

To date, optical conical waves are known to be spon-taneously generated in Kerr media by filamentation andin quadratic media [13–16]. Experimentally, conical waveshave been extensively characterized by far-field measurements[17–20], but clear experimental characterization of the near-field of conical waves remains a difficult task. The three-dimensional (3D) reconstruction technique proposed in Refs.[13,21] was applied to characterize the space-time intensitydistribution of a complex ultrashort pulse that underwentfilamentation in water and showed the need to developalternative methods for the retrieval of weakly decaying tailsof space-time coupled wave packets [15]. Several techniqueswere recently proposed for measuring the space-time am-plitude and phase of complex ultrashort pulses [22–24]. Itwas foreseen from numerical simulations that this informationgives direct access to the characterization of conical wavesin terms of energy flux [25,26]. Although the robustness offilaments is interpreted as resulting from an inward energy flux[27–31], an experimental characterization of the energy fluxassociated with filamentation dynamics has been achieved onlyrecently [32], opening the way to a similar characterization ofspace-time coupled wave packets.

This article proposes the concept of density of energy flux asa tool that gives important information on pulse localization,stationarity, and dynamics. This concept is particularly im-portant in the case of conical waves for which stationarity andlocalization are intimately linked to the energy flux. The energy

1050-2947/2010/81(2)/023810(14) 023810-1 ©2010 The American Physical Society

http://dx.doi.org/10.1103/PhysRevA.81.023810

A. LOTTI, A. COUAIRON, D. FACCIO, AND P. DI TRAPANI PHYSICAL REVIEW A 81, 023810 (2010)

flux is traditionally given by the Poynting vector. However, adiagnostic based on the Poynting vector requires knowledgeof the electric and magnetic fields and the latter is not anexperimentally accessible quantity. Pioneering works showedthat under certain conditions the components of the Poyntingvector can be expressed as functions of the transverse electricfield distribution in space and time of a forward-propagatingparaxial pulse [33–36]. In this article, we propose an alternativeconcept, namely the energy-flux density expressed in thelocal reference frame of the optical pulse under examination,which only requires knowledge of the intensity and phasedistribution of the pulse. We comment on the link between localenergy-flux density and the Poynting vector, and we illustratethe usefulness of energy-flux characterization for wave packetswith space-time coupling by means of the example of linearconical waves.

Section II presents the theoretical building blocks of the ar-ticle: first, the concept of energy-flux density for optical pulseswith envelope satisfying a forward-propagation equation;second, the link to the Poynting vector; and third, the conicalwave packets which will be characterized by their energydensity flux. Section III illustrates the fundamental differencebetween the time-averaged energy flux, usually applied formonochromatic beams, and the local energy flux we proposefor characterizing polychromatic wave packets featured bytemporal localization and space-time coupling. Section IVestablishes a complete characterization of linear conical wavesin terms of local flux components. Section V characterizes thelongitudinal component of the local energy flux. Section VIshows how the concept of energy-flux characterization extendsto nonlinear unbalanced conical waves.

II. DEFINITIONS OF ENERGY DENSITY FLUX

We start by defining the notion of energy density flux. Aswe will see, important information regarding stationarity ofa wave packet may be obtained from the energy density fluxexpressed in the local frame of the wave packet. We will thusdefine the local energy density flux in the laboratory frameas well as in the local frame moving at the group velocityof the wave packet. We consider an optical pulse of centralfrequency ω0 propagating in the forward direction (negativetimes t) in a dispersive medium. The electric field E(x, y, z, t)is decomposed into carrier and envelope as E(x, y, z, t) =E(x, y, z, t) exp(−iω0t + ik0z), where k0 = ω0n(ω0)/c is themodulus of the wave vector at ω0. The linear propagation ofthe pulse is governed by the envelope equation

∂E∂z

+ k′0∂E∂t

= i

2k0∇2

⊥E − ik′′

0

2

∂2E∂t2

, (1)

where E represents the complex amplitude of the envelopeof the electric field. The derivation of Eq. (1) from Maxwellequations relies on several approximations, namely:

1. Scalar approximation: The pulse is assumed to belinearly polarized in a direction transverse to the prop-agation direction z. E is the scalar complex amplitudeof the transverse electric field.

2. Slowly varying envelope approximation: The envelopeshape evolves over a scale typically much larger than

the wavelength. The second-order derivative ∂2E/∂z2

is negligible with respect to k0∂E/∂z.3. Paraxial approximation: |k|/k 1.4. Narrow bandwidth: |ω|/ω0 1, which justifies a

second-order expansion of the dispersive terms k(ω).

The left-hand side of Eq. (1) represents the propagator,where k′

0 ≡ ∂k/∂ω(ω = ω0) represents the inverse of thegroup velocity of the pulse, the time in the laboratory framet represents the longitudinal coordinate, and the propagationdistance z represents the evolution parameter. The first andsecond terms on the right-hand side of Eq. (1) representdiffraction and group velocity dispersion (GVD), respectively;∇2

⊥ = ∂2/∂x2 + ∂2/∂y2 is the Laplacian in transverse co-ordinates, which may as well be expressed with cylindricalcoordinates and k′′

0 ≡ ∂2k/∂ω2(ω = ω0).

A. Derivation of the energy density flux fromthe propagation equation

For convenience, E is assumed to be expressed in W/cm2;thus, |E |2 represents the local intensity of the pulse. It alsorepresents the pulse energy density in the (x, y, t) space sincethe magnetic part of the electromagnetic energy is negligibleand the total pulse energy is given by integration of intensityover the entire space. If a volume V is considered in the(x, y, t) space,

∫V |E |2dxdydt represents the total energy

carried by the pulse in region V and satisfies a conservationequation obtained by multiplying Eq. (1) by E∗, summing theresult with its complex conjugate and performing the volumeintegration. Locally, the energy-conservation equation takes aform analogous to the divergence theorem,

∂|E |2∂z

= −divJ, (2)

where the divergence operator is defined in the (x, y, t) spaceas divF = ∇⊥ · F⊥ + ∂Ft/∂t . This allows for an identificationof the flux of energy density J through the surface S enclosingvolume V:

J⊥ = 1

2ik0[E∗∇⊥E − E∇⊥E∗] (3)

Jt = k′0|E |2 − k′′

0

2i

[E∗ ∂E

∂t− E ∂E∗

∂t

]. (4)

The longitudinal component defined by Eq. (4) includes twocontributions: the first term is associated with the propagationof the pulse at velocity VG = k′

0−1, whereas the second

contribution is intrinsic to the propagation medium andassociated with dispersion. The longitudinal energy densityflux in the local frame of the pulse, associated with the localtime τ = t − k′

0z, thus reads

Jτ = −k′′0

2i

[E∗ ∂E

∂τ− E ∂E∗

∂τ

], (5)

whereas the transverse component is unchanged. In thisframe, the energy density flux entails information on therelative redistribution of energy in the (x, y, τ ) space duringpropagation. If the complex envelope is written in terms ofamplitude and phase E = |E | exp(iφ), each of the energy-fluxcomponents is shown to be proportional to the intensity and to

023810-2

ENERGY-FLUX CHARACTERIZATION OF CONICAL AND . . . PHYSICAL REVIEW A 81, 023810 (2010)

the phase gradient along the corresponding direction:

J = 1

k0|E |2

(∇⊥φ,−k0k

′′0∂φ

∂τ

). (6)

Therefore, if it is possible to have access experimentallyto the intensity and phase distributions of an optical pulse,as recently shown, for example, in Refs. [24,32,37–39], thelongitudinal and transverse components of the energy densityflux vector J are fully determined [32]. As will be seen inSec. II C, this result still holds for a wave packet movingat velocity V different from k′−1

0 , provided corrections aremade to account for the moving frame. Equation (6) alsoshows that the flux is proportional to the intensity distribution,indicating that features of a space-time coupled wave packetmay be undetectable when an (x, y, t) intensity distributionis measured by the 3D mapping technique but visible in theflux distribution. For instance, the weighting effect introducedby the phase gradient may enhance the contrast in the weaktails [15,32].

