Precise and accurate measurements of strong-field photoionisation and a transferrable laser intensity calibration standard

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by  W. C. Wallace

Precise and accurate measurements of strong-field photoionisation and a transferrable
laser intensity calibration standard
W.C. Wallace,1,2 O. Ghafur,1,2 C. Khurmi,1,2 Satya Sainadh U,1,2 J.E. Calvert,1,2 D.E.
Laban,1,2 M.G. Pullen,1,2,∗ K. Bartschat,1,2,3 A.N. Grum-Grzhimailo,4 D. Wells,5
H.M. Quiney,5 X.M. Tong,6 I.V. Litvinyuk,2 R.T. Sang,1,2 and D. Kielpinski1,2,†
1ARC Centre of Excellence for Coherent X-Ray Science,
Griffith University, Brisbane, Queensland, Australia
2Australian Attosecond Science Facility and Centre for Quantum Dynamics,
6
1 Griffith University, Brisbane, Queensland, Australia
0 3Department of Physics and Astronomy, Drake University, Des Moines, Iowa 50311, USA
2 4Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia
5ARC Centre of Excellence for Coherent X-Ray Science,
n
University of Melbourne, Melbourne, Victoria, Australia
a 6Division of Materials Science, Faculty of Pure and Applied Sciences,
J
and Center for Computational Science, University of Tsukuba,
8
1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan
1
(Dated: January 19, 2016)

] Ionization ofatomsandmoleculesinstronglaserfieldsisafundamentalprocessinmanyfieldsof
h
research, especially in theemerging field of attosecond science. So far, demonstrably accurate data
p
have only been acquired for atomic hydrogen (H), a species that is accessible to few investigators.
-
m Here we present measurements of the ionization yield for argon, krypton, and xenon with percent-
level accuracy, calibrated using H, in a laser regime widely used in attosecond science. We derive
o
a transferrable calibration standard for laser peak intensity, accurate to 1.3%, that is based on
t
a a simple reference curve. In addition, our measurements provide a much-needed benchmark for
. testing models of ionisation in noble-gas atoms, such as the widely employed single-active electron
s
c approximation.
i
s
y
h Ionization by strong laser fields drives processes rang- Hence, the accuracy of the H data can be certified by
p ing from attosecond pulse generation [1, 2] to filamenta- their agreement with 3D-TDSE simulations.
[ tion [3] and remote lasing [4]. Measurements of strong- Hereweusethesetechniquestoperformaccuratemea-

1 field ionization have revealed complex and surprising surements of the strong-field ionization yield from three
v qualitative features [5, 6] that can depend sensitively commonly used noble-gas targets, and to derive a trans-
9 on the laser intensity. Precise measurements of strong- ferrable, high-accuracy calibration standard for laser in-
7
fieldionizationarenowbeingusedtoprobefundamental tensity. Our data enable accurate inter-comparisons
3
physics, such as time delays in photoionization [7], but of data taken at various strong-field laboratories; and
4
0 there is substantial evidence that small systematic off- improved simulations of complex phenomena involv-
. sets in these measurements can obscure the results [8]. ing strong-field ionisation, such as filamentation, high-
1
In frequency metrology, measurements of atomic transi- harmonic generation, and laser-induced electron diffrac-
0
6 tions are affected by systematic errors arising from the tion. We use the noble-gas data to derive a calibration
1 AC-Stark shift and laser intensity uncertainty, thereby standard for laser peak intensity, given by a simple ref-
: limitingtheprecisionoftheresult[9]. Accuratereference erence curve, that offers an order of magnitude better
v
i data on strong-field photoionization and laser intensity, accuracy than previous transferrable standards [16, 17].
X
especially in the attosecondscience regime,aretherefore Ourintensitycalibrationstandardappliestoanintensity
r needed for further progress on these questions. regime that is readily transferrable to laboratories using
a
In recent years, our group has used atomic hydrogen few-cycle 800 nm laser systems, including most attosec-
(H) to perform quantitatively accurate strong-field mea- ond science laboratories.
surements that are demonstrably free from systematic The experimental apparatus is detailed in [10, 11]. It
errors [10–12]. These measurements are performed in consists of a well-collimated atomic H beam skimmed
the regime of laser pulse durations and peak intensities from the output of an RF discharge dissociator, which
that are most widely used in attosecond science. As intersectsthefocusofanintensefew-cyclelaser. Theflux
the simplest electronic system, H has long been recog- ofourcustom-constructedatomicsourceisseveralorders
nised as a benchmark species for strong-field physics of magnitude higher than commercial sources [18], facil-
experiments [13, 14]. Direct integration of the three- itating a high signal level. The laser generates pulses of
dimensional time-dependent Schr¨odinger equation (3D- 5.5fs duration(measuredatfull-widthathalf-maximum
TDSE) enables high-accuracy simulation of H and less of intensity) with a central wavelength of 800 nm. Ions
than1%error,withonlyveryminorapproximations[15]. are created in the overlap region between the atomic
2
ous work [10, 11], we perform FVA over the interaction
(a)
105 volume,assuming a Gaussianlaserprofile anda molecu-
larbeamofdiameterd,wheredismuchsmallerthanthe
u.) laserRayleighrangeandmuchlargerthanthetransverse
b. size of the laser beam. Under these conditions, the FVA
eld (ar104 bWeecopmerefsorimndaepweenidgehnttedofleabseta-msquparroepsafigtatoifotnheeff3eDct-Ts D[1S2E].
