Veiled space-time Talbot effect

Murat Yessenov1,∗ Layton A. Hall1,∗ Sergey A. Ponomarenko2,3, and Ayman F. Abouraddy1

1CREOL, The College of Optics & Photonics, University of Central Florida, Orlando, FL 32816, USA2Department of Elect. and Computer Eng., Dalhousie University, Halifax, Nova Scotia B3J 2X4, Canada

3Department of Physics and Atmospheric Science,Dalhousie University, Halifax, Nova Scotia B3H 4R2, Canada

A freely propagating optical field having a periodic transverse spatial profile undergoes periodicaxial revivals – a well-known phenomenon known as the Talbot effect or self-imaging. We showhere that introducing tight spatio-temporal spectral correlations into an ultrafast pulsed opticalfield with a periodic transverse spatial profile eliminates all axial dynamics in physical space whilerevealing a novel space-time Talbot effect that can be observed only when carrying out time-resolvedmeasurements. Indeed, ’time-diffraction’ is observed whereupon the temporal profile of the fieldenvelope at a fixed axial plane corresponds to a segment of the spatial propagation profile of amonochromatic field sharing the initial spatial profile and observed at the same axial plane. Time-averaging, which is intrinsic to observing the intensity, altogether veils this effect.

The long-known Talbot effect [1, 2], or self-imaging [3],refers to a freely propagating paraxial optical field (at awavelength λo) having a periodic transverse spatial pro-file (of period L) undergoing periodic axial revivals (at

planes separated by the Talbot distance zT = 2L2

λo). The

field evolution unique to the Talbot effect can be readilyobserved in physical space. Space-time duality [4] sug-gests that an analogous temporal Talbot effect [5, 6] oc-curs when a periodic train of pulses (of period T ) travelsin a dispersive medium (with dispersion parameter k2):the pulses initially disperse but are subsequently recon-

stituted axially at multiples of zT = T 2

πk2. This temporal

evolution can be observed by a sufficiently fast detec-tor. The Talbot effect has been used in a wide span ofapplications [7] extending most recently from structuredillumination in fluorescence microscopy [8] to temporalcloaking [9] and prime-number decomposition [10].

An implicit – yet fundamental – assumption underly-ing the Talbot effect is the separability of the spatial andtemporal degrees of freedom of the optical field. Imposinga periodic profile in either space or time implies discretiz-ing the corresponding spectrum at multiples of 2π

L or 2πT ,

respectively, which raises a question with regards to theTalbot effect for pulsed optical fields in which the spatialand temporal degrees of freedom are inextricably inter-twined. Specifically, the spectral support domain of so-called ‘space-time’ (ST) wave packets tightly associateseach spatial frequency with a single temporal frequency(or wavelength) [11–13], such that discretization of onedegree of freedom inescapably entails discretization of theother. To date, such pulsed beams [14, 15] have been syn-thesized exclusively with continuous spectra tailored torender them propagation invariant [16–18]. Indeed, it isnow well-established that they are transported rigidly infree space at a fixed, but arbitrary group velocity [19–21].Self-imaging of ST wave packets lacking themselves a pe-riodic structure (thus retaining a continuous spectrum)

∗ These authors contributed equally to this work

was examined theoretically by superposing wave packetsof different phase velocities [22], and realized in spaceby superposing pulsed Bessel beams [23] (self-imaging inboth cases is thus unrelated to the Talbot effect).

We pose here a question regarding the propagation ofST wave packets having periodic transverse spatial pro-files, whereupon the spatial and temporal spectra aresimultaneously discretized on account of their tight asso-ciation. Two mutually exclusive scenarios appear to beon offer. If the spatio-temporal structure undergirdingpropagation-invariance plays the same role with discretespectra as with continuous spectra, then any axial dy-namics will be arrested and self-imaging thwarted. Al-ternatively, spatio-temporal spectral discretization maydisrupt the propagation invariance of periodic ST wavepackets and lead to self-imaging at the Talbot planes.

Here we show that aspects of both contradictory sce-narios are realized by ST wave packets when endowedwith a periodic spatial profile, leading to a remarkablephenomenon: a veiled ST Talbot effect. We find thatno spatial axial dynamics is discernible in the time-averaged intensity, and no axial temporal dynamics is dis-cernible in the space-averaged intensity; Talbot revivalsare altogether absent. This points to the impact of theunique spectral structure of ST wave packets trumpingthat of spectral discretization, whereby the initial pe-riodic spatial profile propagates self-similarly, and mayin fact display no spatial features whatsoever. Never-theless, time-resolved measurements of the spatial pro-file unveil spectral-discretization-induced axial dynamicsthat disrupts propagation invariance: the ‘temporal’ dy-namics at every axial plane recapitulates a different por-tion of the ‘spatial’ dynamics of the traditional Talboteffect, which can be shown to be a new manifestation of‘time-diffraction’ [24–26]. ST revivals occur at the Tal-bot planes with complex axial dynamics enfolding in in-tervening planes, which is veiled from view in physicalspace. Moreover, we encounter an unanticipated effect:the disparity of the transverse period observed in spacefrom that observed in space-time. Although the spatialspectrum is sampled at multiples of 2π

L , the observed pe-

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value L that is observed in space-time. These phenom-ena associated with the veiled ST Talbot effect indicatemore generally the rich repertoire of behavior intrinsic tofields endowed with precise spatio-temporal structures.

We first outline the theoretical basis for the veiled STTalbot effect (see Supplementary Material). Through-out we consider for simplicity optical fields propagatingalong z that are uniform over the transverse coordinatey (i.e., a light-sheet modulated along the transverse co-ordinate x). A monochromatic paraxial field at a fixedtemporal (angular) frequency ω = ωo and wave numberko = ωo

c diffracts along z [Fig. 1(a)]. Writing the field

as E(x, z; t) = ei(koz−ωot)ψx(x, z), the envelope is given

by ψx(x, z) =∫dkxψ(kx) exp i(kxx− k2x

2koz), where kx

is the transverse component of the wave vector (denoted

the spatial frequency), and the spatial spectrum ψ(kx)is the Fourier transform of ψx(x, 0). If the transversefield is periodic with period L, the spatial spectrum is

discretized kx → nkL, ψ(kx) → ψ(nkL) = ψn, and

ψx(x, z) =∑n ψn exp i2πn( xL − n z

zT); n is an integer

and kL = 2πL . Consequently, the initial profile is revived

periodically at the Talbot planes ψx(x,mzT) = ψx(x, 0)for integer m, with rich and striking dynamics enfold-ing between these planes [Fig. 1(b)]. Similar dynamicsis observed when pulsed fields are employed in which thespatial and temporal degrees-of-freedom are separable –as in most mode-locked lasers, which allows kx to be dis-cretized while maintaining Ω=ω − ωo continuous.

