Tracing the phase of focused broadband laser …focus of a few-cycle laser pulse. This is accomplished with two separate but synchronized measurement chambers, see Fig. S1. First we

In the format provided by the authors and unedited.

Tracing the phase of focused broadband laser pulses, Supplementary Information

Dominik Hoff, Michael Kruger, Lothar Maisenbacher, A. M. Sayler, Gerhard G. Paulus, and Peter Hommelhoff

I. MAPPING THE FOCAL PHASE

The goal of this setup is to map the phase within thefocus of a few-cycle laser pulse. This is accomplished withtwo separate but synchronized measurement chambers,see Fig. S1. First we need to determine the carrier-envelope phase (CEP) of each and every 4.0± 0.5 fs pulsein the 4 kHz pulse train. This is necessary as the CEP israndomly varying from shot to shot. Note that one couldalternatively lock the CEP using a variety of techniques.However, we have found the technique of locking andscanning the CEP to be both less robust and less accuratethan measuring the CEP for every laser shot as it variesrandomly and then subsequently sorting and binning thedata1.

Measuring the relative CEP is accomplished using acarrier-envelope phase meter (CEPM), as shown in Sup-plementary Fig. S1. The measurement is relative since wedo not know the absolute relationship between the CEPmeasured in the CEPM, φPM , and the CEP at the focalpoint, (r, z)=(0, 0), in the nanotip chamber, φNT

0 . How-ever, the relationship is constant, i.e. φNT

0 = φPM + δ,where the offset, δ, does not change from shot to shot asall optical elements and the paths to both chambers arefixed. Thus, we know the CEP for every shot at the focalpoint in the nanotip chamber up to a constant offset.Although this is a relative CEP measurement and the

CEP changes randomly from shot to shot, one can stillmap the spatial dependence of the CEP in the nanotipchamber. Specifically, to measure how the CEP evolvesin space near the focal point, one calculates

∆φ(r, z) = φNT (r, z)− φNT0 , (S1)

where both the offset, δ, and the absolute phase of eachshot cancel out. Thus, to simplify the following descrip-tion, the distinction between the relative and absoluteCEP will be dropped.To determine the CEP for different points we move

the nanotip with a 3D-translation stage to those pointsand measure the above-threshold photoemission (ATP)spectrum. The ATP spectrum is used for two primaryreasons. First, the ATP spectrum is very sensitive tothe CEP2, just like its gas-phase counterpart, above-threshold ionization3. Second, the tip significantly en-hances the electric field at the ∼ 10 nm apex of thetip, which ensures that virtually all the emitted elec-trons come from that region. This is not the case forgas jet targets, which are typically much larger and re-sult in electrons emitted from a volume with a range ofintensities and CEPs.Determining the CEP for a single nanotip position is

done in several steps. First, for each laser shot the CEPin the CEPM, φPM , is recorded along side the time-of-flight, TOF, of all electrons emitted by the same shot inthe nanotip chamber. This is done for 10 − 15 minuteswith an average rate of less than one electron per shotand results in a large list of data with � 1,000,000 entriessynchronized by laser shot. Second, this list is sortedaccording to the CEP measured in the CEPM, φPM ,and the flight times are converted to energies using thespectrometer length. Third, this list is binned in phase,φPM , and energy, E, to produce a two-dimensional his-togram of the ATP yield like the inset in SupplementaryFig. S2. Fourth, to more clearly see the CEP depen-dence, we look at the asymmetry, A, in the yield. For alinearly polarized pulse in the atomic physics case, thisis typically defined as the difference between those elec-trons flying left, L, and right, R, divided by the sum,i.e. A = (L−R)/(L+R). However, in the case of thenanotip electrons can only propagate in one direction —away from the tip. Nevertheless, we can still determinethe asymmetry, A, by using the symmetry of the laserpulse, namely the fact that cos(ωt) = − cos(ωt − π).Therefore, we define the asymmetry as a function of theyield, Y , for a given energy-, E, and phase-bin, φ, as

A(E, φPM ) =Y (E, φPM )− Y (E, φPM − π)

Y (E, φPM ) + Y (E, φPM − π). (S2)

As shown in Fig. S3, this allows us to clearly observethe CEP dependence. Fifth, we choose to look at a spe-cific energy region. Namely, we use those electrons withenergies above the cutoff for classically allowed rescat-tered electrons, which occurs above 10 times the pon-deromotive energy, UP , i.e. above ∼ 30 eV in this case.For a detailed discussion of ATI trajectories and featuressee Ref. 4. This region is selected because the asym-metry is strongest and has the least energy dependence,see Fig. S3. Although there are significantly more elec-trons at lower energies, the asymmetry is weaker, changesquickly with energy and is much more dependent on thelaser parameters1. Sixth, A(φPM ) is well described by acosine wave. Therefore, we fit the data integrated overthe aforementioned energy gate, E > 10 UP , at a specificnanotip position with the function