Yi
yieldpredictionstothe Y datausinga functionofthe
H+
form
103 Y3ADDH−+KTDSE PH+(Iest,A,η1)=A·S(η1Iest). (1)
Emp−ADK
75 (b) Here A and η1 are the fit parameters, while S is the
focal-volume averaged and carrier-envelope-phase aver-
%) 50 aged theory model evaluated at the actual laser inten-
E (
S sity I = η I . The independently estimated intensity
D−TD 25 Iest is0obta1ineesdtfrom measurements of the laser parame-
3 ters – waistsize w , averagepower P, pulse durationτ ,
m 0 0 p
o and repetition rate f – via
ce fr−25 rep
en 2P 1
Differ−50 Iest = πw02frepτp. (2)
YH+
ADK
−75 Emp−ADK The η1 fitparameteris a rescalingcoefficientofthe laser
2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 intensity that accounts for the error in I , and permits
est
Intensity (x 1014 W/cm2)
the accurateretrievalofI . TheAfit parameterrescales
0
the yield to account for both the unknown atomic den-
FIG. 1. (a) Experimental data for H+ yield (solid circles),
sity anddetector efficiency, but is irrelevantfor intensity
compared with theoretical predictions from 3D-TDSE (dot-
dashed line), standard ADK (dashed line), and empirically calibration.
corrected ADK (Emp-ADK; dotted line) models. In some Figure 1 illustrates the agreement between the
cases,theerrorbarsonthedataaresmallerthanthesymbols; Y data and the 3D-TDSE simulations, certifying that
H+
see Supplementary Information for error estimates. (b) Per- our data are free of systematic error to within our mea-
centage difference of experimental H+ yield data and both
surementprecisionof2%. A value of η = 0.641 0.007
1
ADK theories from the3D-TDSEsimulations. ±
is found, indicating that we can calibrate the laser peak
intensity to a theory-certified accuracy of better than
beam and the laser beam, and are detected with an 1.1% without systematic error. Normalized residuals
ion time-of-flight (ion-TOF) mass spectrometer. A mi- fromthefitarealsoshownaspercentagedeviationsfrom
crochannel plate (MCP) located at the end of the ion- the 3D-TDSE predictions in Fig. 1. The deviation of η
1
TOFdetectstheionsandoutputsavoltageproportional from its ideal value of 1 is common in high-field experi-
to the ion yield (see Supplementary Information). The ments, where the uncertainty in I can easily approach
est
overlapregion is well-defined, allowing us to account ac- 50%. We note that if the backgroundis not removed, η
1
curately for focal-volume averaging (FVA) effects [12]. shifts by 3.1%, much more than our accuracy of 1.1%.
The measured yield of H+ ions resulting from ionisa- Our data are easily able to discriminate between the
tion of atomic H over a range of laser peak intensities is 3D-TDSE and other commonly used theoretical approx-
shown in Fig. 1. This yield, denoted YH+, is accurately imations. Awell-knownalternative to solvingthe TDSE
measuredby removing contributions arising from ionisa- is the analytic theory of Ammosov, Delone and Krainov
tion of undissociated H2 in the beam, and background (ADK) [19]. Standard ADK theory is intuitive and
H2O vapour. These contaminant signals can be as large straightforward to calculate, but it is known to fail at
as 9.5% of the desired YH+, and hence must be removed intensities near to or exceeding the onset of barrier-
to obtainpercent-levelaccuracy. ErrorsinYH+ accumu- suppressed ionization (BSI) [20]. An empirical correc-
late from MCP voltage baseline subtraction, from short- tion [21] has since been developed to extend the validity
and long-term laser drifts, and from uncertainties in de- of ADK rates to higher intensities. Standard ADK and
termining the dissociation fraction. A detailed account empirically corrected ADK rates, as well as percentage
of the analysis and error estimation is given in the Sup- residuals have also been plotted in Fig. 1 using the A
plementary Information. andη fit parametersobtained fromthe 5.5 fs 3D-TDSE
1
We certify the accuracy of our data by comparison fit. Standard ADK deviates from the data at high in-
with accurate 3D-TDSE calculations. As in our previ- tensities by almost a factor of two, whereas empirically
3
corrected ADK is accurate at the 10% level there. Nev-
TABLE I. Fit parameters used in Eq. (4) for the Ar+, Kr+,
ertheless,thelattermodelisstillclearlyruledoutbythe
and Xe+ gas targets.
data at lower intensities.
.