However, other pulsed-beam configurations have beenrecently explored in which the spatial and temporal de-grees of freedom are non-separable, leading to novel emer-gent behaviors including propagation invariance [16, 17],controllable group velocities [18, 20, 21, 27–30], trans-verse orbital angular momentum [31], ultrafast beamsteering [32], and omni-resonant interactions with planarcavities [33]. We consider here ST wave packets satisfyingthe spectral condition: Ω=(kz − ko)c tan θ, which repre-sents a plane in (kx, kz,

ωc )-space that makes an angle θ

(the spectral tilt angle) with respect to the kz-axis [13].the constraint k2

x + k2z = (ωc )2 imposes a relationship be-

tween the spatial and temporal frequencies in the parax-

ial regime Ωc (1 − cot θ) =

k2x2ko

, leading to a propagation-

invariant (diffraction-free and dispersion-free) envelope,

ψ(x, z; t) =

∫dkxψ(kx)eikxxe−iΩ(t−z/v) = ψ(x, 0; t−z/v),

(1)representing a wave packet traveling rigidly at a group

velocity v = c tan θ [Fig. 1(c)]; here ψ(kx) is theFourier transform of ψ(x, 0, 0). The group velocitycan be tuned by tailoring the spatio-temporal spec-tral support domain to change θ (thus resembling as-pects of tilted pulse fronts [34]). The time-averaged

intensity I(x, z) =∫dt|ψ(x, z; t)|2 =

∫dkx|ψ(kx)|2 +∫

dkxψ(kx)ψ∗(−kx)ei2kxx as recorded by a slow detector(e.g., a camera) is independent of z [Fig. 1(c)]. The fac-

tor of 2 in the exponent of the second term indicates thatI(x, z) is scaled spatially by a factor of 2 with respect tothe time-resolved intensity profile I(x, z; t)= |ψ(x, z; t)|2,a point we will return to shortly. Finally, finite exper-imental resources impose an unavoidable uncertainty inthe association between spatial and temporal frequencies,resulting in a ‘pilot’ envelope accompanying the ST wavepacket that travels at a group velocity of c [35]. This pi-lot envelope limits the temporal window over which theST wave packet can be observed.

When the transverse spatial profile is periodic and thespatial spectrum discretized kx → nkL, the tight as-sociation between spatial and temporal frequencies en-tails that the temporal frequencies are also discretizedΩ→ n2ΩL, ΩL= 2π

zTc

1−cot θ , and the envelope becomes

ψ(x, z; t) =

∞∑

n=−∞ψne

i2πnxL e−i2πn2 z

zT ei2πn2 z−ct

zT(1−cot θ) ,

(2)where we have recast the phase factors to separate outthe (z − ct)-term. The pilot envelope described above(accompanying the ST wave packet but traveling at c)is contracted after spectral discretization, so that we ob-serve a narrower temporal window (traveling at c) of theST wave packet (traveling at v). The pilot envelope thusacts as a limited-width ‘searchlight’ scanning the ST wavepacket axially. Indeed ψ(x,mzT; t − m zT

c ) = ψ(x, 0; t);that is, at the Talbot planes in a reference frame prop-agating at v = c, the initial field distribution at z = 0is retrieved. Furthermore, comparing the formulas forthe monochromatic field and the ST wave packet, wefind that ψ(x, z; t − z

c ) = ψx(x, z + ct1−cot θ ). In other

words, the time-resolved spatial distribution of a ST wavepacket at a fixed plane z corresponds to the axial spatialdistribution of a monochromatic periodic field after re-placing axial displacement with a scaled time variable;which is a manifestation of ‘time diffraction’ [24–26]; seeFig. 1(d). Recording instead the time-averaged intensityI(x, z) with a camera, for instance, yields

I(x, z) =∑

n

|ψn|2 +∑

n

ψnψ∗−n ei2·2π

xL , (3)

which indicates the complete absence of axial dynamics.Furthermore, the spatial frequency has doubled (corre-sponding to half the transverse spacing), so we expecta period of L

2 in I(x, z) in contrast to a period L inI(x, z; t), in addition to a scaling in the size of the fea-tures in each period [Fig. 1(d)].

We have carried out experiments to confirm this pre-dicted veiled ST Talbot effect by utilizing the two-dimensional pulse-shaper developed in Refs. [17, 20, 21]to synthesize superluminal ST wave packets at a spectraltilt angle of θ = 80, corresponding to a group veloc-ity v= tan 80≈ 5.67c, starting with femtosecond pulsesfrom a Ti:sapphire laser (see Supplementary Materialsfor details). Using a spatial light modulator, we sam-ple the spatial spectrum at nkL and concurrently sample

3

FIG. 1. Concept of the veiled ST Talbot effect. (a) The axial evolution of the intensity I(x, z) for a monochromatic (λo =800 nm)plane wave illuminating an aperture of width ∆x = 10 µm. On the left we depict the light-cone k2x + k2z = (ω

c)2 and the

representation of the spectral support domain of the field on its surface. (b) Plot of I(x, z) for a monochromatic field λo =800 nmwith periodic transverse spatial profile of period L= 100 µm and ∆x= 10 µm, exhibiting the traditional Talbot effect withzT = 25 mm. (c) The intensity I(x, z) for a ST wave packet with λo = 800 nm, ∆λ= 2 nm, ∆x= 10 µm, and θ = 80. Thelower panels are the invariant spatio-temporal profiles I(x, z; t) in a frame moving at v at selected axial planes. (d) Same as(c) except that the spatial spectrum is sampled at integer multiples of 2π

L, with L=100 µm; I(x, z) has a transverse period of

L2

= 50 µm rather than 100 µm, and there are no discernible axial changes. The lower panels are I(x, z; t) in a frame movingat c at selected axial planes, thereby revealing veiled periodic ST Talbot revivals. In each panel, the white dashed curve onthe left is the spatial profile at t=0, I(x, z; z

c), which evolves along z and is periodic with period L at z=0, whereas the white

curve on the right is the time-averaged intensity I(x, z), which is independent of z and is periodic with period L2

.

the temporal spectrum at n2ΩL over a temporal band-width of ∆λ ≈ 2 nm, corresponding to temporal fea-tures of width ∆τ ∼2.5 ps and spatial features of width∆x ∼ 25 µm in the spatio-temporal profile. We measureI(x, z) by scanning a CCD camera along z, the spatio-temporal spectrum, and I(x, z; τ − z

c ) measured interfer-ometrically at different axial planes in a reference framemoving at a group velocity of c.