B. Link to the Poynting vector

In classical electromagnetism, the evolution of the energydensity defined as w = ReE · D∗ + B · H∗/2 (where D, B,and H denote the electric displacement field, the magneticinduction, and the magnetic field, respectively) is governed bythe conservation equation

∂w

∂t= −divS, (7)

where S ≡ ReE × H∗/2 denotes the Poynting vector and thedivergence operator follows here from the standard definition:divF = ∇⊥ · F⊥ + ∂Fz/∂z. Several works have shown thatthe transverse component of the Poynting vector correspondsexactly to the expression given by Eq. (3); thus, S⊥ = J⊥[33–36,40] (see also Appendix A) . The longitudinal compo-nent Sz, however, must be interpreted differently from the locallongitudinal energy flux Jτ . The density of electromagneticenergy indeed includes an electric and a magnetic contributionand the Poynting vector represents the energy density fluxfor both contributions in the laboratory frame. In contrast,the energy density flux J accounts for the electric part only.Appendix A shows that the continuity equations [Eqs. (7)and (2)], and therefore Eqs. (3) and (4), can be transformedinto each other. However, the information on the dispersiveproperties of the medium that is contained in the longitudinalcomponent Jτ of the energy density flux is not present in thelongitudinal component of the Poynting vector Sz but in thetime derivative of the electromagnetic energy. Both quantitiesare linked by the following expression, derived in Appendix A:

∂Jt

∂t= ∂

∂z(Sz − |Ex |2) + ∂

∂t(we + wm). (8)

We note that although there is a clear link between J and S,experimental evaluation of energy density flux requires the useof J as defined in this work since it relies on the knowledge ofthe electric field alone. On the other hand, theoretical studiesin which one has direct access to E and H permit the use ofthe energy density flux based on the Poynting vector S [41].

Finally we underline that there is no obvious route forderiving Jτ directly from S without further approximation.

The Poynting vector indeed describes the energy flux whenMaxwell equations describe the propagation of the pulse.Rather, we have shown that the transverse component J⊥and the longitudinal component Jτ are derived from theenvelope-propagation equation and that J⊥ coincides withS⊥. We show in Appendix B that for nondispersive media,a generalization of the longitudinal flux component Jτ may beexpressed as w − Sz/c, whereas the longitudinal componentof the Poynting vector Sz represents the energy densityin the (x, y, t) space. This link to the Poynting vector iscompatible with the expressions for the flux found under theenvelope approximation. We expect that further generalizationis possible in dispersive media; however, both w and Sz requirethe knowledge or measurement of the magnetic field, whereaswithin the approximations considered in this article J can beobtained from the knowledge of the intensity and phase of theenvelope of the electric field only.

C. Linear conical waves, Bessel X pulse, andpulsed Bessel beams

We consider scalar wave packets (WPs) propagating in adispersive medium characterized by a refractive index n(ω),thus satisfying the wave equation expressed in the Fourierdomain, [

∂2

∂z2+ ⊥ + k2(ω)

]E(r, ω, z) = 0, (9)

where k(ω) ≡ ωn(ω)/c. WPs with central frequency ω0 areassumed to be described by a complex envelope (r, τ =t − z/V, z) propagating in the forward direction at velocityV , possibly different from VG = 1/k′

0, and a carrier waveexp(iβz − iω0t), where the carrier wave number β may alsodiffer from k0 and τ denotes the local time in the envelopeframe. In particular, we may define the velocity at which theenergy of the pulse propagates as VE = VG cos θ0, where θ0 is acharacteristic angle of the propagating WP described below. Inthis work the difference between VG and VE from a practicalpoint of view is always negligible; we therefore choose toname superluminal a velocity V greater than the Gaussianvelocity VG and subluminal a velocity V lower than VG. Thepropagation equation governing the envelope spectrum reads[

∂2

∂z2+ 2i

(β +

V

)∂

∂z+ ⊥ + k2

⊥()

](r,, z) = 0,

(10)where

k2⊥() ≡ k2(ω) − (β + /V )2 , (11)

and ≡ ω − ω0 denotes the frequency departure from thecarrier frequency. The general solution to Eq. (10) propagatingin the forward direction with revolution symmetry reads

(r, τ, z) = 1

2π

∫ +∞

−∞d

∫ +∞

0dKKA(K,)J0(Kr)

× exp(iκz − iτ ), (12)

where K is the transverse component of the wave vector,A(K,) denotes an arbitrary function which represents thecomplex angular spectrum of the WP at z = 0, and the

023810-3


longitudinal wave number associated with the envelope reads

κ =(

β +

V

) ⎡⎣√1 + k2

⊥() − K2

(β + /V )2− 1

⎤⎦=

√k2(ω) − K2 −

(β +

V

). (13)

Conical wave packets (CWPs) correspond to specificchoices of the angular spectrum A(K,) = f ()δ(K −k(ω) sin[θ ()]), where f () is a weight function for whichthe WP may be viewed as a superposition of plane waves withfrequency-dependent wave numbers K distributed over conesof angle θ () with respect to the propagation direction z.

The pulsed Bessel beam (PBB) and the Bessel X pulse(BXP) are particular cases of conical waves obtained fromEqs. (12) and (13) for which the plane-wave constituentspresent a constant transverse component of the k vector (PBB,θ () = arcsin[K0/k(ω)]) [42] or a constant propagation angleθ0 for each frequency (BXP, K() = k(ω) sin θ0) [11]. ThePBB is similar to a Bessel beam generated by means of acircular diffraction grating and may be regarded as the productof a Bessel beam and a Gaussian pulse. The BXP can begenerated by sending a Gaussian beam through an axicon andexhibits a two-winged (X-shaped) structure resulting from thesuperposition of an inward and an outward beam [37,42,43].Ideally, the PBB and the BXP have infinite energy due totheir weakly decaying tails, but in practice both are producedwith finite energy beams sent through finite apertures and thusexhibit a central intensity peak that extends over a propagationregion called the Bessel zone (which depends on the spatialapodization of the profile).

Conical waves with a stationary envelope in their referenceframe constitute an important subclass of CWP for which theenvelope does not depend on z. This condition is satisfiedwhen the longitudinal wave number varies linearly withfrequency, which is obtained by choosing the angular spec-trum A(K,) = f ()δ(K − k⊥()) and k⊥() is defined byEq. (11). The envelope then reads

(r, τ ) =∫

k⊥()realf ()J0(k⊥()r) exp(−iτ )d, (14)

where integration is performed over frequencies for whichk⊥() is real.

It is particularly instructive to assume that a small

expansion of the dispersion relation up to second order issufficient to describe the dispersive properties of the medium;thus, k(ω) k0 + k′

0 + k′′02/2 and

k2⊥() = α2

2 + 2α1 + α0, (15)

where α2 = k0k′′0 + k′2

0 − 1/V 2, α1 ≡ k0k′0 − β/V , and α0 =

k20 − β2.