We now presentdemonstrably accurate measurements
ofthephotoionizationyieldofargon,kryptonandxenon, Fit Parameter Ar+ Kr+ Xe+
providing reference data in a regime for which accurate α (arb. units) 2.84 4.24 3.71
simulations are not available. The results for the yield γ (arb. units) -3.03 -2.49 -2.69
of each gas target, denoted YAr+, YKr+, and YXe+are Ic (× 1014 W/cm2) 3.86 ± 0.05 2.06 ± 0.03 1.18 ± 0.03
shown in Fig. 2. The agreement between theory and
experiment for H certifies the accuracy of the noble-
gas yields, as these measurements were performed in the I include the uncertainty in the H+ calibration as well
c
same apparatus, using identical laser parameters. as the fit error, and demonstrate our ability to calibrate
Theoretical ionisation probabilities for the noble-gas theintensityatthe1.3%,1.5%,and2.5%levelsusingAr,
atoms are obtained by solving the 3D-TDSE under the Kr, and Xe as gas targets respectively1. The value of I
c
single-active electron approximation with the second- is insensitive against the removal of any individual data
order-splitoperatormethod inthe energyrepresentation point from the fit dataset, indicating that the model ro-
[22, 23]. The model potentials [21] are calculated by bustlyrepresentsthedataovertheentireintensityrange.
using the density functional theory with self-interaction However,itisimportanttonotethatthe phenomenolog-
correction [24], from which the calculated atomic ioni- icalmodel is introduced purely as a convenience,so that
sation potentials are in good agreement with the mea- the readercan easily carryout intensity calibrationwith
sured ones. The theoretical simulations are subjected to a closed-form analytic fit function. We emphasize that
FVA for comparison with experimental data. As with Eq.(4)isnotassociatedwithanymodeloftheionization
the atomic H data, we also compare standard ADK and physics. Hence, we do not expect it to be valid outside
empiricallycorrectedADKrateswiththenoble-gasdata. the range of intensities studied here.
EachofthetheorymodelsiscomparedtoYNG+(where Equation (4) enables absolute intensity calibration at
the subscript NG denotes one of the noble gases) using the 1.5% level for few-cycle 800 nm laser systems, like
thecalibratedintensityI0 andthefittingmethodEq.(1). those widely used for attosecond science. The cali-
ThefitsandresidualsforeachtargetareshowninFig.2. bration only requires a mass spectrometer, a few-cycle
SincetheintensityisalreadycalibratedbytheYH+ data, laser at 800 nm, and a source of either Ar, Kr, or
the calibrationfactor η1 is fixed to a value of1, whilst A Xe. The intensity range covered by our calibration,
is allowed to vary in order to account for the unknown 1 – 5 1014W/cm2,isusedbymostatomicandmolecu-
×
gasdensity. While the data areaccurateatthe 2%level, larstrong-fieldphysicsexperiments,particularlyattosec-
thetheoreticalpredictionsagreewiththedataonlyatthe ond science experiments. Instructions for using our cali-
tens of percent level, with both ADK rates performing bration are detailed in the Supplementary Information.
poorly. These data therefore pose a direct challenge to Our transferrable calibration standard can be shown
current models which are widely used to predict results to reliably determine the absolute intensity without sys-
from strong-field ionization experiments. tematic errors; in other words, the retrieved intensity
While all three theories disagree with the experimen- can be accurately expressed in the SI unit of W/m2.
tal data, Fig. 2 shows that we can achieve good agree- Previously presented transferrable intensity calibration
mentwiththedatausingthefollowingphenomenological methods[17,25,26]reliedontheoreticalapproximations
model whosesystematicerrorswerenotfullyquantified. There-
movalof systematic errors,i.e., offsets between the mea-
PN(2GD+)(Iest;A,η2)=A·Sphenom(η2Iest), (3) sured value and the “true value” of the measured quan-
tity [27], is crucial for accurate measurement. There-
where fore, while these previous methods provide relative cali-
brations of the intensities in the interactionregion,their
−1/2
exp α η I /I
− 2 est c relationshipto the SI system, or any other standardsys-
S (η I )= . (4)
phenom 2 est (cid:16)1+ (cid:0)η2Iest/Ic(cid:1)γ (cid:17) tem of units, remains unclear.
