Electron momentum distributions and photoelectron spectra of atoms driven by intense spatially inhomogeneous field

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by  M. F. Ciappina

Electron momentum distributions and photoelectron spectra of atoms driven by
intense spatially inhomogeneous field
M. F. Ciappina1,2, J. A. P´erez-Hern´andez3, T. Shaaran1, L. Roso 3, and M. Lewenstein1,4
1ICFO-Institut de Ci`encies Fot`oniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain
2Department of Physics, Auburn University, Auburn, Alabama 36849, USA
3Centro de L´aseres Pulsados, CLPU, Parque Cient´ıfico, 37185 Villamayor, Salamanca, Spain and
4ICREA-Instituci´o Catalana de Recerca i Estudis Avanc¸ats, Lluis Companys 23, 08010 Barcelona, Spain
Weusethreedimensionaltime-dependentSchr¨odingerequation(3D–TDSE)tocalculate angular
3 electron momentum distributions and photoelectron spectra of atoms driven by spatially inhomo-
1 geneous fields. An example for such inhomogeneous fields is the locally enhanced field induced
0
by resonant plasmons, appearing at surfaces of metallic nanoparticles, nanotips and gold bow-tie
2
shape nanostructures. Our studies show that the inhomogeneity of the laser electric field plays an

n importantroleintheabovethresholdionization processinthetunnelingregime,causingsignificant
a modifications to the electron momentum distributions and photoelectron spectra, while its effects
J inthemultiphotonregimeappeartobenegligible. Indeed,throughtunnelingATIprocess, onecan

1 obtain higher energy electrons as well as high degree of asymmetry in the momentum space map.
2 In this study we consider near infrared laser fields with intensities in the mid- 1014 W/cm2 range
and we use linear approximation to describe their spatial dependence. We show that in this case

] it is possible to drive electrons with energies in the near-keV regime. Furthermore, we study how
h
the carrier envelope phase influences the emission of ATI photoelectrons for few-cycle pulses. Our
p
quantummechanical calculations are fully supported by their classical counterparts.
-
m
PACSnumbers: 42.65.Ky,78.67.Bf,32.80.Rm
o
t
a
. I. INTRODUCTION analyzedwasthecomplexemissionpatternpresentinthe
s
two-dimensional momentum plane, parallel and perpen-
c
i dicular to the laser polarizationaxis, of the laser-ionized
s The process known as above-threshold ionization
y (ATI), in which an atom or molecule absorbs more pho- electrondistributions near threshold [11]. It was also in-
h vestigated how these patterns evolve as the laser-matter
tons than the minimum number requiredto single ionize
p process change from the multiphoton to the tunneling
it, has been a subject of intensive studies during the last
[ regimes [10].
decades (see e.g. [1] and references therein). The first

1 experimentalrealizationwasmade atthe endof the sev- The main difference between a few-cycle pulse and
v enties [2] and since then there has been a truly amazing a multicycle one is the strong dependence of the laser
6
progressinunderstandingofthenon-perturbativenature electric field on the so-called Carrier Envelope Phase
0
of ATI. The recent advances in laser technology made (CEP) [12, 13]. The electric field in a few-cycle pulse
9
4 possible to routinely generate laser pulses with few cy- can be characterized by its duration and by the CEP.
. cles of duration, which allows control of the atomic and The influence of CEP has been experimentally observed
1
molecular processes in their natural time scales, i.e. in inhigh-harmonicgeneration(HHG)[14],theemissiondi-
0
3 the range of (sub)-femto to attoseconds. In addition, rection of electrons from atoms [15], and in the yield of
1 these short laser sources find an extensive range of ap- nonsequential double ionization [16]. In order to have a
: plications in basic science, such as controlling molecular better control of the system on an attosecond temporal
v
motions andchemical reactions [3, 4]. Furthermore,the scale it is, therefore, important to find reliable and di-
i
X few-cycles pulses provide the fundamental pillar in the rect schemes to measure the absolute phase of few-cycle
r generation of high-order harmonics and the creation of pulses.