In this case, the stationary CWPs have been obtainedby Porras and Di Trapani [44] and were recently studiedin a nonparaxial framework by Malaguti and Trillo [8].A comprehensive analysis of angular dispersion curves ofstationary CWPs in a linear dispersive medium is givenin Ref. [45]. In the following treatment we will refer tothe notation of Ref. [8]. Stationary CWPs are characterizedby two parameters: the angular aperture θ0 of the beam

V/c

θ 0 ( rad )

-A

B+

B-

D+D-

B-

0 0.1 0.2 0.3 0.40.7

0.72

0.74

0.76

0.78

A+a

c

b

0

3

6

-2 0 2Ω

k⊥

0

3

6

-2 0 2Ω

k⊥

0

3

6

-2 0 2Ω

k⊥

FIG. 1. (Color online) Parameter space (θ0, V ) for stationaryCWP in water and a central wavelength of 527 nm. The differentsectors labeled A, B, and D are bounded by the dashed curves whereα2

1 = α2α0 and α2 = 0 (line c), on which the shape of the angularspectrum k⊥() [Eq. (15)] changes. A, X waves (hyperbolic) withangle gap; B, X waves (hyperbolic) with frequency gap; D, O waves(elliptic). The ± labels indicate that the central frequency of the k⊥()curve is upshifted (+) or downshifted (−) with respect to ω0. Thecurves a (V = VG cos θ0) and b (V = VG/ cos θ0) determine lociaround which stationary CWPs with sufficiently narrow bandwidthshare the features of PBBs or BXPs, respectively. The insets showthe qualitative shape of the dispersion curve [Eq. (15)] for B−, A−,and D−(k⊥ in µm−1, in rad/fs).

at the carrier frequency which is linked to the longitudinalwave number β = k0 cos θ0 and the velocity V of the CWP.Following Malaguti and Trillo, Fig. 1 shows the different typesof stationary conical waves in the plane (θ0, V ), which wewill use to illustrate the use of energy flux as a tool forcharacterizing stationary CWPs. This classification is basedon a second-order approximation of chromatic dispersion, butspectral features and the characterization of the energy fluxremain qualitatively similar even if the full dispersion curveis retained, as in all numerical results in this article. A singleexception exists when the spectrum of the WP is centeredaround a zero of the GVD, which will not be treated in thisarticle. The line c is defined by α2 = 0. The region above linec corresponds to X-wave solutions (α2 > 0). In particular,regions A± (α2

1 − α2α0 < 0 or k′0 cos θ0 − √

k0k′′0 sin θ0

V −1 k′0 cos θ0 + √

k0k′′0 sin θ0) correspond to hyperbolic so-

lutions of Eq. (15) with a wave number (angle) gap, whileregions B± (α2

1 − α2α0 > 0 or shaded areas in Fig. 1) corre-spond to hyperbolic solutions with a frequency (wavelength)gap. The region below line c corresponds to O-wave-likesolutions (α2 < 0), which present elliptic k⊥() profiles andare analogous to the solutions found in the case of anomalousGVD [42,44,46].

In general the PPB and the BXP are not stationary solutionsin the strict sense. However, it is possible to find somelimit case in which stationary CWPs may be consideredas belonging to the family of PBB or BXP within thissecond-order expansion. From Eq. (15), it can be seen thata constant k⊥ (thus a PBB profile) is obtained for α1 = 0 and

023810-4


α2 = 0, which results in a single couple of velocity and angularaperture V = VG cos θ0 and tan2 θ0 = k0k

′′0/k′2

0 , respectively,corresponding to the intersection of lines a and c in Fig. 1.However, the weight function f () may be sufficiently narrowto neglect α2

2 in Eq. (15), while the condition α1 = 0just gives the velocity featuring a PBB so that the wholecurve a (V = VG cos θ0) characterizes PBB-like solutions.Similarly, the angular aperture at a given frequency of thestationary CWP presented in Fig. 1 is given by cos θ () =(β + /V )/(k0 + k′

0 + k′′02/2). This shows that stationary

conical waves with strictly constant cone angles (BXP) arenever obtained in condensed media. However, if the bandwidthof the weight function in Eq. (14) allows us to neglect k′′

02

in front of k0 + k′0, we obtain stationary CWPs with almost

constant cone angle θ0 for the velocity V ∼ VG cos θ0 featuringsolutions of the BXP type (line b in Fig. 1).

Finally, within the framework of the second-order expan-sion of k(ω) and the slowly varying envelope approximation[ ∂2

∂z2 β ∂∂z

in Eq. (10)], the propagation equation for theenvelope (r, τ, z) can be put in the form of Eq. (1):

2iβ∂

∂z+ ⊥ − α2

∂2

∂τ 2+ 2iα1

∂

∂τ+ α0 = 0. (16)

Therefore, the associated energy-flux components are ex-pressed as

J⊥ = 1

2iβ

[∗∇⊥ − ∇⊥∗] , (17)

Jτ = − α2

2iβ

[∗ ∂

∂τ−

∂∗

∂τ

]+ α1

β||2. (18)

III. MONOCHROMATIC VS POLYCHROMATIC ENERGYDENSITY FLUX

A. Time-integrated flux

The energy density flux given in Eqs. (3) and (5) isparticularly well suited to characterize polychromatic wavepackets such as those presented in Sec. II C, which exhibitspace-time couplings and/or angular dispersion. In most casesstudied in the literature, optical wave packets are supposed tobe describable in terms of separated variables with uncoupledbeam and pulse dynamics. This is the case for the Gaussianpulse (GP) obtained by focusing a laser beam with a lens withfocal length f in a dispersive medium. The linear propagationof a GP is well known and may be described in termsof the beam and pulse parameters solely depending on thepropagation distance, with the origin (z = 0) at the focus ofthe lens [47]:

E(r, t, z) = E0w0T

1/20

w(z)T 1/2(z)exp

[− r2

w(z)2− τ 2

T (z)2

]× exp

[ik0

r2

2F (z)− iζ (z) − i

τ 2

2k′′0L(z)

], (19)

where the beam waist, beam curvature, Gouy shift, pulseduration, and pulse chirp are given, respectively, by

w(z) = w0(1 + z2/z2

R

)1/2,

F (z) = z(1 + z2

R/z2),

ζ (z) = arctan(z/zR),

T (z) = Tp

[1 + (z + d)2/z2

GVD

]1/2,

L(z) = (z + d)[1 + z2

GVD/(z + d)2].Here, a flat temporal phase is assumed at the beginning of thepropagation z = −d defined by f = −d − z2

R/d. The quantityzR = k0w

20/2 denotes the Rayleigh length and zGVD = T 2

p /2k′′0

the dispersion length. The pulse duration at the focus T0

is linked to the initial pulse duration Tp by the relationT0 = Tp(1 + d2/z2

GVD)1/2. From Eq. (19), it is readily seenthat variables are separated: the temporal (or longitudinal) andthe transverse dynamics remain completely uncoupled.

When experimental characterization deals with the spatialdynamics, the temporal dynamics is assumed to be frozen andthe pulse is considered as quasimonochromatic. In this case, acharacterization in terms of energy flux amounts to consideringthe transverse component of the flux only. As we have seen,it is proportional to the intensity of the wave packet and tothe transverse gradient of the wave-packet phase. If the latterquantity is assumed not to depend on time, the time-integratedflux can be approximated as

J⊥(r, z) =∫

J⊥(r, τ, z)dτ ≡ 1

k0

∫|E(r, τ, z)|2∇⊥φ(r, z)dτ

∼ 1

k0F(r, z)∇⊥φ(r, z), (20)

where F(r, z) ≡ ∫ |E(r, τ, z)|2dτ denotes the fluence of thebeam which is the experimentally accessible quantity.

As we shall show, a fundamental difference exists betweenthe various notions of energy flux; namely, J⊥(r, z) gives usinformation about the fluence in the monochromatic case.In contrast, the transverse energy flux J⊥(r, τ, z) for eachpropagation distance gives information on the spatial andtemporal reorganization of energy within the wave packet,which applies for polychromatic cases as well. The differencewill be shown by considering simple situations—a GP, anapodized PBB, and an apodized BXP coming from an idealaxicon—which exhibit different space-time coupling andcannot be distinguished from their characterization by meansof J⊥(r, z), whereas the energy-flux characterization withJ⊥(r, τ, z) clearly shows their distinct features.