A(E > 10 UP ;φPM ) = aposco · cos(φPM + φpos

co ) , (S3)

where the position-dependent cutoff asymmetry phaseand amplitude are φpos

co and aposco , respectively. This en-tire procedure allows us to represent the local phase witha single value, φpos

co . The uncertainty of which resultsfrom adding the error in the cosine fit (statistical) andfrom the gate positioning (systematic) in quadrature to≈ 150mrad in average.

Tracing the phase of focused broadband laser pulses

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2

FIG. S1. Overview of the experimental setup. At the beam splitter (BS) an amplified 4 fs, 700 nm central wavelength,horizontally polarized pulse train from a laser system with a hollow core fibre compressor is split 65:35% towards an time-of-flight-spectrometer (TOF) arm, and a CEP-meter arm (CEPM), respectively. Fused silica wedges (W) independentlycontrol and minimize the dispersion in each arm. The spatial mode of the beam to the experiment is cleaned with a spatialfrequency filter (SFF) and its intensity is adjusted by a neutral density filter (ND). The beam is focused inside the ultrahighvacuum chamber with a 90◦ off-axis parabolic mirror (OAP, f = 50mm) on the nanotip. The nanotip is mounted on a xyz-nanopositioning stage that also carries a gas jet (GJ) for Xe measurements and two knife-edges (KE) for focus characterisation.The TOF spectrometer records the flight time of photoelectrons with a microchannel-plate (MCP) that is then convertedinto kinetic energy. CCD cameras in front (CCD1) and behind the chamber (CCD2) were used for spectral beam profilecharacterization and the knife-edge measurements, respectively. The MCP and phosphor screen was observed via a mirror anda camera and was used for tip diagnostics.

With all this in place we can now map the phase evo-lution of the focused pulses with the nanotip. For ex-ample, assume that one moves the nanotip to the focus,(r, z)=(0, 0), measures the ATP spectrum, and retrievesthe corresponding φpos

co (r=0, z=0) value. Then if the na-notip is moved to another position, (r, z), all the laserparameters are kept constant, and nothing is changed ineither of the beam paths; then one would expect to mea-sure exactly the same ATP spectrum and resulting φpos

co

value only with a shift in phase equal to the phase dif-ference between the two points in space, ∆φ. In otherwords,

φposco (r, z) = φpos

co (0, 0) + ∆φ(r, z). (S4)

In this way one can map the phase evolution, ∆φ(r, z),of the focused laser pulses with respect to the focal pointas shown in Figs. 3 and 4, where the uncertainty due tothe referencing is also added in quadrature to a sum of≈ 0.2 rad.

In practice there are several additional complications.In the example above, all the laser parameters, except thephase, were identical at each point. However, the inten-sity within the focal region is spatially dependent and theATP spectrum from the nanotip is intensity dependent.Thus, we adjusted the intensity of the beam going to

the nanotip with a neutral density filter for each positionwhile the fraction of the beam going into the CEPM re-mained untouched to maintain the reference phase. Theintensity was adjusted to match the emission rate seen atthe focal point and thus ensure that the local intensityat each new nanotip position was the same. We foundthat the error potentially caused by a small movement ofthe ND-filter is well below 0.14 rad and hence negligible.Further, to minimize drifts of the relative phase due toambient fluctuations, periodic measurements at the focalpoint, (r, z) = (0, 0), were made to account for drifts inthe set-up.

In order to determine the focus position in space, thenanotip translation stage also carries two perpendicularknife-edges to perform a measurement of the focal profile,revealing the focus position with a precision of about50 µm. To further fine tune the tip position, it was movedto the point at which the photoelectron emission wasmaximized. Finally, the focal point was defined to bestsymmetrize the recorded phase values φpos

co (0, z) as thephase profile is an odd function in z.