Ourcalibrationisrelativelyinsensitivetolaserparam-
(cid:0) (cid:1)
Here A and η are the same fit parameters as described etersotherthanpeakintensity. Asshowninourprevious
2
in Eq. (1). The coefficients α and γ are set by fitting work [11], variations of the pulse duration by 10% may
Eq. (4) to the 3D-TDSE for each gas target. The value cause a rescaling of the overall yield, but cause < 1%
ofI wasdeterminedfromthe dataofFig. 2byfixingη shifts in the retrieved intensity. Similarly, the calibra-
c 2
to a value of 1, and substituting I with our I values tion is not overly sensitive to the precise form of the
est 0
obtainedfromtheH+ fit. Thevaluesoftheseparameters beam profile: we achieve good theory-experiment agree-
are shown in Table I for each gas target. Our values for ment for H with beam M2 factors as high as 1.5. In
4
(a) (b) (c)
101
+Ar yield (arb. u.)100 +Kr yield (arb. u.)110001 +e yield (arb. u.)101
X
1400−1 (d) A3AEPmDhrD+e−Kp nDT−oDaAmtSDaEeKnological 20 (e) K3AEPmDhrD+e−Kp nDT−oDaAmtSDaEeKnological 12000 (f) X3AEPmDheDe−+Kpn TD−oDAamStDaEeKnological
%) %) 10 %) 10
+Difference from Ar data (−−−264200000 3AEmDD−KpT−DASDEK +Difference from Kr data (−−−−−54321000000 3AEmDD−KpT−DASDEK +Difference from Xe data (−−−−−54321000000 3AEmDD−KpT−DASDEK
−80 1.5 2 2.5 3 3.5 4Phenom4e.n5ological −60 1.5 2 2.5 3 3.5 4Phenom4en.o5logical 1.5 2 2.5 3 3.5 4Phenom4en.5ological
Intensity (x 1014 W/cm2) Intensity (x 1014 W/cm2) Intensity (x 1014 W/cm2)
FIG. 2. (Color online) Top row: intensity-calibrated experimental data (black circles) for (a) Ar+, (b) Kr+, and (c) Xe+
compared with theoretical predictionsfrom 3D-TDSE(blue,dot-dashedline), ADK(red,dashedline), andEmp-ADK(green,
dotted line) models, as well as the phenomenological model given by Eq. (3) (cyan, solid line). The only fit parameter is
the overall rescaling of yield, A. In some cases, the error bars on the data are smaller than the symbols. Bottom row:
normalized residuals for (d) Ar+, (e) Kr+, and (f) Xe+, for each theoretical prediction, shown as percentage deviations from
theexperimental yield data for the respectivegas targets.
most experiments with molecular beams, including ours, constructaphenomenologicalfittingfunctionforthenew
the focal volume averagingis independent of M2 so long data, to be used in the same way as Eq. (3) above. A
asthetransverseintensitydistributionisGaussianandis method for intensity calibration for apparatus, in which
constant within the interaction region. As long as these anatomic beam is not employed,has also been provided
conditions are satisfied, Eq. (4) is expected to hold even in the Supplementary Information.
for much larger values of M2. While the calibration can We have presented photoionization yield measure-
presently only be used for wavelengthsnear 800 nm, our ments with an accuracy that improves on previous mea-
simulations show that changes of the wavelength by 50 surements by an order of magnitude. Our data are ob-
nm affect the retrieved intensity by < 1%. This lack of tained in a regime of laser pulse duration and inten-
sensitivityisexpectedsinceourlaserbandwidthis>200 sity that is widely used for attosecond science, and can
nm. Finally, the calibration can tolerate laser pulse en- be used to benchmark measurement techniques in that
ergy fluctuations of at least 0.7% (root-mean-square),as field. Themeasurementsarecertifiedatthepercentlevel
independently measured on a photodiode. We achieve throughthe observationoftheory-experimentagreement
good theory-experiment agreement for H at this level of for H. Using the noble-gas data presented here, other
fluctuation. laboratories can verify the accuracy of their measure-
Other laboratories can transfer our calibration stan- ments, calibrate their apparatus, and obtain similarly
dardtolasersofwidelydifferentpulsedurationsorwave- accurate data for other atomic and molecular species.
lengths, without the use ofanH source,by the following Inthe meantime, ourdataprovideaccuratereferencefor
procedure. (i) Calibrate the intensity of a standard few- simulationsofstrong-fieldphenomenainvolvingfew-cycle
cycle laser near 800 nm by measuring one of Ar+, Kr+, ionization. Finally, we have presented a transferrable
or Xe+ photoion yields. (ii) Measure the ratios of the calibration of laser intensity that provides an order-of-
beam parameters used in Eq. (2) between the new and magnitudeaccuracyimprovement. Thestandardisread-
standardlaser. Theseratioscanbemeasuredmuchmore ilyaccessibletolaboratoriesusingfew-cycle800nmlasers
accurately than the parameters themselves. From these and can be further transferred to other laser systems,
ratios, derive an absolute calibration of the new laser’s enabling the correct measurement and interpretation of
intensity. (iii)MeasureAr+,Kr+, orXe+ photoionyield intensity-sensitive phenomena in strong-field ionization.
as a function of the new laser’s intensity. The data for
This work was supported by the United States Air
the new laser are known to be accurate, since they are
Force Office of Scientific Research under Grant FA2386-
referenced to our data by steps (i) and (ii). Finally, (iv)
12-1-4025andbytheAustralianResearchCouncil(ARC)
5
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1 Perthe recommendations ofthe Joint Committee forGuides in
Metrology[27],wehaveherestatedthe1σstandarddeviationfor
Ic,i.e.,the68%confidenceinterval. Therefore,onanyparticular
reproduction of our experiment, the retrieved intensity has a
32%probabilityoffallingoutsidethe68%confidenceintervalfor
purelystatisticalreasons.