We first sample the spatial spectrum at multiples of 2πL

with L= 80 µm [Fig. 2(a)]. Nevertheless, the measuredintensity [Fig. 2(a)] shows a periodic transverse profileof period L

2 = 40 µm rather than L = 80 µm, and notrace of the Talbot effect is observable. Indeed, the ax-ial propagation is invariant just as in previous studies ofST wave packets having continuous spectra. However,time-resolved measurements reveal a different picture al-together. The dynamics in the measured spatio-temporalprofiles I(x, z; τ − z

c ) [Fig. 2(b)] is completely veiled inthe time-averaged measurements [Fig. 2(a)]. We note therevivals at z=zT and 2zT, and those at z=0.5zT, 1.5zT,and 2.5zT, all with a transverse period of L = 80 µm.The patterns are shifted along x by L

2 midway between

the Talbot planes, whereupon the ‘dark’ and ‘bright’ fea-tures are exchanged with respect to the Talbot planes. Incontrast, if the spatio-temporal spectrum remains contin-uous without sampling, both the intensity I(x, z) and thespatio-temporal profile I(x, z; τ) lack periodicity and areinvariant at all axial planes (Supplementary Material).

To resolve the dynamics within a single Talbot length,we increase the period to L=160 µm (zT =64 mm) by in-creasing the spectral sampling rate. The measured inten-sity I(x, z) [Fig. 2(c)] remains invariant with z, shows noaxial dynamics, and has a transverse period L

2 = 80 µmrather than L = 160 µm. The spatio-temporal profilesof the ST wave packet measured at 5 planes within aTalbot length at z = 0.15zT, 0.25zT, 0.35zT, 0.5zT, and0.65zT reveal complex dynamics in which energy is ex-changed between the peaks in the profile [Fig. 2(d)]. Atz = 0.5zT the transverse period is L = 160 µm, and atz = 0.25zT is L

2 = 80 µm albeit with less contrast withrespect to the Talbot planes. At intermediate planes thespatio-temporal profile takes on quasi-random distribu-tions before returning to more recognizable profiles atTalbot half- and quarter-lengths. Once again, this dy-

4

FIG. 2. Measurements of veiled ST Talbot revivals. (a) Themeasured I(x, z) for a ST wave packet whose spatial spectrumis sampled at multiples of 2π

L(shown on the right); L=80 µm,

zT = 16 mm, θ= 80, λo ≈ 799.1 nm, ∆λ= 2 nm, and ∆x=10 µm. The transverse period is L

2= 40 µm, and no axial

dynamics is discernible. (b) Measured I(x, z; τ) at z=0.5zT,zT, 1.5zT, 2zT, and 2.5zT. The transverse period is L=80 µm,∆x≈20− 30 µm, and ∆τ≈2.5 ps. The white dotted line is aguide for the eye at x=0. (c) Same as (a) but with L=160 µm(zT = 64 µm). The observed transverse period is L

2= 80 µm.

(d) Measured I(x, z; τ) at z = 0.1zT, 0.25zT, 0.35zT, 0.5zT,and 0.65zT; ∆x ≈ 25 − 30 µm, ∆τ ≈ 2.5 − 3 ps, and themeasured transverse period is L=160 mm at z=0.5zT.

namics is fully hidden from view in the spatial intensity[Fig. 2(c)]. These experimental observations are all inexcellent agreement with simulations of ST wave packetsafter discretizing the spatio-temporal spectrum (Supple-mentary Material, Supplementary Movies 1 and 2).

The veiled nature of the ST Talbot effect can bebrought out even further by blocking half the spatial

spectrum; ψn = 0 when n < 0. The intensity becomes

a constant I(x, z) =∑∞n=0 |ψn|2 independent of x and

z. Measurements of I(x, z) are shown in Fig. 3(a) us-ing the same parameters from Fig. 2(a) except for theone-sided spectrum. The spatio-temporal profiles mea-sured at different axial planes reveal the underlying pe-riodic transverse axial structure that is altogether veiledin the intensity and verify the axial dynamics associated

FIG. 3. Measurements of the veiled ST Talbot revivals fora one-sided spatial spectrum. (a) The measured intensityI(x, z) with the same parameters in Fig. 2(a) (L=80 µm) ex-cept that only positive spatial frequencies are retained (shownon right). There are no discernible transverse spatial fea-tures altogether. (b) Measured spatio-temporal profiles atz=0.5zT, 1.25zT, 2zT, 2.25zT, and 2.5zT, where zT =16 mm,∆x≈20 − 30 µm, and ∆τ≈2 − 3 ps.

with time-diffraction. The ST Talbot effect is seen in theself-imaging revivals at z = 1.25zT and 2.25zT, and atz=0.5zT and 2.5zT (see Supplementary Movie 3).

The factor of 2 in the time-averaged intensity (Eq. 3)with respect to spatio-temporal envelope (Eq. 2) appearsregularly in correlation functions in the context of opti-cal coherence after averaging over a statistical ensemble.However, the underlying stochastic fields are inaccessi-ble, and the correlation functions are the only observ-ables. Here, we capture the intensity after time-averaging(which is entangled with the spatial degree of freedom[12]) and the underlying spatio-temporal profile prior totime-averaging, thus allowing an unambiguous observa-tion of the impact of this spatial-scaling factor in chang-ing the transverse period in the two distinct domains.

The unique behaviors observed and recent advancesachieved using ST wave packets rely on inculcating a pre-cise spatio-temporal structure into the optical field. Thecurrent work raises the question regarding which featuresof ST wave packets survive spectral discretization. Wehave shown that propagation invariance is maintainedin the time-averaged intensity after spectral discretiza-tion. Nevertheless, the underlying field exhibits complexdynamics in space-time associated with time-diffractionthat reveals self-imaging Talbot revivals, all of which iscompletely veiled in the spatial intensity profile. Thetime-resolved dynamics reported here is reminiscent ofquantum carpets [36] despite the absence of an index-confining structure. Although we have couched this phe-nomenon in terms of optical fields, it is applicable to anyother waves, such as acoustics or matter waves.

Acknowledgments. This work was funded by the U.S.Office of Naval Research, contract N00014-17-1-2458.