a
isolated extreme ultraviolet (XUV) pulses [5, 6]. The investigation of ATI generated by few-cycle driv-
The appearance of COLTRIMS experiments (see ing laser pulses plays a key role in the CEP character-
e.g. [7] and references therein) offered an unprecedented ization due to the sensitivity of the energy and angle-
possibility of performing stringent tests on the differ- resolved photoelectron spectra to the value of the laser
ent theoretical approaches. On one side, this is because electric field absolute phase [17, 18]. Consequently, the
the imaging of the vectorial momentum distributions of behavior of the laser-ionized electrons renders the ATI
the reactionfragmentsare easily accessible,while onthe phenomenon a very valuable tool for laser pulse charac-
other they are particularly sensitive to various details of terization. To determine the CEP of a few-cycle laser
the theory. COLTRIMS were primarily developed for pulse, it is essential to record the difference between the
the study of few-body dynamics induced by particle im- yield of electrons ionized for different emission angles.
pact, i.e. electrons and ions, but the extension to scruti- Through this technique one can analysis the so-called
nize and tackle laser-induced processes was natural (see backward-forwardasymmetry in order to obtain the ab-
e.g [8–10]). Among the features whichweretheoretically solute CEP [19]. Furthermore, it appears that the high
2
energy region of the photoelectron spectra is most sen- inhomogeneityofthefieldmodifiesboththetwoelectron
sitive to the absolute CEP and consequently electrons momentum distributions and the photoelectron spectra;
with large kinetic energy are needed in order to describe we examine also the influence of the CEP parameter.
it [1, 20]. Furthermore, our quantum mechanical results are com-
Recent experiments using a combination of plasmonic pared with classical calculations of the kinetic energy of
nanostructures and rare gases have demonstrated that the electron.
the harmonic cutoff of the gases could be extended fur- This article is organized as follows. In the next sec-
ther than in conventional situations by using the field, tion, we present our theoretical approach to model ATI
locallyenhancedduetothecouplingofalaserpulsewith producedby spatiallynonhomogeneousfields,withmain
a nanosystems [21, 22]. In such nanosystems, due to the emphasis on the extraction of the electron angular mo-
strong confinement of the plasmonics spots and the dis- mentumdistributionsstartingfromtheTDSEoutcomes.
tortion of the electric field by the surface plasmons, the Subsequently, in Sec. III, we apply our method to com-
locally enhanced field is not spatially homogeneous in pute the electron momentum distributions and energy-
the region,where the electrondynamics take place. One resolved photoelectron spectra of hydrogen atom using
should note, however, that the outcome of the experi- few-cycle laser pulses for both homogeneous and inho-
ments of Ref. [21], in which a combination of gold bow- mogeneousfields,consideringtunnelingandmultiphoton
tie nanostructureandargongasforgenerationHHGwas regimes. Furthermore, we solve the classical equations
used, has been recently under intense scrutiny [23, 24]. of motion of an electronin an oscillating inhomogeneous
Inaddition,recently,insteadofatomsormoleculesingas electricfieldtosupportourquantummechanicalmethod.
phase solid state nanostructures have been employed as Finally, in Sec. IV, we conclude our contributions with a
a target to study the photoelectron emission by few in- short summary and outlook.
tenselaserpulses[25,26]. Thislaserdrivenphenomenon,
called Above Threshold Photoemission (ATP), has re-
II. THEORETICAL APPROACH
ceivedspecialattentionduetothenoveltyoftheinvolved
physics and potential applications. In ATP process, the
emittedelectronshaveenergyfarbeyondtheusualcutoff TostudythepropertiesoftheATIphenomenondriven
for noble gases (see e.g. [26–31]). Furthermore, the pho- by spatial nonhomogeneous fields, we solve the three di-
toelectrons emitted from these nanosources are sensitive mensional Time Dependent Schr¨odinger Equation (3D-
to the CEP, and consequently this fact plays an impor- TDSE)inthelengthgauge. Theelectronmomentumdis-
tant role in the angle and energy resolved photoelectron tributionandenergy-resolvedphotoelectronspectraofan
spectra [25, 26, 32, 33]. atom are calculated from the time propagatedelectronic
wave function.
From theoretical point of view, the fundamental as-
Our calculations are based on spherical harmonics ex-
sumption behind strong field phenomena, that the laser
pansion, Ym, considering only the m = 0 terms due to
field is spatially homogeneous in the region where the l
the cylindrical symmetry of the problem. The Crank-
electron dynamics takes place [34, 35], is not any more
Nicholson method, which is based on a splitting of the
valid for the locally enhanced plasmonic field. Indeed,
time-evolution operator that preserves the norm of the
insuchsystemthe drivenelectricfield,andconsequently
wave function, is used as the numerical technique. We
the Lorentz force the electron feels, will also depend on
consider the field to be linearly polarized along the z
position. Uptonow,therehavebeenveryfewstudiesin-
axis and the variation of the electric field is linear with
vestigating the strong field phenomena in such spatially
respect to the position. As a result, the coupling V (r,t)
inhomogeneous fields [36–42]. All of these studies have l
between the atomic system and the electromagnetic ra-
demonstrated that the spatial dependency of the field
diation reads
strongly modifies the laser-driven phenomena that ap-
r
pear in such circumstances.