1
Supplementary information: Precise and accurate measurements of strong-field
photoionisation and a transferrable laser intensity calibration standard
This Supplementary Information details the methods 0.9
used for acquisition and analysis of the data that are H+
0.8
presented in this Letter, alongside detailed instructions
on how to calibrate the laser peak intensity according to 0.7
Eq. (3) of the manuscript.
olts)0.6
v
m (0.5
DATA ACQUISITION or
ef
v
a0.4
w
Dataareacquiredusingahome-builtiontime-of-flight OF
(ion-TOF) mass spectrometer of standard design. Ions n−T0.3
o
createdintheextractionregionviathelaser–H-beamin- I0.2 H2+
teraction are accelerated in a 250 V/cm extraction field
0.1
before passing through a 1.7 mm slit into a 95 mm long
HO+
field-freedriftregion,whereintheionsdisperseaccording 2
0
to their mass-to-charge ratio. The drift region is termi- 0.5 1 1.5 2 2.5 3
Arrival time (µs)
natedwith a groundedstainless-steelmeshwith anopen
FIG. S1. Typical ion-TOF waveform obtained with a single
area ratio of 70%. An 800 V/cm field is used to post- lasershotatacalibratedpeakintensityof4.72±0.05×1014
acceleratetheionsintoamicro-channelplate(MCP)de- W/cm2. Shaded regions represent the window of integration
tector (Hamamatsu F9890-31). for each ion species (H+ in blue, H+2 in red, and H2O+ in
For each laser shot, the voltage waveform V(t) from green).
the output MCP (model: Hamamatsu F9890-31, 450 ps
Atypicalion-TOFwaveformisshowninFig.S1. This
resolution) is recorded via a PCIe digitizer card (model:
waveform was obtained with a calibrated peak laser in-
Agilent U1084A, 250 ps resolution). A LabView-based
tensity of 4.72 0.05 1014 W/cm2 and illustrates
acquisitionprogramsumsupeachshot-to-shotwaveform
foradurationof10s,correspondingto1 104lasershots. the presence of ±three io×n species: H+, H+2 from resid-
A 10 s acquisition is obtained for eac×h setting of the ualH2 presentin the beam, andH2O+ frombackground
gas in the vacuum chamber. Evaluation of the yield of
RFdischargedissociatorsource(dischargeON,discharge
eachionspeciesisperformedbyintegratingtheion-TOF
OFF, background BCK, and noble gas NG), and for a
waveforminthetime intervalboundedbythe shadedre-
number of different estimated laser intensities I . For
est
gions of interest. Ringing in the waveform is observed,
the case when the dissociator is ON, excited species for
alongside a non-zero baseline. Errors in the evaluation
atomic H and molecular H are generated, however the
2
of the yield due to baseline subtraction and ringing are
excited state lifetimes are an order of magnitude shorter
discussed below in Sec. .
than the flight time from the source to the laser-matter
To isolate the yield of H+ ions resulting from ionisa-
interaction region. Any metastable atomic H is rapidly
tion of atomic H we combine individual waveform mea-
quenched by the electric fields present in the apparatus.
surements as:
Depending on the acquisition, the ion yield Y for three
doviffeerarellnytieslpdeciisesca(lcHu+la,teHd+2a,sN: G+) were measured. The YH+ =YHO+N−YHBC+K−(1−µ)(YHO+FF−YHBC+K). (S2)
Here the superscriptdenotes the setting of the discharge
t2 (BCKforwhentheH2gasisoff,OFFforwhentheH2gas
Ysetting = V(t)dt. (S1) is on but the discharge is off, and ON for when the dis-
species
Zt1 charge is on); and the subscript denotes the ion species.
It is not a requirement that the detection efficiency of The µ term represents the dissociation fraction and is
the detection system be equal for all species, only that the percentage of H molecules dissociated when the H
2
thedetectionefficiencyforaparticularionspeciesiscon- discharge source is turned ON. It is given by:
stantacrossalllaserintensities,whichitis. However,this
YON YBCK
condition canbe broken,for example, when the detector µ=1 H+2 − H+2 . (S3)
is saturated or when space-charge effects occur. We op- − YOFF YBCK
H+ − H+
erateourdetectorbelowsaturation;andbothouratomic 2 2
beamdensity andbackgroundgasdensity inthe interac- Determining the yieldfromthe noblegasis muchsim-
tion region are low enough to avoid space-chargeeffects. pler. For example, the Kr+ ion yield (denoted Y )
Kr+
2
from photoionisation of krypton atoms was determined 40
bycombiningtheNGandBCKmeasurementsaccording
35
to:
30
Y =YNG YBCK, (S4) olts)
Kr+ Kr+ − Kr+ -30 v25
1
x
whereKrdenotesthatkryptongas(ratherthanH2)flows m (20
throughtheRFdissociatorsource. Thedischargesetting or
vef15
is OFF when measuring noble-gas yields. a
w
F
O10
T
−
n
ERROR ANALYSIS Io 5
0
The overall uncertainty in the photoionisation yield
−5 t t
measurements has four error contributions: 1 2
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
1. Estimation of yield from the MCP voltage wave- Arrival time (µs)
form
FIG. S2. Example of waveform ringing at high laser peak
2. Short-term drift (within an individual measure- intensities. Theringingextendsfrom theendoftheH+ peak
(blue)untiltheonsetoftheH+peak(red). Thetimeinterval
ment) 2
boundedbytheblueboxisontheorderof250ns. Increasing
3. Long-term drift (between the ON, OFF, and BCK the width of the time interval does not affect the overall H+
yield at the 0.05% level from one half-cycle of the ringing to
measurements)
thenext.