5

[1] H. F. Talbot, Philos. Mag. 9, 401 (1836).[2] M. V. Berry and S. Klein, J. Mod. Opt. 43, 2139 (1996).[3] W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967).[4] B. H. Kolner, IEEE J. Quantum Electron. 30, 1951

(1994).[5] T. Jannson and J. Jannson, J. Opt. Soc. Am 71, 1373

(1981).[6] F. Mitschke and U. Morgner, Opt. Photon. News 9, 45

(1998).[7] J. Wen, Y. Zhang, and M. Xiao, Adv. Opt. Photon. 5,

83 (2013).[8] S. Chowdhury, J. Chen, and J. A. Izatt,

arXiv:1801.03540 (2018).[9] B. Li, X. Wang, J. Kang, Y. Wei, T. Yung, and K. K. Y.

Wong, Opt. Lett. 42, 767 (2017).[10] K. Pelka, J. Graf, T. Mehringer, and J. von Zanthier,

Opt. Express 26, 15009 (2018).[11] R. Donnelly and R. Ziolkowski, Proc. R. Soc. Lond. A

440, 541 (1993).[12] H. E. Kondakci, M. A. Alonso, and A. F. Abouraddy,

Opt. Lett. 44, 2645 (2019).[13] M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F.

Abouraddy, Phys. Rev. A 99, 023856 (2019).[14] J. Turunen and A. T. Friberg, Prog. Opt. 54, 1 (2010).[15] H. E. Hernandez-Figueroa, E. Recami, and M. Zamboni-

Rached, eds., Non-diffracting Waves (Wiley-VCH, 2014).[16] P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997).[17] H. E. Kondakci and A. F. Abouraddy, Nat. Photon. 11,

733 (2017).[18] L. J. Wong and I. Kaminer, ACS Photon. 4, 2257 (2017).[19] J. Salo and M. M. Salomaa, J. Opt. A 3, 366 (2001).[20] H. E. Kondakci and A. F. Abouraddy, Nat. Commun.

10, 929 (2019).

[21] B. Bhaduri, M. Yessenov, and A. F. Abouraddy, Optica6, 139 (2019).

[22] K. Reivelt, Opt. Express 10, 360 (2002).[23] M. Bock, A. Treffer, and R. Grunwald, Opt. Lett. 42,

2374 (2017).[24] S. Longhi, Opt. Express 12, 935 (2004).[25] M. A. Porras, Opt. Lett. 42, 4679 (2017).[26] H. E. Kondakci and A. F. Abouraddy, Phys. Rev. Lett.

120, 163901 (2018).[27] A. Sainte-Marie, O. Gobert, and F. Quere, Optica 4,

1298 (2017).[28] D. H. Froula, D. Turnbull, A. S. Davies, T. J. Kessler,

D. Haberberger, J. P. Palastro, S.-W. Bahk, I. A. Begi-shev, R. Boni, S. Bucht, J. Katz, and J. L. Shaw, Nat.Photon. 12, 262 (2018).

[29] S. W. Jolly, O. Gobert, A. Jeandet, and F. Quere, Opt.Express 28, 4888 (2020).

[30] B. Bhaduri, M. Yessenov, and A. F. Abouraddy,arXiv:1912.13341 (2019).

[31] A. Chong, C. Wan, J. Chen, and Q. Zhan, Nat. Photon.14 (2020).

[32] A. M. Shaltout, K. G. Lagoudakis, J. van de Groep, S. J.K. J. V. V. M. Shalaev, and M. L. Brongersma, Science365, 374 (2019).

[33] A. Shiri, M. Yessenov, R. Aravindakshan, and A. F.Abouraddy, Opt. Lett. 45, 1774 (2020).

[34] J. A. Fulop and J. Hebling, in Recent Optical and Pho-tonic Technologies, edited by K. Y. Kim (InTech, 2010).

[35] M. Yessenov, B. Bhaduri, L. Mach, D. Mardani, H. E.Kondakci, M. A. Alonso, G. A. Atia, and A. F.Abouraddy, Opt. Express 27, 12443 (2019).

[36] A. E. Kaplan, P. Stifter, K. A. H. van Leeuwen, W. E.Lamb, Jr., and W. P. Schleich, Phys. Scr. 76, 93 (1998).

Veiled space-time Talbot effect: Supplementary material

Murat Yessenov1,∗ Layton A. Hall1,∗ Sergey A. Ponomarenko2,3, and Ayman F. Abouraddy1

1CREOL, The College of Optics & Photonics, University of Central Florida, Orlando, FL 32816, USA2Department of Elect. and Computer Eng., Dalhousie University, Halifax, Nova Scotia B3J 2X4, Canada

3Department of Physics and Atmospheric Science,Dalhousie University, Halifax, Nova Scotia B3H 4R2, Canada

S1. TRADITIONAL TALBOT EFFECT

A. Monochromatic field

Consider a monochromatic field at a temporal fre-quency ω = ωo, E(x, z; t) = ei(koz−ωot)ψx(x, z), with an

envelope given by ψx(x, z) =∫dkxψ(kx)eikxxei(kz−ko)z;

here the spatial spectrum ψ(kx) is the Fourier transformof ψx(x, 0), ko = ωo

c = 2πλo

is the wave number correspond-

ing to the temporal frequency ωo (wavelength λo), and cis the speed of light in vacuum. The dispersion relation-ship in free space enforces the relationship k2

z =k2o − k2

x.

In the paraxial limit, we have kz − ko ≈ − k2x2ko

, and theenvelope takes the form:

ψx(x, z) =

∫dkxψ(kx)eikxxe

−i k2x

2koz. (S1)

If the field at z= 0 is periodic along x with period L,ψx(x + mL, 0) =ψx(x, 0) for integer m, then the spatialspectrum is discrete and sampled at periods kL= 2π

L , so

that kx→nkL and ψ(kx)→ ψ(nkL) = ψn, where n is aninteger, and the envelope becomes

ψx(x, z) =∞∑

n=−∞ψne

i2πnxL e−i2πn2 z

zT , (S2)

where zT = 2L2

λois the Talbot length. Therefore, whenever

z=mzT (for integer m), we have

ψx(x,mzT) =

∞∑

n=−∞ψne

i2πnxL = ψx(x, 0). (S3)

That is, at the Talbot planes z=mzT, periodic revivals ofthe initial field distribution occur. The intensity I(x, z)=|ψx(x, z)|2 has the same period L along x at all the Talbotplanes. The evolution of the Talbot revivals are explicitin physical space.