To map the focal phase profile in two-dimensions wedid not use a cartesian grid. The radial scale of the laserin the focus is defined by the beam width, w(z). Thus,for each z-position we chose to follow the hyperbolae at






3

FIG. S2. Closeup of the measurement. A horizontallypolarized, broadband, pulsed Gaussian beam (bright orange)is focused in space and triggers photo-electron emission froma metal nanotip (grey, red sphere indicates electron). Theelectron energy is measured via the time-of-flight method.The acquired kinetic energy of the electrons depends on thecarrier phase at the maximum of the laser pulse envelope(carrier-envelope phase, CEP, see insets, vertical orientationhere for illustration only). We record the photo-electron ki-netic energy as a function of the CEP, which was simul-taneously determined for each and every laser shot with astereo above-threshold ionization CEP-meter. Accumulationof these detected electron events results in an energy- andCEP-dependent spectrum as shown in the background. Themetal nanotip is positioned at several points in the beam pathwhere the CEP is extracted from the photo-electron spectra.With these values, the spatially dependent CEP evolution canbe determined, illustrated for instance by the blue on-axiscurve.

w(z)/√2, w(z), and w(z) ·

√2 (Supplementary Eq. S6).

Note that this difference in coordinate system choices canbe seen in Fig. 1, which is shown in standard cylindricalcoordinates, and Fig. 4 and S5, which are displayed inscaled coordinates, r/w(z). Finally, to fit the measuredpoints, obtain values for g0, and produce the maps dis-played here, Eq. 2 was used and the zR-parameter wasset to the value obtained by the knife-edge measurement.

II. EXPERIMENTAL DETAILS

A schematic of the experimental setup is shown inFig. S1. For these measurements we used an up-graded commercial laser system (Femtopower CompactPro HP/HRTM) and a hollow-core fibre compressor witha central wavelength tunable from 680 to 750 nm. Thissystem produces a horizontally polarized few-cycle pulsetrain at 4 kHz with a randomly varying CEP. This beamis split 65:35% between the nanotip chamber and theCEP-meter, respectively, and the dispersion in both armsis controlled and minimized independently with fused

FIG. S3. Determination of the phase in a photoelec-tron spectrum. Photoelectron asymmetry as a function ofrelative CEP-meter phase and energy. In order to determinethe phase, we chose a gate-region including all electrons withenergies above the cutoff for classically allowed rescatteredelectron trajectories (black).

silica wedges by optimising on the asymmetry in theATI-signal of Xe gas, see Ref. 5. With a transformlimited pulse duration of 3.7 fs, actual pulse durationsof 3.75± 0.25 fs are achieved, thus providing practicallychirp-free pulses as assumed for the model used6.

The intensity of the beam going to the nanotip cham-ber is independently controlled using a reflective neutral-density filter. It is also spatially cleaned with an aper-ture. The beam is focused onto the nanotip with a f =50mm, 90◦ off-axis parabolic mirror inside the ultra-highvacuum chamber, hence minimizing the effect of aberra-tions on the CEP as discussed in Refs. 6, 7. A backgroundpressure of ∼ 10−9 mbar is reached. The effective inten-sity at the tip’s apex reaches 3.5± 2.0× 1013 W/cm2 in-cluding the field enhancement at nanometric metal tips8.

The tip is mounted on a 3D-translation stage (Smar-Act), which also carries two knife-edges for focus charac-terisation in x- and y-directions. The z-positioning un-certainty is less than 1µmwhile the uncertainty in r/w(z)is less than 0.1, where the radial uncertainty scales withw(z). Note that the translation stage also carries the gasnozzle, which was used to optimize the pulse compres-sion by evaluating the photo-electron emission from Xe.A microchannel-plate-detector with a phosphor screen isused for tip diagnostics using field emission and field ionmicroscopy9,10. CCDs in front (CCD1) and behind thechamber (CCD2) are used for spectrally resolved beamprofile characterisation and the knife-edge focus size mea-surements. We obtained an M2 of 1.2 for the beam, henceassuming a TEM00 mode as used in Porras’ calculationsis a good approximation.

Two different tips were used for the one-dimensionalmeasurements of the phase profile along z, which is dis-played in Fig. 3. Namely, data sets I and II were madewith a tungsten and a gold tip, respectively. The re-producibility of the measurement with two different tipsshows the independence of the measurement on the tip






4

material. Note that this is expected as the material andtip shape dependent phase retardation effects are pre-dicted to lead only to a phase offset11 which is eliminatedhere as the data for each tip is referenced to the phase atthe focal point for that tip.

III. FUNDAMENTAL EQUATIONS

For the focus of a Gaussian beam, the intensity incylindrical coordinates r (radius) and z (propagationaxis) is given by

I(r, z) = |E(r, z)|2 = I0

(w0

w(z)

)2

e−2( rw(z) )

2

, (S5)

with the peak intensity I0 and the 1/e2 waist radius w0

(at z = 0). The hyperbola of the local radius is definedas

w(z) = w0

√1 +

(z

zR

)2

, (S6)

with zR being the Rayleigh length:

zR =π · w0

2

λ=

ω · w02

2c, (S7)

where λ is the wavelength, ω the angular frequency andc the speed of light.