4. Errorin the dissociationfractionestimate entering
Eq. (S2).
interval over which each ion peak was integrated is cho-
ThesecontributionsaresummarisedinFig.2ofthemain sen such that there is less than 0.05% difference in the
text. The error in yield estimation arose from back- overall yield when the time interval is extended by one-
ground subtraction and ringing. Short-term drift errors half cycle of the ringing. The residual error is negligible
were calculated via a modified Allan deviation analysis in comparisonto the other contributions. Hence, we dis-
[S2]. Long-term drift errors were estimated from mea- cardits contributiontothe overallerrorinthe yield,but
surements of the power spectral density (PSD) of the for completeness still present it here.
noise [S3]. Errors in the dissociation fraction were in-
ferred from the errors in the yield measurements. The
Short-term drift
long-term drift was the dominant error source, with a
factor of four higher contribution than the other error
sources. We use the IEEE-recommended [S1] method of over-
lappingAllanvarianceanalysis[S2]tomeasuretheshort-
term variance in the time domain. To quantify the vari-
Yield estimation ance we first recorded 800 consecutive ion-TOF wave-
forms, each averaged over 3000 laser shots, for a total
The raw waveform acquired from the multichannel time interval of approximately 2400 seconds. During
plate has a non-zero baseline, owing to a small but non- this time interval, no experimental parameters were ad-
negligible DC voltage offset at the input to the PCIe justed. Next, the yield of each ion species was extracted
digitizer card. The voltage baseline was removed by av- from each ion-TOF waveform via the baseline subtrac-
eragingthe waveformovera550ns timeintervalpriorto tionandintegrationmethodoutlinedearlier. Theresult-
the onsetofthe firstionpeak,andsubtractingthis aver- ing time series measurements of ion yield versus time,
age from the entire waveform. The error in the baseline Ysseptetciinegs(t), taken at low (IL = 1.7 × 1014 W/cm2) and
subtraction was calculated using the standard deviation high (I = 3.9 1014 W/cm2) peak laser intensities,
H
×
of the corrected baseline, and is denoted σ (I). are shown in Fig. S3 for the case of YON. The analysis
B H+
Waveform ringing was observed at high laser peak in- describedbelowwascarriedoutalsoforallthenoble-gas
tensities, as illustrated in Fig. S2. This ringing is due to yields, but here we only discuss YON measurements for
H+
a small impedance mismatch between the MCP output simplicity.
and the PCIe digitizer input. The ringing constitutes a We perform Allan variance analysis on both low- and
real and observable output current, and therefore con- high-intensitytimeseriesmeasurementsaccordingtothe
tributesto theoverallyieldsignalforeachion. The time methodgivenin[S2]. Theresultofthisanalysisisshown
3
5.4 2.5
(a)
(a)
5.3
-110 a.u.) 555...012 (%) n, σAl 21..05
x o
ONY (t) (H+ 44..89 eviati
Yield, 4.7 Allan d 1.0
4.6
0.5
4.5
0
(b)
(b)
7.2 1.0
0.9
7.1 %) 0.8
ONY(t) (a.u.)H+ 76..09 ation, σ (Al 00..67
Yield, 6.8 n devi 00..45
a
6.7 All 0.3
0.2
6.6
0 500 1000 1500 2000 2500 0.1
Time (s) 100 101 102 103
Averaging time, τ (s)
FIG. S3. Time series measurements of ion yields for YON
H+
acquiredatlaserpeakintensitiesof(a)1.7and(b)3.9×1014
FIG. S4. Overlapping Allan deviation shown as percentage
W/cm2. Yield units are arbitrary, but are kept consistent errorinYON atlaserpeakintensitiesof(a)1.7and(b)3.9×
between thetwo datasets. 1014 W/cmH+2.
in Fig. S4. The y-axis shows the percentage error in- time varying signal y(t) is given by:
curred for an averagingtime of τ seconds. In our exper-
1
iment, the τ = 10 s averagingtime for a single ion-TOF S (ω)= lim Y (ω)2, (S5)
YY T
waveform acquisition corresponds to a measured Allan T→∞2T| |
deviationerror,σ , inYON oflessthan 1.4%(0.6%)for
Al H+ where Y (ω) is the Fourier transform of y(t):
T
low(high)laserpeakintensity. Ideally,one wouldlike to
measuretheintensity-dependentAllandeviation,σAl(I), T
across all intensities, but as seen below, the short-term YT(ω)= y(t)e−iωtdt. (S6)
errorcontributionisnegligibleincomparisontothelong- Z−T
term error contribution. In practice, Eq. (SS6) is true only if y(t) is a signal with
zero mean, such that:
y(t) = Y(t) Y(t) .