B. Energy flux in the monochromatic case

For the GP, the components of J can be calculated directlyfrom Eq. (19):

Jr = |E(r, τ, z)|2 r

F (z)(21)

Jτ = |E(r, τ, z)|2 τ

L(z). (22)

From this expression, it is readily seen that J⊥ = ∫Jrdτ =

F(r, z) rF (z) . Thus, a GP is associated with an inward flux before

the focus of the lens [F (z) < 0] and an outward flux after thefocus.

For the PBB and the BXP, the flux is calculated numericallyby considering propagation through water for WP with centralfrequency at λ0 = 800 nm. The GP parameters are w0 =0.03 cm, Tp = 80 fs, and f = 4 cm. We chose PBB and

023810-5


z (cm)0 2 4 6

1

2

-4 -2 0 2 4z (cm)

r (m

m) 0.4

0.2

0

z (cm)0 2 4 6

1

2(b) PBB (c) BXP(a) GP

FIG. 2. (Color online) Time-integrated transverse flux profiles vspropagation distance for (a) a focused Gaussian pulse, (b) a pulsedBessel beam, and (c) a Bessel X pulse. Red (blue) indicates outward(inward) flux with respect to the propagation axis.

BXP with the same propagation angle θ0 = 2o at the centralwavelength λ0 = 800 nm, temporal duration Tp = 80 fs, andGaussian apodization with w0 = 0.1 cm to mimic a realsituation with finite energy.

Figure 2 shows the time-integrated transverse flux J⊥(r, z)as a function of the propagation distance z (evolution pa-rameter) and radial coordinate for the three cases underexamination. The focused Gaussian case [Fig. 2(a)] presentsinward-directed (negative) flux before the beam waist positionat z = 0 and outward-directed (positive) beyond the focus, dueto linear diffraction. As can be seen in Figs. 2(b) and 2(c), thecases of the PBB and the BXP cannot be distinguished by theircharacterization in terms of time-integrated flux. Both exhibitan integrated flux qualitatively similar to that of a GP, withan initial inward flux, followed by a region (correspondingapproximatively to the center of the Bessel zone) in whichthe flux vanishes, and finally an outward flux. This behavioris strictly connected to the fact that the beams are spatiallyapodized, as will be discussed later. For longer propagationdistances, the flux becomes ring-shaped (in the x-y plane),which corresponds to the far field of a PBB or a BXP.

C. Energy flux in the polychromatic case

The representation at fixed z of Jr in the (r, τ ) domainallows us to discriminate between the three cases. Figure 3shows the change in the transverse flux distribution for the GP,PBB, and BXP as a function of the propagation distance.

Before the focus of the GP or at the very input of the Besselzone [Figs. 3(a)–3(c)], the GP transverse flux does not presentany spatiotemporal feature, indicating that the pulse as a wholeis focusing, since Jr < 0. The PBB and the BXP present aninward energy flux, featured by the intensity distribution ofthe wave packet itself. Figures 3(d)–3(f) show the three casesaround the beam waist or the center of the Bessel zone. The GPtransverse flux [Fig. 3(d)] is considerably smaller with respectto the previous case since the WP is approaching the beamwaist.

The BXP flux [Fig. 3(f)] presents two wings: one in theleading part of the pulse with inward-directed flux and onein the trailing part with outward-directed flux. This behaviormay be simply understood by means of Fig. 4(a) as a resultof the conical nature of this wave packet. The PBB can beregarded as a degenerate BXP, in which propagation angle andpulse tilt angle are equal, so that the inward-flux and outward-flux wings spatially overlap. Figures 3(g)–3(i) show the fluxcharacterizing the GP, PBB, and BXP at a greater propagationdistance. The energy flux is directed outward for all wave

-4 -2 2 4

x 10-5

0 -10 -6 2 6

x 10-5

-2 102-2

x 10-5

0

1.2

0.8

0.4

-200 0 200

(b)

-200 0 200

1.2

0.8

0.4

(c)

-200 0 200

r (m

m) 0.4

0.20.1

0.3

(a)

-200 0 200

r (m

m) 0.4

0.20.1

0.3

(d)

-200 0 200T (fs)

r (m

m) 0.4

0.20.1

0.3

(g)

-200 0 200

1.2

0.8

0.4

(f)1.2

0.8

0.4

-200 0 200

(e)

-200 0 200T (fs)

1.2

0.8

0.4

(i)1.2

0.8

0.4

-200 0 200T (fs)

(h)

FIG. 3. (Color online) Transverse flux Jr for different wavepackets at different propagation distances. Panels (a), (d), and (g)refer to the GP 4 cm before the beam waist, near the beam waist,and 4 cm after that point, respectively. Panels (b), (e), and (h) referto a PBB obtained by multiplying a Gaussian profile by a radiallydependent phase at the beginning of propagation, near the center ofthe Bessel zone, and toward the end of this zone, respectively. Panels(c), (f), and (i) refer to the same situation as panels (b), (e), and (h) forthe BXP wave-packet profile. The color scale of Jr is in arbitrary unitssince the intensities were normalized to the maximum value reachedduring propagation. The black contour plots show the intensity overtwo decades.

packets and exhibits features of the intensity distribution inthe far-field region, which corresponds to a diverging Gaussianbeam for the GP and a diverging ring for the PBB and the BXP.

zθ

(a)

zθ

δ(b)

FIG. 4. (Color online) Simple representation of the propagationof the expected transverse flux for the propagation of (a) a BXPand (b) a rotationally symmetric tilted pulse. The red (blue) colorcorresponds to flux directed upward (downward) with respect to thevertical axis.

023810-6


IV. STATIONARY ENVELOPE WAVES

A. Normal GVD case: X waves

As previously stated, stationary CWPs may be described bymeans of Eq. (14). Malaguti and Trillo have derived closed-form analytical solutions for stationary CWPs with specificspectral weight functions f () [8]. From these solutions andEqs. (17) and (18), we evaluated the energy flux characterizingCWPs: All the solutions of Ref. [8] with velocity equalto the energy velocity VE = VG cos θ0 and corresponding tosymmetrically distributed spectral weight turn out to haveboth transverse and longitudinal flux components equal to 0.The case of an asymmetric spectral weight leads to analyticalsolutions featuring a single branch of an X wave with frequencygap in region B of Fig. 1.

In this section we will show the flux of stationary CWPswith asymmetric spectral weight, evaluated by means ofEq. (14). It is important to emphasize that we used a fulldispersion relation k(ω), extrapolated from the data of [48], inthe definition of these WPs [Eq. (11)]: The expression for theflux has been derived in the second-order approximation andgives the main contribution. Higher-order terms become moreimportant in a spectral region corresponding to zero GVD.

Figure 5 shows the transverse flux Jr for an X wavecorresponding to region B of Fig. 1, with central wavelengthλ0 = 527 nm, reference angle θ0 = 0 rad, Gaussian spectralweight (Tp = 80 fs), and Gaussian spatial apodization (w0 =0.5 mm), propagating in water when the velocity of the pulsepeak is subluminal [Fig. 5(a), V = 0.9995VG] or superluminal[Fig. 5(b), V = 1.0005VG] with respect to the Gaussian groupvelocity VG = 1/k′

0 = 220720072 m/s. The flux profile issimilar to that of the BXP case.

It may be noticed that the inward-directed flux wing appearsin the leading part of the pulse or the trailing one accordingto the superluminal or subluminal propagation of the envelopepeak. This behavior is not in contradiction with the fact thatthe overall pulse is moving in the forward direction and it maybe simply understood by considering that the propagation ofthe central peak is the result of the cylindrically symmetricsuperposition of tilted pulses, as already pointed out in [42]and schematically represented in Fig. 4(b).