V (r,t)= dr′·E(r′,t)=E z(1+βz)f(t)sin(ωt+φ),
For homogeneous driving field, up to know, different l Z 0
numericalandanalyticalapproacheshavebeenemployed (1)
to calculate the ATI (see e.g. [1, 43–47] and references where E0, ω and φ are the laser electric field amplitude,
therein). Inthisarticle,weextendthestudiesofourpre- the central frequency and characterizes the carrier enve-
viouspaper[40]byapplyingthenumericalsolutionofthe lopephase(CEP),respectively. Theparameterβ defines
time-dependent Schr¨odinger Equation (TDSE) in three the ‘strength’ of the inhomogeneity and has units of in-
dimensions to calculate the angular electron momentum verse length (see also [36–38]). For modeling the short
distributionsandphotoelectronspectraofATI drivenby laserpulsesinEq.(1),weuseasin-squaredenvelopef(t)
spatially inhomogeneous fields, considering both tunnel- of the form
ing and multiphotonregimes. The spatialdependence of
ωt
the field is considered to be linear. We mainly focus on f(t)=sin2 , (2)
(cid:18)2n (cid:19)
studyingATIofhydrogenatoms,butourscheme,within p
the single active electron approximation,can be directly where n is the total number of optical cycles. As a re-
p
applied to any complex atom. We demonstrate how the sult,thetotaldurationofthelaserpulsewillbeT =n τ
p p
3
whereτ =2π/ω isthelaserperiod. Wealsoassumethat γ . 1 (γ = I /2U , where U = I/4ω2 is the pon-
p p p
before switch on of the laser (t = −∞) the target atom deromotive enpergy and Ip the ionization potential), and
(hydrogen) is in its ground state (1s), whose analytic multiphoton regime, for which the Keldysh parameters
form can be found in standard textbooks. Within the is γ >> 1. Furthermore, we confirm how in the tun-
single active electron approximation, however, our nu- neling regime the CEP, joint with the spatial nonhomo-
merical scheme is tunable to treat any complex atom by geneities, modify in a particular way both the energy-
choosingtheadequateeffective(Hartree-Fock)potential, resolved photoelectron spectra and the two-dimensional
and finding the ground state by the means of numerical electron momentum distributions as we have shown in
diagonalization. our previous contribution [40]. On the other hand, we
The ATI spectrum is calculatedusing the time depen- show that in the multiphoton regime (γ >>1), the spa-
dentwavefunctionmethoddevelopedbySchaferandKu- tial nonhomogeneous character of the laser electric field
lander (see [48] for more details). As a preliminary test, hardly affects the analyzed quantities. We also want to
forensuringtheconsistenceofournumericalsimulations, point out, however, that the frontier between the tunnel
wehavecheckedoutourcalculationswiththeresultspre- and multiphoton regimes appears to be a controversial
viously obtained in Ref. [48]. The comparison confirms and diffuse issue [51, 52].
the high degree of accuracy of our calculations as shown
in Fig. 1.
A. Tunneling regime
We commence by investigating the tunneling regime.
For this case, we employ a four-cycle (total duration 10
fs) sin-squared laser pulse with wavelength λ = 800 nm
and two different intensities, namely I = 1.140× 1014
W/cm2 and I =5.0544×1014 W/cm2. These twointen-
sities give values for the laser electric field of E =0.057
0
a.u. andE =0.12a.u.,respectively. Forallthecaseswe
0
chosefourdifferentvaluesfortheparameterthatcharac-
terizes the inhomogeneity strength, namely, β = 0 (ho-
mogeneous case), 0.002,0.003 and 0.005. In addition we
also vary the carrier envelope phase φ in Eq. (1), tak-
ing φ = 0, φ = π/2, φ = π and φ = 3π/2. For all
the above mentioned cases, we calculate the the energy-
FIG.1. (Coloronline)Photoelectronspectrumresultingfrom
resolvedphotoelectronspectra. The results areshownin
our 3D TDSE simulations (in red) and superimposed (in
Figures2 and3 for I =1.140×1014 W/cm2 (γ =1)and
black) with the ATI results calculated by Schafer and Ku-
I =5.0544×1014 W/cm2 (γ =0.475), respectively.