∗ These authors contributed equally to this work

B. Pulsed field

If the field is pulsed E(x, z; t) = ei(koz−ωot)ψ(x, z; t),then the envelope takes the form

ψ(x, z; t) =

∫∫dkxdΩ ψ(kx,Ω)eikxxei(kz−ko)ze−iΩt,

(S4)where Ω = ω − ωo is the temporal frequency mea-sured respect to the fixed frequency ωo, and the spatio-

temporal spectrum ψ(kx,Ω) is the 2D Fourier transformof ψ(x, 0; t) with respect to x and t. For most laser pulseswe can make the assumption that the spatio-temporalspectrum is separable with respect to the spatial and

temporal degrees of freedom ψ(kx,Ω)≈ ψx(kx)ψt(Ω), in

which case ψ(x, 0; t) ≈ ψx(x)ψt(t), where ψx(kx) if the

Fourier transform of ψx(x) and ψt(Ω) is the Fourier trans-form of ψt(t). Besides the separability of the field in spaceand time at z = 0, the field remains approximately sep-arable along z if space-time coupling remains minimal,which requires narrow spatial and temporal bandwidths.

Within these limits, we have kz − ko≈ Ωc −

k2x2ko

, leadingto an envelope of the form

ψ(x, z; t)=

∫dkxψ(kx)eikxxe

−i k2x

2koz∫dΩψt(Ω)e−iΩ(t−z/c)

= ψx(x, z)ψt(t− z/c), (S5)

where the term ψx(x, z) is the same as that for amonochromatic beam in Eq. S1, and ψt(t−z/c) is a pulsetemporal envelope traveling at a group velocity c.

Because of the separability of the spatio-temporal spec-trum with respect to kx and Ω, we can discretize the spa-tial spectrum along kx while maintaining Ω a continuousvariable. Introducing a periodic spatial structure withperiod L along x, and utilizing the same nomenclaturefrom the previous subsection, results in a spatial envelopegiven by

ψx(x, z) =

∞∑

n=−∞ψne

i2πnxL e−i2πn2 z

zT , (S6)

which is identical to that of the periodic monochro-matic field in Eq. S2. Consequently, ψ(x,mzT; t) =ψx(x, 0)ψt(t − mzT/c); that is, the transverse periodicspatial pattern undergoes revivals at the Talbot planesaccompanied by a temporal envelope propagating at agroup velocity c independently of the spatial evolutionof the field. Once again, the spatial evolution associated

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with the Talbot effect is explicit in physical space forpulsed fields.

S2. SPACE-TIME TALBOT EFFECT

A. ST wave packets with continuousspatio-temporal spectrum

Because of the tight association between the spatialand temporal frequencies intrinsic to ST wave packets,the spatio-temporal spectrum is no longer a function ofkx and Ω. Rather, its dimensionality is reduced, and theenvelope takes the form

ψ(x, z; t) =

∫dkxψ(kx)eikxxe−i(kz−ko)ze−iΩt, (S7)

where Ω is no longer an independent variable, but de-pends on kx. There are two constraints on the valuesof kx, kz and Ω: (1) the dispersion relationship in freespace k2

x + k2z =(ωc )2, and (2) the definition of the tilted

spectral plane Ω=(kz−ko)c tan θ, where θ is the spectraltilt angle made by the plane with respect to the kz-axis.Therefore, the envelope can be written as

ψ(x, z; t) =

∫dkxψ(kx)eikxxe−iΩ(t−z/v) = ψ(x, 0; t−z/v).

(S8)The wave packet travels rigidly in free space at a groupvelocity v=c tan θ.

In anticipation of the discretization of kx when con-sidering the ST Talbot effect, we express the envelopeby an integral with the integrand expressed explicitly interms of kx. Implementing the same paraxial and nar-rowband approximations as in the traditional pulsed case

described above, we have kz − ko ≈ Ωc −

k2x2ko

. The addi-tional constraint associated with the spectral plane canbe used to eliminate kz, so that we have

Ω

c(1− cot θ) =

k2x

2ko. (S9)

For a plane-wave pulse (kx = 0, θ= 45), the dispersionrelationship is the light-line Ω = c(kz − ko), which in-dicates that the group velocity is c. The existence of aspatial spectrum (kx 6=0, θ 6=45) leads to deviation awayfrom this light-line, and the tight association between thespatial frequencies and temporal frequencies according toEq. S9 changes the group velocity from v=c to v=c tan θ,as seen in Eq. S8. We now rewrite the envelope of theST wave packet explicity in terms of kx,

ψ(x, z; t) =

∫dkx ψ(kx) eikxx e

−i k2x

2kozeik2x2ko

11−cot θ (z−ct)

.

(S10)In contrast to Eq. S5 that applies to a traditional pulsedbeam, the integral here does not separate into a prod-uct of functions of space and time. In anticipation of the

analysis below, note that the exponent with the termk2x2ko

,which is responsible for the change in the group velocity,will be neutralized at the Talbot planes once the spatialspectrum is discretized. We thus expect that the groupvelocity of a ST wave packet with periodic transverse dis-tribution will no longer be c tan θ, but will revert insteadto c.

We have two domains of interest that must be delin-eated here (see Fig. Supplementary Figure 1). The first isthe subluminal regime θ<45, whereupon v < c and theterm 1− cot θ is negative, so that Eq. S9 enforces Ω<0.In other words, ωo is the maximum temporal frequencyin the spectrum of a subluminal ST wave packet (ω<ωo).The second regime is superluminal with θ > 45, v > c,and 1 − cot θ > 0, so that Eq. S9 enforces Ω > 0. Inother words, ωo is the minimum temporal frequency inthe spectrum of a superluminal ST wave packet (ω>ωo).

We have assumed perfect correlations between the spa-tial and temporal frequencies; that is, each spatial fre-quency kx is ideally associated with a single temporalfrequency ω (or wavelength λ). Such an ideal scenariorequires infinite energy [1]. Any finite realistic systemproducing a finite-energy ST wave packet introduces anunavoidable ‘fuzziness’ in the association between thespatial and temporal frequencies that we term the spec-tral uncertainty δω (δλ on a wavelength scale) [2]; thespectral uncertainty is typically much smaller than thanthe temporal bandwidth δλ∆λ. As a consequence, theST wave packet has a finite propagation distance [3]. Therealistic finite-energy ST wave packet can be decomposedinto a product of an ideal ST wave packet traveling at agroup velocity v and a so-called ‘pilot’ wave packet thattravels at a group velocity c. The temporal width of thispilot envelope is the inverse of the temporal uncertaintyδω [2]. Because of the difference in group velocities of theST wave packet and the pilot envelope,v= c tan θ and c,respectively, the propagation distance Lmax of the STwave packet is related to this difference and the spectraluncertainty [2].