Spatial apodization limits the propagation range in whichthe wave packet is effectively stationary: the behavior is

-500 0 500T (fs)

0.2

0.4

0.62

0

-2

x 10-4

-500 0 500T (fs)

0.2

0.4

0.6

r (m

m)

(a) (b)515 540

0.1

λ (nm)k⊥ (

1/µm

)

515 540

0.1

λ (nm)k⊥ (

1/µm

)

FIG. 5. (Color online) Transverse flux Jr for stationary CWPin the X-wave case for different propagation velocities of theenvelope peak. (a) V = 0.9995VG; (b) V = 1.0005VG. The graycontour plots show the intensity over four decades. The insets showthe corresponding spectral λ, k⊥ features. Color scale in arbitraryunits.

-4x 10

-2

0

2

-400 0 400

0.2

0.4

0.6

T (fs)-400 0 400

0.2

0.4

0.6

T (fs)

r (m

m)

(a) (b)1210 1380

0.1

λ (nm)k⊥ (

1/µm

)

1210 1380

0.1

λ (nm)k⊥ (

1/µ m

)

FIG. 6. (Color online) Transverse flux Jr for stationary CWP ofthe O-wave type for different propagation velocities of the envelopepeak. (a) V = 0.9995VG; (b) V = 1.0005VG. The gray contourplots show the intensity over four decades. The insets show thecorresponding spectral λ, k⊥ features. Color scale in arbitrary units.

qualitatively similar to the BXP case. After propagation towardthe end of the Bessel zone, a dominant wing carrying anoutward-directed flux is obtained in the trailing or leadingpart of the pulse depending on the peak velocity (superluminalin the first case and subluminal in the second case).

B. Anomalous GVD case: O waves

Stationary CWPs in media with anomalous dispersiontypically feature elliptical spectral k⊥() profiles called Owaves [44,49]. Similar stationary CWPs are also obtainedin the case of normal dispersion [8] in region D in Fig. 1;however, these profiles are either highly nonparaxial or arealmost identical to a single-branch X-wave profile if we remainin a paraxial frame.

In this section we will study the O-wave case within theparaxial approximation by considering the propagation in amedium with anomalous GVD (k′′

0 < 0). As in the previoussection, we will consider the full dispersion relation in thedefinition of the wave packet and an asymmetrical spectralweight.

Figure 6 shows the transverse flux Jr for this WPpropagating in water at central wavelength λ0 = 1300 nmand corresponding angle θ0 = 0.005 rad, Gaussian spectralweight (Tp = 40 fs), Gaussian spatial apodization (w0 =0.6 mm), peak velocity [Fig. 6(a) V = 0.9995 VG and Fig. 6(b)V = 1.0005 VG] greater and lower than VG = 222930323m/s, respectively. Although the near-field profile of the wavepacket under examination is definitely different from that ofthe X-wave case, it still exhibits an almost identical flux profileand we are still able to predict super- or subluminal velocityby looking at the position of the inward-directed energywing.

V. LONGITUDINAL COMPONENT ANDENERGY STREAM

Up to now, we considered only the transverse componentof the energy flux J, which coincides with the transversecomponent of the Poynting vector and does not depend onthe velocity of the wave packet under examination. In thissection, we show that the longitudinal component of the energyflux also constitutes an important source of information thatwe exploit in two different forms: First, we consider the

023810-7


T (fs)-200 0 200

0.4

0.20.1

0.3

T (fs)-200 0 200

r (m

m) 0.4

0.20.1

0.3

T (fs)-200 0 200

0.4

0.20.1

0.3

(a) (b) (c)

FIG. 7. (Color online) Longitudinal flux Jτ for a GP propagatingin different regimes. (a) Linear propagation with normal GVD.(b) Nonlinear propagation (high intensity) with anomalous GVD(λ0 = 1300 nm in water). (c) Nonlinear propagation with anomalousGVD at lower intensity with respect to case (b). The color scaleof Jτ indicates flux toward positive (negative) τ values in red(blue). The gray contour plots show intensity contours over twodecades.

longitudinal energy flux in itself, and second, we associateit with the transverse component to obtain the direction of theenergy flux for general wave packets. To our knowledge, thiswas attempted only once with the Poynting vector associatedwith localized waves [41]. Here we consider the energy fluxin the frame moving at the velocity of the envelope peak ofspecific WP to diagnose the energy redistribution occurringwithin the WP.

A. Longitudinal component of the energy flux

In the frame moving at VG = 1/k′0, the longitudinal flux

component Jτ is related to the chromatic dispersion of thepulse, as indicated by Eq. (5). In both cases of normal andanomalous GVD, a Gaussian pulse with an initial flat phasedevelops during propagation a longitudinal flux componentdistinguished by two side lobes in the leading and in the trailingpart of the pulse: the leading lobe is directed toward negativetimes and the trailing lobe toward positive times [Fig. 7(a)].This indicates that the pulse is broadening along the temporalcoordinate. In contrast, if the pulse has an initial negativechirp and propagates in a normally dispersive medium, eachside lobe corresponds to a longitudinal flux with opposite sign,indicating pulse compression during propagation, as expectedfrom the action of normal GVD.

The case of nonlinear propagation in a Kerr medium withanomalous dispersion is particularly instructive. Although wederived the energy flux from linear propagation equationsin Sec. II, the expression for J also holds for nonlinearpropagation and in the presence of losses, as we will illustratein Sec. VI. Self phase modulation induced by the opticalKerr effect generates new frequencies in the regions of thepulse with the strongest intensity gradients. In particular, afocusing Kerr nonlinearity generates upshifted (downshifted)frequencies with respect to the central frequency in the de-creasing (increasing) part of the pulse. In conjunction with withanomalous GVD, this effect is responsible for pulse shorteningduring propagation. Figure 7(b) illustrates the longitudinal fluxobtained in this situation: Each lobe of Jτ exhibits an oppositesign with respect to the linear propagation case. Figure 7(c)depicts the longitudinal flux associated with the nonlinearpropagation of a GP with lower energy and pulse intensitythan that in Fig. 7(b). In this case, self phase modulation isweaker and the new frequencies are not able to overcome

0 10 20 30 40 50z (cm)

-800

400

800

T (

fs)

(a) (b)

0

-400

0 10 20 30 40 50z (cm)

-800

400

800

0

-400

FIG. 8. (Color online) (a) Linear scale of the intensity plot and(b) longitudinal flux Jτ vs propagation distance z for a Gaussian pulsewith strong quadratic and cubic frequency chirp. For both figures, thetemporal axis refers to the local frame moving at VG = 1/k′

0. Thecolor scale of Jτ indicates flux toward positive (negative) τ values inred (blue).

linear dispersion over the whole pulse. Nonlinear effects thusdominate in the central high-intensity core and the flux pointsinward, thus leading to a local temporal compression, whilethe surrounding low-intensity regions behave almost linearlywith an outward flux. This leads to the progressive distortionof the pulse.

In analogy with the evolution of the temporally averagedtransverse flux depicted in Figs. 2(a)–2(c), the evolution of theradially averaged longitudinal flux Jτ may be regarded as thepropagation of a pulse inside an optical fiber.