−h i
Long-term drift
The calculatedPSD for the YON(t) time series measure-
H+
mentsisshowninFig.S5onalog-logplot. Asexpected,
Errorsdue tolong-termdrifts areestimatedvia power
thelow-frequencyPSDatlaserpeakintensityI follows
spectral density (PSD) analysis [S3] of the YON(t) time H
H+ a power law, which takes the form of:
seriesmeasurements. Briefly,the PSDofatime seriesor
signal gives the distribution of power contained within A(I )
H
S (f,I )= +B(I ), (S7)
that signal per unit frequency. The PSD SYY(ω) of a YY H fγ H
4
10-2 requiredforasingleion-TOFwaveform. Inourmeasure-
(a)
ments, f f , where f is the frequency below which
2 c c
≤
the B(I) contribution to the PSD is negligible. Thus
z)
H 10-3 the B(I) term in Eq. (SS7) has minimal effect on the
2y (V/ measured variance. We measure σP in YHO+N to be 1.4%
sit (0.6%) for low (high) laser peak intensity.
n10-4
e Ideally, one would like to measure the long-term
d
al drift errors at each laser intensity independently, but
ectr such measurements are impractical owing to their time-
p 10-5
er s consuming nature. Consideration of the different exper-
w imental parameters reveals that the absolute contribu-
o
P tions (as opposed to relative contributions) to the error
10-6
YO+N scalemostquicklywithfluctuationsinatomicbeamden-
H
sity and least quickly with fluctuations in laser peak in-
Power law fit
tensity. Fromtheseobservations,andthePSDcalculated
(b) atI andI previously,wedevelopageneralerrormodel
L H
100 which provides the best estimate of the upper-bounds
on A(I) and B(I) for use in calculating the intensity-
Hz) dependent PSD SYY(f,I).
2nsity (V/10-1 tshcaeAletrenIgoio≤nfaoIstLfe,irnAtteh(nIasLnit)ieidssYgk/invdoeIwn2n,bywtoiItLhbe≤thnIees≤geliIgdHirbi,flteAs,(aaInr)idscianinng
e
d | |
ctral 10-2 fAro(Im)flmuacytusactaiolensnionflaassteerrinthteannsitYy.(IA)t2,inwteinthsittiheissIsHca≤linIg,
e | |
p arising from drift in inlet gas pressure and dissociation
s
wer 1 0-3 efficiency. We derive similar bounds for B(I), except
o YON B(I I ) is set equal to B(I ).
P H+ ≤ L L
Fromtheseobservations,weobtainthefollowingpiece-
Power law fit
wise upper limits on both A(I) and B(I):
10-4
10-4 10-3 10-2 10-1
Frequency (Hz)
0 I I
L
≤
FIG. S5. Power spectral densities calculated from the time 2
Eosefqr(.iae(s)S1mS.77e)aasanurdree(smbh)eonw3t.n9o×afsY1sHO0o+1Nli4daWcbql/auccimrke2dl.inaTetsh.leaspeorwpeeralkawinfitetnssfirtoimes A(I)=A(IH)"YY′(′(IIH))#2 IL ≤I ≤IH (S9)
A(I ) Y(I) I I
w4.h8e6re A0.(1I6H) =105−.937V±2/0H.3z8×an1d0−γ9=V22/.H43zγ−10,.0B8(I(aH.u).=).  H "Y(IH)# H ≤
At lo±w laser×peak intensity IL, we find th±at SYY(f,IL) B(IL) I ≤IL
Vlias2wp/uHfirtzeltayondwthhAeit(PeILSnD)ocieosnenaswibslitetehsnutBsw(tIiotLh)inz=teerro8p..o4Ul2ast±iengb0.et4th3wee×peon1w0te−hre4- B(I)=B(IH)"YY′(′(IIH))#22 IL ≤I ≤IH (S10)
discrete sample frequencies provided by the time-series B(I ) Y(I) I I,
dnpaoretbsalee.n-gtAeadsgahyieiner,led.aAdsairmtealiilaabbrulteanfeoasrtlyimtshiasetewsaaoksfetphoeefrlfobonrrmegv-etidteyrmoinsdntrhiofett where Y′(I) = dHY/"dYI(.IH)# H ≤
error σ is found by integrating the PSD over the fre- We note that for B(I), shot-noise can be considered
P
quency range relevant to the time required to perform as a scaling option for intensities IL I IH, since
≤ ≤
our measurements [S3]: theB(I)termofthePSDhasthefrequency-independent
power spectrum characteristic of white noise. This op-
f2 tion is not present for the A(I) term since that term is
σ2 =2π S (f)df. (S8)
P YY not frequency-independent. However, it turns out that
Zf1 dY/dI2 scalesinnearlythe samewayasshotnoiseover
| |
Heref isthefrequencycorrespondingtoperiodrequired the relevant intensity range. Since the contribution of
1
to acquire all ion-TOF waveforms, and f is the period B(I) to the total variance is small in any case, we find
2
5
the focus in comparison to the Rayleigh range of the
Long−term drift
Short−term drift beam. As an off-axis parabolic mirror was used to fo-
10−1 VDoislstaogciea tbioasne flrinacet siounb trµaction cus the beam (as opposed to a lens), the only diffracting
effects that could arise are fromthe 0.5 mm viewporton
our experimental chamber. Any possible effects on the
10−2
position of the foci for different wavelengths are negligi-
or
err ble owing to the thin-ness of the viewport with respect
al to the 750 mm focal length of the parabolic mirror.