Observations made by a camera (or other ‘slow’ de-tector that cannot resolve the ST wave packet in time)scanned along the propagation direction z correspondto the time-averaged intensity I(x, z) =

∫dt|ψ(x, z; t)|2,

which takes the form

I(x, z) =

∫dkx |ψ(kx)|2 +

∫dkx ψ(kx)ψ∗(−kx)ei2kxx.

(S11)Several pertinent observations with regards to Eq. S11will be needed subsequently. First, Eq. S11 is altogetherindependent of z; i.e., there are no observable axial dy-namics in the time-averaged intensity evolution. Theprecise association between the axial wave numbers andtemporal frequencies according to Ω = (kz − ko)c tan θlead to the time-averaging process wiping out any z-dependence. Second, the first term in Eq. S11 is a con-stant background term that is independent of the fielddistribution. Third, the second term in Eq. S11 is re-sponsible for the transverse spatial variation in the time-

3

averaged intensity. If the spatial spectrum is an even

function ψ(−x)= ψ(x), then the field takes the form of apeaked function atop the background term; whereas an

odd function ψ(−kx) = ψ(kx) produces a dip below thebackground. Fourth, note the factor of 2 that appears inthe exponent in the second term in Eq. S11. This factorsignifies a reduction of transverse spatial scale by 2, whichroutinely appears in coherence theory. However, becausethe underlying field is not observable when using inco-herent sources, this factor has not been of critical impor-tance to date. We show below that this factor of 2 leadsto an interesting feature of the ST Talbot effect. Finally,

if the spatial spectrum is one-sided (i.e., ψ(kx)=0 whenkx<0), then the time-averaged intensity is reduced sim-

ply to the constant background I(x, z) =∫∞

0dkx|ψ(kx)|2

with no observable spatial variation.

B. ST wave packets with discretized spectrum

If a transverse periodic spatial pattern along x with pe-riod L is imposed at z= 0, then the spatial spectrum issampled at kx → nkL where kL= 2π

L as before. However,because the spatial and temporal frequencies are related,discretizing the spatial frequencies kx entails that thetemporal frequencies Ω are also discretized. However, be-cause Ω is related to kx quadratically as shown in Eq. S9,it is sampled at Ω → n2ΩL, where ΩL= 2π

zTc

1−cot θ . Theenvelope of the wave packet now takes the form:

ψ(x, z; t) =

∞∑

n=−∞ψne

i2πnxL e−i2πn2 z

zT ei2πn2 1

1−cot θz−ctzT .

(S12)

Note that the term e−i2πn2 z

zT vanishes at the Talbotplanes z = mzT, which indicates (as anticipated above)that the wave packet propagates at a group velocity v=crather than v = c tan θ as for its continuous-spectrumcounterpart.

We have separated out the term containing z−ct in theequation above. We show below in the description of thephase patterns for synthesizing periodic ST wave pack-ets that the spectral discretization increases the spectraluncertainty δω, thereby reducing the width of the pilotenvelope. The observed spatio-temporal profile will thusbe the portion of the ST wave packet revealed by thisnarrower pilot envelope [2]. Writing the ST wave packetenvelope in terms of z − ct therefore highlights the ob-servable spatio-temporal profile.

At the Talbot planes (z = mzT), the envelope in areference frame traveling at a group velocity v= c (ct=mzT) takes the form

ψ(x,mzT; mzTc ) =

∞∑

n=−∞ψne

i2πnxL = ψ(x, 0; 0), (S13)

indicating perfect revivals of the initial field at the Talbotplanes. The envelope observed in time at the Talbot

FIG. Supplementary Figure 1. Spatio-temporal spectral sup-port domain of subluminal and superluminal ST wave packetsprojected onto the (kx,

Ωc

)-plane. The spectrum is discretized

kx → nkL and Ω → n2ΩL.

planes (ct=mzT + cτ) has the form

ψ(x,mzT; mzTc + τ) =

∞∑

n=−∞ψne

i2πnxL e−i2πn2 1

1−cot θcτzT

= ψ(x, 0; τ), (S14)

so that the time-resolved profile at the Talbot planes (inthe moving reference frame) corresponds to the profileat the initial plane z = 0. Note that this equation isidentical to the Talbot effect effect as displayed by themonochromatic field or the spatial part of the pulsed field(when the spatial and temporal degrees of freedom areseparable),

ψ(x, z;

z

c+ τ)

= ψx

(x, z +

cτ

1− cot θ

). (S15)

The axial coordinate z is replaced by z= cτ1−cot θ . In other

words, the profile observed in space along the axis in thetraditional Talbot configuration is now observed in thetime domain at fixed axial planes.

The time-averaged intensity I(x, z)=∫dt|ψ(x, z; t)|2 is

given by

I(x, z) =∞∑

n=−∞|ψn|2 +

∞∑

n−∞ψnψ

∗−n ei2·2πn

xL . (S16)

Note the absence of any dependence on z, so thatthe underlying complex dynamics associated with time-diffraction is altogether veiled in physical space. Alsonote the appearance of the factor of 2 in the exponent,which indicates that the transverse spatial period ob-served in physical space is L/2 rather than L. There-fore, although the spatial spectrum is sampled at integer

4

FIG. Supplementary Figure 2. (a) Calculated time-averagedintensity I(x, z) using the same experimental parameters fromFig. 2(a) in the main text: ∆λ= 2 nm, θ= 80, L= 80 µm,and ∆x=10 µm. (b) The spatio-temporal profiles calculatedat particular axial planes in (a) extending over two Talbotlengths. The dotted white line is a guide for the eye. (c)Same as (a), but using the same parameters as in Fig. 2(c) inthe main text: L= 160 µm and ∆x= 8 µm. (d) The spatio-temporal profiles calculated at particular axial planes in (c)within a single Talbot period.

multiples of kL= 2πL and the time-resolved measurements

of the envelope profile reveals a periodic distribution ofperiod L, observing the field with a slow detector re-veals a periodic distribution of period L

2 . In other words,twice the number of peaks will be observed in the time-averaged intensity than in the time-resolved profile, withhalf of those peaks not corresponding to underlying time-resolved peaks. Finally, when the spatial spectrum is

one-sided (ψn=0 if n<0), then the time-averaged inten-

sity is a constant I(x, z)=∑∞n=0 |ψn|2 with no observable

spatial variations.

FIG. Supplementary Figure 3. (a) Calculated time-averagedintensity I(x, z) using the same experimental parameters fromFig. 3(a) in the main text: ∆λ= 2 nm, θ= 80, L= 80 µm,and ∆x=14 µm. (b) The spatio-temporal profiles calculatedat particular axial planes in (a) extending over two Talbotlengths. The dotted white line is a guide for the eye.