Figures 8(a) and 8(b) show the intensity and the longitudinalfluxJτ versus propagation distance z for the linear propagationof a pulse with an initially Gaussian spectral distributionand strong quadratic and cubic phase chirp. The dispersionparameters correspond to propagation in water at a centralwavelength of 527 nm. In this simulation we considered thatonly dispersive effects occur during propagation and neglecteddiffraction. The intensity pattern depicts an intense temporalpeak whose velocity v varies during propagation, followingthe relation 1/v = 1/VG + dτ/dz. In the local frame movingat VG, the instantaneous velocity of this peak is given bythe first derivative of the (τ, z) intensity distribution curve.The curvature of this intensity distribution thus represents thepeak acceleration: in Fig. 8(a), the peak exhibits a constantdeceleration. The peak velocity changes from superluminalfor the first ∼20 cm to subluminal afterward. This behavior isgoverned by the cubic chirp, while the distance z at which thepeak travels at VG (i.e., the z position for which dτ/dz = 0) isdetermined by the quadratic chirp. Note that a cubic chirp ofopposite sign would lead to a constant acceleration, due to thechange of the curvature sign. The longitudinal component ofJ shown in Fig. 8(b) illustrates the deceleration property: Weobserve in correspondence of the main peak a net flux directedtoward negative (positive) times in the superluminal (sublumi-nal) region. However, the overall pulse presents a broad seriesof secondary intensity peaks associated with a flux of oppositesign. This indicates, as expected, that the center of mass of theentire pulse energy does not change during propagation in spiteof the peak deceleration. Furthermore, for large propagationdistances [rightmost part of Fig. 8(b)], the flux pattern becomessimilar to that obtained for a GP subject to normal GVD[see for example Fig. 7(a)].

023810-8


-500 0 500

0.2

0.4

0.6

r (m

m)

-500 0 500

0.2

0.4

0.6

-400 0 400T (fs)

0.2

0.4

0.6

r (m

m)

-400 0 400T (fs)

0.2

0.4

0.6

(a) (b)

(c) (d)

FIG. 9. (Color online) Streamline plot of vector J for station-ary envelope wave packets. (a, b) subluminal and superluminalfrequency-gap X waves, respectively, as in Fig. 5. (c) SubluminalO wave as in Fig. 6(a). (d) Superluminal wave-number-gap X wavewith parameters like the ones of Fig. 5(b), but Tp = 60 fs and θ0 =0.005 rad. The contour plots show the intensity over five decades.

B. Streamline representation of the flux

The longitudinal component is also useful for envisioningthe direction of the energy density flux. In this context,the choice of the reference system is important; that is, thedirection of the energy flux depends on the moving framechosen to represent this quantity. For stationary-envelopeWPs, a natural choice is the reference system moving atthe peak velocity of the pulse, as expressed in Eqs. (17)and (18). Figure 9 illustrates four different cases of CWPswith stationary envelope. Figures 9(a), 9(b), and 9(c) showthe streamline plots of vector J for the previously considered(Secs. IV A and IV B) subluminal and superluminal X waveswith a frequency gap (as in regions B± of Fig. 1) andsubluminal O wave (as in region D− of Fig. 1), respectively.Figure 9(d) shows the energy-flux streamlines in the cases ofa superluminal X wave with a wave number gap (region Aof Fig. 1). The most important feature is that the energy flowsalong conical surfaces from the outer region toward the centralpeak and then streams outward again to the surrounding region.Moreover, along the temporal (longitudinal) coordinate, theflow is directed toward negative (positive) times for subluminal(superluminal) stationary conical wave packets. This resultis in keeping with the fact that the peak results from aconstructive interference supported by the surrounding energyreservoir: Even if the longitudinal velocity of the envelopepeak V is superluminal (larger than VG), the energy is actuallytransported along z at velocity VE = VG cos θ0. As a lastremark, it is worth noticing that the condition divJ = 0,which characterizes stationarity of the intensity profile of theenvelope during its paraxial propagation along z, is satisfied byapplying Eqs. (17) and (18) to any analytical solutions obtainedby Malaguti and Trillo. This result is remarkable since thesesolutions are fully nonparaxial, whereas Eqs. (17) and (18)are obtained in a paraxial framework. This is actually due tothe truncature of dispersive terms at second order for k2

⊥() inthe Malaguti and Trillo solutions.

We now give a geometrical interpretation of the the energy-flux streamlines in different regimes of GVD, highlightinguncovered links with the phase fronts.

By using a decomposition of the complex amplitude ≡|| exp(iφ) into a real amplitude and phase and by means ofthe local longitudinal coordinate transformation ξ = −Vmτ ,where Vm ≡ (|α2|)−1/2, the components [Eqs. (17) and (18)]of the flux read

J⊥ = 1

β||2∇⊥φ(r, ξ, z) (23)

Jξ = − sgn(α2)

β||2

[∂φ

∂ξ(r, ξ, z) + sgn(α2)α1√|α2|

]. (24)

The last term in Eq. (24) may be absorbed in the phaseby using φ′(r, ξ, z) = φ(r, ξ, z) + sgn(α2)α1√|α2| ξ . The flux is then

expressed as the product of the intensity ||2 by vector eJ ≡β−1(∇⊥φ′,−sgn(α2)∂φ′/∂ξ ). Both the transverse and longi-tudinal components are then proportional to the phase gradientin the corresponding direction. Only in the case of an effectiveanomalous dispersion (α2 < 0) is the flux vector propor-tional to the phase gradient: (J⊥, Jξ ) = β−1||2∇φ′(r, ξ, z),where ∇ denotes the gradient in the (x, y, ξ ) space. In thiscase, the vector (J⊥, Jξ ) is orthogonal to the isocontour forthe phase φ′(r, ξ, z). This is shown, for example, in Fig. 10(d),where we plot eJ = J/|ψ |2 as a vector field and the phase φ′as a color plot [50] for a GP centered at λ0 = 1300 nm in water(anomalous GVD regime).

For α2 > 0, vector eJ represents the symmetric of thenormal to the phase front with respect to the axis ξ = 0. Thisis a geometric property that follows from the definition of theflux. Consequently, the energy flux is such that phase frontsparallel to the loci r = ±ξ are associated with an energy fluxparallel to the phase front. In the case of a GP, the possibleregimes are represented in Figs. 10(a) and 10(b), where eJ

is plotted over cos φ′ for different propagation distances z ofthe same GP having zR = zGVD. Figure 10(a) refers to a shortpropagation distance (z < zR, zGVD): the iso-φ′ curves do not

-1 -0.5 0 0.5 1

1

2

3

r (m

m)

-2 -1 0 1 2

0.4

0.8

1.2

1.6

-2 -1 0 1 2

0.4

0.8

1.2

1.6

r (m

m)

ξ (mm)-1.5 -0.5 0.5 1.5

0.5

1.5

2.5

3.5

ξ (mm)

(d)(c)

(a) (b)

FIG. 10. (Color online) Vector eJ superimposed to the plot ofcos φ′ for different cases. GP in normal GVD regime with zR = zGVD

for short (a) and long (b) propagation distances. Note the differencesin axis limits. (c) Subluminal frequency-gap X-wave profile. (d) GPin anomalous GVD regime.

023810-9


follow r = ±ξ and eJ are neither orthogonal nor parallel tothe phase fronts, but the symmetric vectors with respect tothe axis ξ = 0 are clearly orthogonal to phase fronts. Figure10(b) refers to a longer propagation distance (z zR, zGVD):since in this case the iso-phase curves asymptotically approachr = ±ξ [within a second-order approximation of k(ω)], thevector field eJ and thus the flux J are asymptotically parallelto phase fronts along r = ±ξ . This is also the case for theX wave with a frequency gap and an asymmetric spectralweight obtained from the analytical formula (24) of Ref. [8],as shown in Fig. 10(c) (subluminal case): the phase frontsalign asymptotically along r = ±ξ , even if in this region theintensity is very low, and thus eJ and the energy flux becomeasymptotically parallel to phase fronts.

This geometrical construction of flux streamlines from thephase fronts illustrates the link between the determination ofthe phase and that of the energy density flux for an optical WPpropagating in a dispersive medium.