For a focused monochromatic Gaussian beam the localelectric field is expressed via

E(r, z) = E0,0w0

w(z)· e−(

rw(z) )

2

· e−ik r2

2R(z) · e−i(kz+∆ψ(z)),

(S8)

where E0,0 is the electric field amplitude at (r = 0, z = 0);k = 2π/λ is the wave number; R(z) is the local radius ofcurvature and ∆ψ(z) = − arctan z/zR is the Gouy phase,compare Eq. 1.

Assuming a Gaussian pulse envelope, the temporalevolution of the electric field of laser pulses at a givenpoint in space is commonly described by the pulse dura-tion τ (intensity full-width-at-half-maximum), the cen-tral angular carrier-frequency ω0, the carrier-envelopephase φ (CEP) and the local peak electric field ampli-tude E0,r,z as

E(t) = E0,r,z e−2 ln 2(t/τ)2 · e−i(ω0t+φ), (S9)

which is the derivative of the vector potential ∂A(t)/∂t(compare inset in Fig. S2)12. The vector potential is im-portant for electron propagation, in particular attosecond

FIG. S4. Rayleigh length of the input beam as functionof optical frequency. Rayleigh length of the input beam asa function of angular frequency as calculated from opticallyfiltered beam profile measurements in front of the vacuumchamber (inset: example profile of the unfiltered beam). Thethree measurements shown here were taken along the tip mea-surements shown in the main text, the Porras factor, g0, canbe estimated in a first-order approximation from fitting linearcurves to the data (coloured lines) and evaluating the nor-malized slope (see text). We retrieve g0 = −2.0 ± 1.0 (greensquares: on-axis (I)), g0 = −1.4±0.4 (blue dots: on-axis (II))and g0 = −1.2± 0.4 (purple triangles: off-axis). These valuesare comparable to the values for the Porras factor that weobtain from a direct measurement of the focal phase inside ofthe vacuum chamber, namely g0 = −2.1 ± 0.2 (on-axis (I)),g0 = −1.8± 0.3 (on-axis (II)) and g0 = −1.2± 0.3 (off-axis).

Measurement g0 from nanotip g0 from spectral profileI −2.1± 0.2 −2.0± 1.0II −1.8± 0.3 −1.4± 0.4off-axis −1.2± 0.3 −1.2± 0.4

TABLE S1. Summary of the determined g0-values

streaking where the mapping of time to electron energyis crucial13.Under conditions where chirp is not negligible, the on-

axis focal CEP-shift can be amended with a relative chirpterm Cr, which yields for focussing with a mirror6

∆φ(z) = − arctan

(z

zR

)+

g0 ·[1 + 2Cr

zzR

]

zzR

+ zRz

. (S10)

IV. INDIRECT METHOD FORDETERMINATION OF THE SPATIAL PHASE

PROFILE

Using a nanotip in the way described and demon-strated here is a direct way to map the phase in the focusof a broadband pulse with unprecedented precision. Ad-ditionally, our measurements show that the formulationof the focal phase evolution in terms of the Porras factoris applicable. However, this method is a complex, timeconsuming, and challenging measurement. For these rea-






5

FIG. S5. Stereogram of the CEP focus profile. The CEP measurement off the optical axis from Fig. 4 in stereoscopic/3Dformat. To watch, mind the following instructions to achieve parallel freeviewing. Size the figure on your reading means toabout 11 − 13 cm width, which corresponds to twice the average human eye separation. Start by holding the graph close toyour eyes and ignore the focussing for the moment, just relax the eyes. Look “through” the graph into infinity so that youreyes are parallel. Now slowly move the graph away from your eyes while maintaining the parallel state of your eyes. When thedistance approaches about 25− 30 cm your eyes should see the graph automatically sharp and three-dimensional.

sons we have also implemented an indirect method togain insight into the spatial phase profile of the focusby estimating the Porras factor g0 of the input beam.As seen in Equation 3, the Porras factor g0, dependson the frequency-dependent Rayleigh length of the in-put beam, ZR(ω), which in turn can be calculated fromthe frequency-dependent input waist. Therefore, if onecan determine the frequency-dependent input waist, onecan use this for a first-order approximation of the spatialphase profile in the focus.