S3. SIMULATIONS OF THE SPACE-TIMETALBOT EFFECT

A. Simulations corresponding to the configurationsin the main text

We have calculated the time-averaged intensity I(x, z)and the spatio-temporal profile I(x, z; t) at the axialplanes corresponding to the measurements reported inFig. 2 and Fig. 3 in the main text. We make use ofthe theoretical model for ST wave packets developed inRefs. [2, 4], which includes the finite spectral uncertaintythat sets a limit on the propagation distance. The simu-lations corresponding to Fig. 2 in the main text in which aperiodic profile is imposed on the ST wave packet throughspectral discretization are plotted in Supplementary Fig-ure 2. The simulations corresponding to Fig. 3 in themain text in which only one half of the discretized spatialspectrum is taken into consideration, thus eliminatingand spatial features in the time-averaged intensity, areplotted in Supplementary Figure 3. In all cases, we ob-serve excellent agreement between the experimental ob-servations and the simulation results.

B. Simulations for the effect of the spectral tiltangle

In the experiments reported in the main text, we madeuse of a spectral tilt angle θ=80 corresponding to a su-perluminal group velocity of v = c tan 80 = 5.67c. Wepresent here simulations of the veiled ST Talbot effectfor other values of the spectral tilt angle spanning thesubluminal and superluminal regimes. In Supplemen-tary Figure 4 we plot simulations for subluminal ST wavepackets for θ= 35 (v= 0.7c) and θ= 40 (v= 0.84c); in

5

FIG. Supplementary Figure 4. Simulations of the veiled Tal-bot ST effect for subluminal ST wave packets with λo =800 nm, ∆λ = 2 nm, and L = 160 µm. (a) The spectraltilt angle is θ = 35. Top row shows the time-averaged in-tensity I(x, z) over an axial propagation distance of 64 mm,and the discretized spatio-temporal spectral intensity. Thelower panels are the calculated spatio-temporal intensity pro-files calculated at z = 0, 16, 32, and 64 mm. (b) Same as(a) for θ = 40. The transverse spatial period in I(x, z) isL2

=80 µm for both values of θ.

Supplementary Figure 5 we plot simulations for superlu-minal ST wave packets for θ=60 (v=1.73c) and θ=80

(v=5.67c) for comparison. The spectral uncertainty usedin the calculations is δλ=15 pm.

In all cases, we note the self-revivals at the Talbotplanes, the shift by L

2 between the Talbot planes and themid-Talbot planes, and the doubling of the transverseperiod at the quarter-Talbot planes. The only impact ofthe spectral tilt angle is a change in the temporal scaleat each axial plane.

FIG. Supplementary Figure 5. Simulations of the veiledTalbot ST effect for superluminal ST wave packets withλo = 800 nm, ∆λ = 2 nm, and L = 160 µm. (a) The spec-tral tilt angle is θ = 60. Top row shows the time-averagedintensity I(x, z) over an axial propagation distance of 64 mm,and the discretized spatio-temporal spectral intensity. Thelower panels are the calculated spatio-temporal intensity pro-files calculated at z = 0, 16, 32, and 64 mm. (b) Same as(a) for θ = 80. The transverse spatial period in I(x, z) isL2

=80 µm for both values of θ.

C. Description of the Supporting movies

Three Supplementary Movies have been provided thatoffer a complete picture of the evolution of the veiled STTalbot effect. Each frame in he movie combines two pan-els. The top panel is corresponds to the time-averagedintensity I(x, z), which shows the diffraction-free behav-ior of the ST wave packet and a transverse period of L

2rather than L. A vertical line identifying an axial plane zmoves from the left (the source) to the right. The lowerpanel is the spatio-temporal profile I(x, z; t) correspond-ing to the axial plane z identified simultaneously in theupper panel. The temporal window over which the profileis shown moves at a group velocity of c.

Parameters common to the three movies are the tem-

6

FIG. Supplementary Figure 6. Schematic depiction of theoptical arrangement for the synthesis of ST wave packets. Theacronyms for the optical components are all explicated in thelegend. The setup consists of three sections highlighted withdifferent background colors: (1) ST field synthesis consistingof a cylindrical lens in a 2f configuration with a diffractiongrating and SLM; (2) ST spectral analysis, which comprises asingle spherical lens and CCD camera; and (3) time-resolvedmeasurements performed via a Mach-Zehnder interferometerthat superposes a reference pulse and the synthesized ST wavepacket to measure its time-resolved spatial profile.

poral bandwidth ∆λ = 2 nm and the wavelength λo =800 nm, which are selected to match the experimentalconfiguration. The spectral uncertainty used in the cal-culations is δλ=15 pm.

Supplementary Movie 1: This movie shows a contin-uous simulation of the veiled ST Talbot effect along theaxis of a ST wave packet having a periodic transverseprofile at z=0. Here we have L=100 µm and a spectraltilt angle of θ=80. The simulations follow the center ofthe wave packet over a full Talbot length of zT =25 mm.

Supplementary Movie 2: This movie shows a con-tinuous simulation of the veiled ST Talbot effect alongthe axis of a ST wave packet similar to that depicted inSupplementary Movie 1 (L = 100 µm), except that thespectral tilt angle has now changed to θ=60. The sim-ulation follows the center of the pulse over a distance ofzT =25 mm.

Supplementary Movie 3: This movie shows a con-tinuous simulation of the veiled ST Talbot effect alongthe axis of a ST wave packet similar to that depictedin Supplementary Movie 1 (spectral tilt angle θ = 80)except that the spatial spectrum is one-sided. That is,only positive spatial frequencies are allowed, thus result-ing in a time-averaged spatial profile with no observablespatial features. Here we have L= 80 µm, and the sim-ulations follow the center of the pulse over a distance ofzT =16 mm.

S4. EXPERIMENTAL CONFIGURATION

A. Experimental setup

The experimental setup for synthesizing ST wavepacket is similar to that used in our previous work[5, 6], and combines elements from ultrafast pulse shap-ing and spatial beam modulation. Plane-wave pulsesat a wavelength of ∼ 800 nm from a Ti:sapphire laser(Tsunami Spectra Physics, ∼ 100-fs pulsewidth) are inci-dent on a diffraction grating (Newport 10HG1200-800-1)followed by a cylindrical lens in a 2 − f configuration;see Fig. Supplementary Figure 6. At the Fourier planeof the lens, where the spectrum of the optical wave ismapped along the horizontal direction (y-axis), we im-part a two-dimensional phase distribution to the spec-trally resolved wave front using a reflective phase-onlyspatial light modulator (SLM; Hamamatsu X10468-02).In contrast to our previous work, here the SLM impartsa phase pattern that is discretized along the wavelengthdimension [2, 5, 7]. Such a pattern corresponds to a dis-crete set of paired spatial and temporal frequencies. Thefield with the discrete spatio-temporal spectral supportdomain is retro-reflected and the pulse is reconstituted atthe diffraction grating, thereby generating the ST wavepacket.