VI. NONLINEARITY AND ABSORPTION

In the case of nonlinear propagation, expressions (3) and(5) of flux density determined in Sec. II remain valid. As anexample, we consider the nonlinear propagation of conicalwave packets such as those formed spontaneously duringfilamentation in water, which is described by the nonlinearenvelope equation including nonlinear source terms for theoptical Kerr effect with instantaneous and delayed (Ramaneffect) contributions, multiphoton absorption, and plasmaeffects (absorption and defocusing) following ionization ofthe medium [16,51]. In this case, Eq. (2) becomes

∂|E |2∂z

+ divJ = −β(K)|E |2K − σρ|E |2, (25)

where the terms on the right-hand side of Eq. (25) representthe density of energy losses by multiphoton absorption andplasma absorption, respectively (see Ref. [25] for details).The parameters are the cross section β(K), where K denotesthe number of photons involved in the multiphoton process,the cross section for inverse Bremsstrahlung σ , and ρ denotesthe plasma density. Note that linear absorption can be treatedsimilarly (K ≡ 1).

We therefore have an extension of the energy flux also inthe nonlinear case and in the presence of absorption. In thiscase the continuity equation contains additional terms whichrepresent the amount of energy transferred from the opticalpulse to the medium.

Recently, nonlinear conical wave-packet profiles whichmaintain an intensity-invariant shape in the presence ofnonlinear losses have been discovered [52,53]. Numericalsolutions called unbalanced nonlinear Bessel beams (UBBs)and unbalanced nonlinear O waves (UBOs) have been foundfor the monochromatic case and the polychromatic case in theregime of anomalous GVD, respectively.

Following the procedure of Ref. [53], we numericallyevaluated the near-field profiles of UBBs and UBOs and theassociated energy flux.

Figure 11 shows the streamline representation of J for anUBO. The parameters were chosen to represent dispersion andnonlinear losses for water at λ0 = 1300 nm. The energy flux

-15 -10 -5 0 5 10 15 (fs)T

r(µm

)

0

0

10

10

FIG. 11. (Color online) Streamline plot of the energy flux J fora nonlinear O wave (UBO) with stationary envelope propagatingat VG = k′−1

0 . The streamlines are superimposed to the intensitydistribution of the UBO plotted over four decades.

is directed toward the center of the pulse corresponding to thepeaks of intensity distribution and multiphoton absorption. Thecentral core, that is, the intense part of the wave, is thereforecontinuously replenished by the conical nature of the pulseand this permits the stationary behavior in spite of losses.The energy flux carried by the conical tail actually exactlycompensates the amount of losses in the intense core.

VII. DISCUSSION AND CONCLUSIONS

Since there is a close relation between the energy flux andthe phase fronts, information on the energy flux constitutesa powerful tool for predicting if and how energy will beredistributed during propagation of the wave packet underexamination. We have seen that knowledge of the magneticfield, which would be necessary to build the Poynting vector,is not necessary with the expressions for the energy flux derivedin the present work. We emphasize here the approximations inwhich this study is valid.

First, it relies on a paraxial approximation. This maybe overcome by means of an effective correction to thelongitudinal component of the main wave vector, as inEq. (16), which leads to similar types of expressions for thecomponents of the energy flux [Eqs. (17) and (18)]. However,in experimental situations it is not always clear which angleshould be considered as θ0. In the paraxial case we may reduceto the assumption β = k0.

Second, it is based on a second-order expansion of k(ω)around ω0. It is possible to include higher-orders terms.Without going into the details, we give here the third-ordercorrection to the longitudinal component of the energy flux:

Jτ = ik′′0

2

(E∗ ∂E

∂τ− E ∂E∗

∂τ

)+ k′′′

0

6

(E∗ ∂2E

∂τ 2+ E ∂2E∗

∂τ 2− ∂E∗

∂τ

∂E∂τ

). (26)

Other approximations are linked to the fact that we haveneglected high-order operators in the propagation equation,such as the space-time focusing operator (which takes intoaccount the fact that for each frequency there is a different

023810-10


Rayleigh range) or self-steepening operator in the nonlinearcase. These operators would give corrections to the fluxexpressions derived in this article, which can be obtainedperturbatively as far as space-time coupling is concerned.The expressions obtained in the paraxial case represent agood approximation for wave packets with sufficiently narrowfrequency bands; however, the corrections to the flux wouldbe relevant for highly nonparaxial wave packets, single-cyclepulses or WPs with central frequency close to a zero in theGVD dispersion curve.

It has been shown that an envelope equation correctedwith space-time focusing and self-steepening terms, that is,the nonlinear envelope equation, still properly describes thepropagation of pulses at single-cycle durations [54]. Theexpression for the flux we have derived does not account forthese terms and is therefore not expected to be accurate aswe approach the single-cycle limit. A first possibility to makethese expressions more accurate for few-cycle pulses is toderive corrections due to the additional space-time focusingterm, in the linear case, by treating this term perturbatively asfor the derivation of the nonlinear envelope equation. However,this procedure would yield expressions for the flux which aredependent on the choice of the carrier frequency and thus of k0,whereas these quantities lose their significance for single-cyclepulses or large bandwidths. A second possibility would be toresort to unidirectional propagation equations such as thoseproposed by Kolesik and Moloney [55], which consider thepropagation of the electric field for any pulse duration andwould yield expressions for the flux that are independent ofany reference frequency or wave number. The group velocity ofthe pulse, however, enters the propagation equation expressedin moving frame [56]. It is thus expected to enter the derivedexpression for the flux as well as the dispersive properties of themedium. Finally, an elegant possibility to obtain expressionsfor the flux that are independent of reference frequencies orwave number, valid for few-cycle pulses, and explicitly linkedto the Poynting vector and electromagnetic energy densityis indicated in Appendix B in the case of propagation in anondispersive medium. A similar derivation for a dispersivemedium goes beyond the scope of the present article but wouldbe the easiest way to generalize the notion of energy densityflux for few-cycle pulses and in particular to apply it to conicalwaves carrying a large spectrum.

In conclusion, we defined a concept of energy densityflux which is slightly different from that of the Poyntingvector since it mainly relies on the propagation equationfor the complex amplitude of the wave packet and does notrequire knowledge of the magnetic field. The energy fluxis intimately connected to the knowledge of amplitude andphase of the electric field in the local reference frame (x, y, τ )of the pulse under examination. In the case of pulses withstrong spatiotemporal coupling, near-field measurements maybe quite difficult to interpret [15]. The phase information,if available, is in general even less clear. We have shownthat an analysis of optical pulses by means of the energydensity flux J is a valuable tool for extracting and representingthis phase information when dispersion of the medium canbe truncated at second order. In particular, we have deeplyanalyzed the case of CWPs for stationary and nonstationarypulses and we have extended the procedure to the nonlinear

case. The energy density flux was shown to be a useful toolfor retrieving information about the nature of the pulse andits evolution during propagation. The most appealing featureof this tool is the possibility to adopt it in an experimentalframework, as introduced in [32], thanks to the developmentof spatiotemporal intensity and phase retrieval techniques.

ACKNOWLEDGMENTS

Fruitful discussions with M. Cessenat are gratefullyacknowledged.