We have also implemented this technique of using themeasured frequency-dependent input beam profile to cal-culate ZR(ω) and the g0-value of our beam geometry con-currently with the three nanotip measurements presentedin the text. These examples are shown in Fig. S4 wherethe Rayleigh length vs. frequency is drawn. As can beseen the Rayleigh lengths are many meters, so estimatingthe waist by measuring the size of the collimated beamis reasonable. The input beam was measured at the fre-quencies denoted by the points and fit in first order witha linear function, to derive the Porras factor, g0, whichin turn describes the phase profile in the focus. The Por-ras factor measured in this way can then be comparedwith the Porras factor extracted from the behaviour ofthe CEP measured with the nanotip. Considering thepioneering character of the method we find fair agree-ment between the two independent determinations. Inthe main text, and going in the opposite direction, wehave used the phase profile in the focus measured with

the nanotip to determine the Porras factor, g0, which inturn reflects the frequency-dependent input beam geom-etry.

These two directions are summarised by this scheme:

Focal CEP-profile

(masured by nanotip) ⇒

Focal CEP-profile

(estimated) ⇐

g0

Spectral input

⇒ (calculated)

Spectral input

⇐ (measured)

The g0 values together with the ones from the nano-tip measurement are summarised in Tab. S1. Unlike thenanotip method, this method does not directly measurethe spatial phase profile in the focus. Nonetheless, therelative simplicity of the technique is attractive and maybe used in the future to infer the focal properties. First,since the beam parameters such as beam radius and beamwaist size are typically on the millimetre rather than onthe micrometer scale like in the focal region, it is easyto measure them with CCD sensors. Second, one candetermine the wavelength-dependence using easily avail-able optical band-pass filters. However, as this is just afirst-order approximation and the uncertainties are large,future precision studies of this technique might also showthe need to include higher order terms from the expansionof ZR(ω), depending on the light source. Further workis clearly needed and warranted to explore the efficacy ofthis promising technique.

1 Sayler, A. M. et al. Accurate determination of absolutecarrier-envelope phase dependence using photo-ionization.Opt. Lett. 40, 3137–3140 (2015).

2 Kruger, M., Schenk, M., Forster, M. & Hommelhoff, P.Attosecond physics in photoemission from a metal nanotip.J. Phys. B: At., Mol. Opt. Physics 45, 074006 (2012).






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3 Paulus, G. G. et al. Measurement of the phase of few-cyclelaser pulses. Phys. Rev. Lett. 91, 253004 (2003).

4 Milosevic, D., Paulus, G., Bauer, D. & Becker, W. Above-threshold ionization by few-cycle pulses. J. Phys. B: At.Mol. Opt. Physics 39, R203 (2006).

5 Sayler, A. et al. Real-time pulse length measurementof few-cycle laser pulses using above-threshold ionization.Opt. Expr. 19, 4464–4471 (2011).

6 Porras, M. A., Major, B. & Horvath, Z. L. Carrier-envelopephase shift of few-cycle pulses along the focus of lenses andmirrors beyond the nonreshaping pulse approximation: theeffect of pulse chirp. J. Opt. Soc. Am. B 29, 3271–3276(2012).

7 Porras, M. A., Horvath, Z. L. & Major, B. On the useof lenses to focus few-cycle pulses with controlled carrier–envelope phase. Appl. Phys. B 108, 521–531 (2012).

8 Thomas, S., Kruger, M., Forster, M., Schenk, M. & Hom-melhoff, P. Probing of optical near-fields by electron rescat-tering on the 1 nm scale. Nano Lett. 13, 4790–4794 (2013).

9 Muller, E. W. Field ion microscopy. Science 149, 591–601(1965).

10 Tsong, T. T. Atom-Probe Field Ion Microscopy: Field IonEmission, and Surfaces and Interfaces at Atomic Resolu-tion (Cambridge University Press, 2005).

11 Thomas, S., Wachter, G., Lemell, C., Burgdorfer, J. &Hommelhoff, P. Large optical field enhancement for nan-

otips with large opening angles. New J. Phys. 17, 063010(2015).

12 Diels, J.-C. & Rudolph, W. Ultrashort laser pulse phenom-ena (Academic press, 2006).

13 Kienberger, R. et al. Atomic transient recorder. Nature427, 817 (2004).

FIG. S6. Projection of Figure 4. The CEP measurementoff the optical axis from Fig. 4 projected in the z-r-plane.The contour map is the fit of Eq. 2 and the circles are themeasured points.






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Tracing the phase of focused broadband laser …focus of a few-cycle laser pulse. This is accomplished with two separate but synchronized measurement chambers, see Fig. S1. First we