B. Designing the SLM phase pattern

In order to associate the desired kx value to each wave-length, a diffraction grating is used to spread the tempo-ral spectrum in space, and a cylindrical lens collimatesthe separated wavelengths along the y-axis as shown inFig. Supplementary Figure 7. Each wavelength occupiesa column on the SLM (x-axis) where a linear phase pat-tern of the form kxx (modulo 2π) is implemented, whichcorresponds to assigning the spatial frequency kx to thiswavelength. An example corresponding to θ = 80 isshown in Fig. Supplementary Figure 7(a) for a continuoustemporal spectrum. In the case of a discrete spectrum,kx → n 2π

L and Ω→ n2ΩL (n integer), which is achievedby activating only specific columns on the SLM (corre-sponding to the desired frequencies n2ΩL) and along eachcolumn we implement the phase associated with the cor-responding spatial frequencies nkL. To provide sufficientoptical signal, we select a finite width for each columnof the phase pattern along y for each wavelength. Thephase pattern along x is maintained the same across thewidth of each column (fixed spatial frequency, nkL). Thewidth of the columns increases the spectral uncertaintyδω, thereby reducing the temporal width of the pilot en-velope as described above [2].

7

FIG. Supplementary Figure 7. (a) Two-dimensional phase pattern imparted by the SLM to the incident spectrally resolvedwave front. The wavelengths are resolved along the horizontal axis, and the phase in each column is kxx (a phase factor eikxx)with kx(Ω) selected to realize θ= 80. (b) The phase pattern from (a) discretized along the temporal frequency (wavelength)quadratically Ω → n2ΩL. Consequently, the spatial frequencies in the columns take on discrete values kx → nkL. In both (a)and (b), λo =800 nm, ∆λ=2 nm, and θ=80, and in (b) we have L=160 µm.

FIG. Supplementary Figure 8. (a) Measured time-averaged intensity I(x, z) for a ST wave packet having the same parametersas Fig. 2 in the main text except that the spatio-temporal spectrum is continuous rather than discrete. The measured spatio-temporal spectral intensity is plotted on the right. (b) Measured spatio-temporal profiles I(x, z; τ) in a moving reference framemoving at a group velocity v and not c, obtained at different axial planes z (taken at 5-mm axial intervals). The spatio-temporalprofile does not change axially in contrast to the periodic ST wave packets resulting from spectral discretization.

C. Characterization

The synthesized ST field is characterized in three dif-ferent domains.

1. Fourier-space analysis. The spatio-temporalspectrum is characterized in Fourier space (kx, λ)to ensure that the desired spatio-temporal spec-tral support domain is indeed implemented by theSLM. By sampling the spectrally resolved field of

the retro-reflected field from the SLM and imple-menting a spatial Fourier transform via the spheri-cal lens in a 2−f configuration, we resolve the fieldin (kx, λ) space at the focal plane of the lens, whereit is captured by a CCD camera (CCD1; TheImag-ingSource, DMK 33UX178); see Fig. Supplemen-tary Figure 6. Using this approach, we obtained thedata in the main text in Fig. 2(a,c) and Fig. 3(a).

2. Spatial intensity profiling in physical space.To record the time-averaged intensity I(x, z), we

8

scan a CCD camera (CCD2; TheImagingSource,DMK 27BUP027) along the propagation axis z; seeFig. Supplementary Figure 6. Using this approach,we obtained the data in the main text in Fig. 2(a,c)and Fig. 3(a).

3. Time-resolved spatial profile. Using a Mach-Zehnder interferometer in which with a short ref-erence laser pulse (100-fs pulsewidth) from theTi:sapphire laser directed to a reference arm con-taining an optical delay line, and the ST wavepacket synthesized in the second arm, interferenceof the two wave packets occurring when they over-lap in space and time reveals high-visibility spa-tially resolved fringes. From this we obtain thetime-resolved intensity I(x, z; τ) at locations of in-terest along z-axis. Using this approach, we ob-tained the data in the main text in Fig. 2(b,d) andFig. 3(b).

In the main text (Fig. 2 and Fig. 3), we presented mea-surements for ST wave packets having a periodic trans-verse profile that result from spectral discretization. Themeasurements in the text demonstrated the veiled STTalbot effect. Specifically, the measured spatio-temporalprofiles I(x, z; τ − z

c ) was measured in a reference framemoving at a group velocity of c, and the profiles changedalong the propagation axis z. We present here for com-parison measurements of the same ST wave packet in ab-sence of spectral discretization. We make use of the sameparameters for the ST wave packet in Fig. 2 in the maintext: ∆λ= 2 nm, λo = 800 nm, and θ = 80. The mea-surements are plotted in Supplementary Figure 8. Thetransverse profile of the time-averaged intensity I(x, z)(Supplementary Figure 8A) is no longer periodic. Cru-cially, the spatio-temporal profiles (Supplementary Fig-ure 8B) are invariant along z in contradistinction to theaxially varying results when the spectrum is discretizedand the transverse profile is periodic. Furthermore, themeasurements are carried out in a reference frame mov-ing at a group velocity of v=c tan θ rather than c.

[1] A. Sezginer, J. Appl. Phys. 57, 678 (1985).[2] M. Yessenov, B. Bhaduri, L. Mach, D. Mardani, H. E.

Kondakci, M. A. Alonso, G. A. Atia, and A. F.Abouraddy, Opt. Express 27, 12443 (2019).

[3] H. E. Kondakci, M. A. Alonso, and A. F. Abouraddy,Opt. Lett. 44, 2645 (2019).

[4] M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F.

Abouraddy, Phys. Rev. A 99, 023856 (2019).[5] H. E. Kondakci and A. F. Abouraddy, Nat. Photon. 11,

733 (2017).[6] M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F.

Abouraddy, Opt. Photon. News 30, 34 (2019).[7] M. Yessenov, B. Bhaduri, P. J. Delfyett, and A. F.

Abouraddy, arXiv:1910.05616 (2019).