APPENDIX A: LINK BETWEEN THE ENERGY DENSITYFLUX AND THE POYNTING VECTOR

We start by considering the electric field as mainly directedalong the x axis and with zero y component (|E| Ex). FromMaxwell equations in the absence of free charges we have

divD = 0 (A1)

and in the Fourier space (only temporal transformation),D(ω) = ε0n

2(ω)E(ω). Equation (A1) may be written as

n2 ∂Ex

∂x+ ∂

∂z(n2Ez) = 0. (A2)

It is possible now to treat this equation perturbatively,considering E = E exp[ik0z − iω0t]. We obtain, consideringonly the dominant terms,

E =Ex ; 0; − 1

ik0

∂Ex

∂x

. (A3)

The magnetic field is obtained with a perturbative approachstarting from the Maxwell equation:

∂B∂t

= −rotE. (A4)

By also considering B = B exp[ik0z − iω0t], we obtain

B =

0;k0

ω0Ex − i

k0

ω20

∂Ex

∂t− i

ω0

∂Ex

∂z;

i

ω0

∂Ex

∂y

. (A5)

The Poynting vector in the case of a rapidly oscillating fieldis

〈S〉 = 〈E × H〉 = 1

2µ0Re

E × B∗. (A6)

With the preceding expressions,

Sx = 1

2µ0Re

− i

ω0Ex

∗ ∂Ex

∂x

, (A7)

Sy = 1

2µ0Re

+ i

ω0Ex

∂Ex∗

∂y

, (A8)

Sz = 1

2µ0Re

k0

ω0|Ex |2 + ik0

ω20

Ex

∂Ex∗

∂t+ i

ω0Ex

∂Ex∗

∂z

. (A9)

S is always given in W/m2. In the preceding equationsthe electric field is given in V/m. However, in derivingthe evolution equation for the intensity from that for theenvelope [Eq. (2)], we assumed that fields were in (W/m2)1/2.If we consider such a change of units in these equations, by

023810-11


mean of a factor ε0cn0, the preceding equations become

Sx = Re

− i

k0Ex

∗ ∂Ex

∂x

, (A10)

Sy = Re

+ i

k0Ex

∂Ex∗

∂y

, (A11)

Sz = Re

|Ex |2 + i

ω0Ex

∂Ex∗

∂t+ i

k0Ex

∂Ex∗

∂z

. (A12)

These equations show that the transverse components of theenergy-flux vector J, defined starting from the paraxial prop-agation equation, coincide with the transverse components ofthe Poynting vector S. The longitudinal components are alsorelated and the link is obtained by considering the continuityequation which produced the definition of J, derived fromthe propagation equation, and the continuity equation of thePoynting vector. These two equations read as

∂|E |2∂z

= −divJ = −∇⊥J⊥ − ∂Jt

∂t,

∂

∂t(we + wm) = −divS = −∇⊥S⊥ − ∂Sz

∂z,

where we and wm represent electric and magnetic energy den-sity, respectively. Since J⊥ = S⊥ within the approximationsinvolved, we thus identify

∂Jt

∂t= ∂

∂z(Sz − |Ex |2) + ∂

∂t(we + wm). (A13)

APPENDIX B: ENERGY FLUX WITHOUTENVELOPE APPROXIMATION

We have emphasized that the validity range of the expres-sions for the energy density flux derived in this work mainlyarise from the envelope approximation. We indicate here apossible route to generalize the expressions for the energydensity flux to the case of few-cycle pulses propagating in anondispersive medium. We link the generalized flux with thePoynting vector.

A. Potential function

Green and Wolf have shown that any electromagnetic fieldmay be rigorously derived from a single, generally complex,scalar wave function V (r, t) [33]. Under the gauge condition∇ · A = 0, where A denotes the usual vector potential. The realand imaginary parts of the scalar complex function V simplyrepresent the two components of A that are orthogonal to thewave vector. The function V allows a simple derivation of themomentum density g(r, t) and the energy density w(r, t) ofthe field. In a homogeneous isotropic medium, these quantitieswere shown to be represented by expressions analogous to theformulas for the probability current and the probability densityin quantum mechanics:

w = ε0

2

[∇V · ∇V ∗ + 1

c2

∂V

∂t

∂V ∗

∂t

], (B1)

S = cg = −ε0

2

[∂V ∗

∂t∇V + ∂V

∂t∇V ∗

]. (B2)

These quantities satisfy the usual conservation law,

∂w

∂t= −∇ · S, (B3)

and S is the standard Poynting vector.

B. Generalized flux for single-cycle pulses

Equation (B3) can be rewritten in a form similar to thatobtained from a unidirectional propagation equation where theevolution variable is z and time t plays the role of a longitudinalcoordinate:

∂Sz

∂z+ ∂w

∂t= −∇⊥ · S⊥. (B4)

By a change of reference frame ξ = z, τ = t − z/c, thederivatives are transformed as ∂/∂t = ∂/∂τ and ∂/∂z =∂/∂ξ − (1/c)∂/∂τ . Equation (B4) becomes

∂Sz

∂ξ= −∇⊥ · S⊥ − ∂

∂τ(w − Sz/c). (B5)

This is the generalization of the energy-conservation equation,without envelope approximation, which also establishes thelink with the usual Poynting vector. The left-hand sidedescribes the evolution of the longitudinal component of thePoynting vector Sz with respect to the evolution variable ξ ;it may be regarded as the energy density in the (x, y, τ )space since it represents the intensity of the optical field. Theright-hand side is the divergence (∇ · v = ∇⊥ · v⊥ + ∂vτ /∂τ )of the flux vector with components (S⊥, w − Sz/c) in theperpendicular and longitudinal τ directions, respectively.The longitudinal flux component is proportional to the dif-ference between the usual density of electromagnetic energyw and the longitudinal component of the Poynting vector. Thetransverse component of the flux is the same as that of thePoynting vector S⊥, whether the envelope approximation ismade or not. The flux components do not depend on the choiceof a carrier frequency or wave number. These expressions wereobtained in a medium without dispersion and would requireto be further generalized in dispersive media. Although thisfull generalization is beyond the scope of the article, one cannote that Eq. (B5) is similar to Eq. (2), which was derivedin a dispersive medium under the envelope approximation.The generalized flux is expected to depend on the dispersiveproperties of the medium since the electromagnetic energyw depends on the dielectric permittivity. We finally notethat the fact that Sz and w − Sz/c can be interpreted as anenergy density in the (x, y, τ ) space and a longitudinal fluxcomponent, respectively, is not obvious; thus, we show in thenext section that these expressions allow us to retrieve the fluxexpressions obtained under the envelope approximation.

C. Link with the envelope formulation

In order to show the connection with the envelope formu-lation, we assume that the field, and hence the potential V ,can be expressed as V = U exp[i(φ − ω0t + k0z)], where ω0

and k0 are the frequency and wave number of the carrier wave,U is a real envelope, and φ the phase of the potential V . The

023810-12


energy density and energy-flux components then read

Sz = −ε0

2

(∂V ∗

∂t

∂V

∂z+ ∂V

∂t

∂V ∗

∂z

)= ε0

U 2

[ω0k0 − ∂φ

∂t

∂φ

∂z− k0

∂φ

∂t

+ω0∂φ

∂z

]− ∂U

∂t

∂U

∂z

, (B6)

S⊥ = −ε0

2

(∂V ∗

∂t∇⊥V + ∂V

∂t∇⊥V ∗

)= ε0

[U 2

(ω0 − ∂φ

∂t

)∇⊥φ − ∂U

∂t∇⊥U

], (B7)

w − Sz/c = ε0

2

[∇⊥V · ∇⊥V ∗ + ∂V

∂ξ

∂V ∗

∂ξ

]= ε0

2

(∇⊥U )2 +

(∂U

∂ξ

)2

+U 2

[(∇⊥φ)2 +

(∂φ

∂ξ

)2]

. (B8)

The preceding equations are still exact. (No envelope approxi-mation was made.) In the envelope approximation, some termsin the preceding equations can be neglected. For example,(∂tU )(∂zU ) ω0k0U

2. The energy density and flux then read

Sz ∼ ε0ω0k0U2, (B9)

S⊥ ∼ ε0ω0U2(∇⊥φ), (B10)

∂

∂τ(w − gz) ∇⊥ · S⊥. (B11)

By introducing these expressions in Eq. (B5), we obtain

k0∂

∂ξU 2 = −∇⊥ · [U 2(∇⊥φ)]. (B12)

Since E = −∂A/∂t , in the envelope approximation, E =iω0A. The complex potential V just contains the componentsof the vector potential A; hence, U is proportional to E . Theconservation equation [Eq. (B12)] then becomes similar toEqs. (2) and (6) in the limit of a nondispersive medium. Theexpressions we have obtained (and the associated geometricalproperties) are therefore valid in the limit of the envelopeapproximation.

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energy-ﬂux characterization of conical and space-time

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