on10−3
cti
a
Fr
10−4 METHOD FOR INTENSITY CALIBRATION
This subsection outlines the method required to cali-
10−5 bratethepeaklaserintensityforanyexperimentalappa-
ratus utilizing a few-cycle laser and krypton gas target.
1.5 2 2.5 3 3.5 4 4.5
Calibrated Intensity (x 1014 W/cm2)
FIG.S6. FractionalerrorcontributionsforYH+ showninlog Take the data
scale. The primary contribution is long-term drift error that
accumulatesoverour1hourtotalacquisition time. Thekink
MeasurethesingleionisationyieldfromtheKr Kr+
in long-term drift error arises from our highly conservative
→
modeling of drift error between 1.7 and 3.9 × 1014 W/cm2. process as a function of Iest. Calculate Iestusing Eq. (2)
from the manuscript, which we restate for convenience:
that the difference in scaling relations leads to a negligi-
2P 1
ble effect on the long-term drift error. Iest = πw2f τ , (2)
We calculate the intensity-dependent long-term drift 0 rep p
error σ (I) by calculating the intensity-dependent PSD
P whereP isthelaseraveragepower,w isthebeamwaist,
0
using the piecewise bounding conditions for A(I) and
f is the repetition rate and τ is the pulse duration.
rep p
B(I) from Eqs. (SS9) and (SS10) respectively, and sub-
As long as the relative intensities are well known, the
stituting them into Eq. (SS7). The intensity-dependent
actual values for I only need to be known roughly, so
est
PSD is subsequently integrated according to Eq. (SS8)
that the subsequent least-squares fit converges properly.
to arrive at σ (I).
P The example Y data shown in Table I are actual ex-
Kr+
All three intensity-dependent errors, σB(I), σAl(I), perimental data obtained using our apparatus.
andσ (I)arecombinedinquadraturetoobtaintheover-
P
allerrorin the yield, σ , for a givenapparatusset-
Ysetting
species
tingandionspecies(e.g. σYON,σYOFF). Thefinalerrors Determine which fit equation is most suitable
H+ H+
inY andY ,fromEq.(1)and(3)ofthemanuscript,
H+ Kr+
respectively,arethenpropagatedthroughfromtheerrors If the Kr target lies well within the Rayleigh range of
ofeachindividualσYsetting term. FigureS6illustratesthe the laser focus, the appropriate fit equation is Eq. (3)
species
contributions of the errors in the determination of Y . from the manuscript, which we restate for convenience:
H+
P(2D)(I ;A,η )=A S (η I ), (3)
Kr+ est 2 · phenom 2 est
BEAM PROPAGATION CONSIDERATIONS
where
As stated in the manuscript, we achieve good agree- −1/2
exp α η I /I
ment with M2 values of 1.5, close to the theoretical − 2 est c
S (η I )= . (4)
limit of 1. Two separate measurements of the M2 value phenom 2 est (cid:16)1+ (cid:0)η2Iest/Ic(cid:1)γ (cid:17)
wereobtainedbeforeandaftertheexperimentalchamber
(cid:0) (cid:1)
using a CCD beam profiler, and Gaussian propagation Here A and η are fit parameters, α = 4.24, γ = 2.49
2
−
wasconfirmed. Further beamprofile measurementswere and Ic = 2.06 0.03 1014 W/cm2. In this situation,
± ×
made with a series of 50 nm bandpass filters extending the cylindrically symmetric geometry of the interaction
across the range of our pulse bandwidth (e.g. 700-750 regionnegatestheneedforfocal-volumeaveraging(FVA)
nm, 750-800 nm). Results showed that our beam had along the propagation direction of the laser.
Gaussian propagation properties across the entire spec- However, if the gas density is uniform within the en-
tral bandwidth, with negligible effect on the position of tirety of the experimental apparatus, as opposed to an