lander in Ref. [48]. The laser wavelength is λ=532 nm and
theintensityisI =2×1013 W/cm2 (seeFig.1in[48]formore
details. The superimposed plot has been extracted from Fig. !"#$
!!" ’() !!"#""$ ’*)
1 of this cited reference. !"#’
/0102 !"#&
Forcalculatingtheenergy-resolvedphotoelectronspec- ()*+,-(. !!""#!#%"
tra P(E) and two-dimensional electron distributions
!"#!$
H(p,θ) we use the window function approach developed !"#$ !!"#""% ’+) !!"#""& ’,)
!"#’
bteolyecSctacrhlocanufleasrpte[e4c8at,rna4g9l[e]5.-0rTe],shoailsvntedodoitlahnradesperbeneseeernngtyws-irdaeeslsoytlveupesdefdop,rhwbooattrohd- ()*+,-(./0102 !!!"""#!##%&"
with respect to the usual projection methods.
!"#!$" $" ’" &" %" !""! "! #! $! %! &!!
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III. RESULTS FIG.2. (Coloronline)Energy-resolvedphotoelectronspectra
P(E) calculated using the 3D-TDSE for an hydrogen atom
(Ip =−0.5 a.u.). The laser parameters are I = 1.140×1014
Inthis section,we calculateboth energy-resolvedpho- W/cm2 (E0 =0.057 a.u.) and λ=800 nm. We have used a
toelectron spectra P(E) and two-dimensional electron sin-squaredshapedpulsewithatotaldurationoffouroptical
momentum distributions in order to investigate the in- cycles (10 fs). (a) β =0 (homogeneous case), (b) β =0.002,
fluence of the inhomogeneities of the field and the sensi- (c)β =0.003 and(d)β =0.005. Inallthepanels,blackline:
tivity of these two measurable quantities to the different φ = 0; blue line φ = π/2; green line: φ = π and red line:
laserparameters,especiallytothecarrierenvelopephase φ=3π/2.
(CEP). The investigations are carried out for both the
tunneling regime, for which the Keldysh parameters is For the homogeneous case, the spectra exhibits the
4
electronspectra. Itmeansfortacklingthisproblemboth
physical mechanisms should be included in theoretical
model. Inthatsense,the3D-TDSE,whichcanbeconsid-
eredas anexact approachto the ATI problemfor atoms
in the single active electron approximation, is the ad-
equate tool to predict the P(E) in the whole range of
electronenergies. Onthe other hand,the mostenergetic
electrons, i.e., those with kinetic energies E ≫ 2U ,
k p
are commonly used to characterizethe CEP of few-cycle
pulses. Consequently, a correct description of the elec-
tron rescattering mechanism is needed.
2(a) −2 2(b) −2
−3 −3
k (a.u.)r1 −−54k (a.u.)r1 −−54
−6 −6
−02 −1 0 1 2 −7 −02 −1 0 1 2 −7
FIG. 3. (Color online) Idem Fig. 2 but for a laser intensity kz (a.u.) kz (a.u.)
I =5.404×1014 W/cm2 (E0=0.12 a.u.). 2(c) −2 2(d) −2
−3 −3
k (a.u.)r1 −−54k (a.u.)r1 −−54
usual distinct behavior, namely, the 2Up cutoff (≈ 13.6 −6 −6
eV in Fig. 2(a) and ≈ 57.4 eV in Fig. 3(a)) and the −02 −1 0 1 2 −7 −02 −1 0 1 2 −7
kz (a.u.) kz (a.u.)
10U cutoff (≈ 68 eV in Fig. 2(a) and ≈ 300 eV in
p
Fig. 3(a)). The former cutoff corresponds to those elec- FIG. 5. (Color online) Idem Fig. 4 but for φ=π/2
trons that, once ionized, never return to the atomic core
(the so-called direct electrons), while the latter one cor-
For the spatially nonhomogeneous cases, the positions
responds to the electrons that, once ionized, return to
of the direct and the rescattered electron cutoffs are
the core and elastically rescatter (the so-called rescat-
extended towards larger energies. For the rescattered
tered electrons). Classically, it is a well established ar-
electrons, this extension is very noticeable. In fact for
guments that the maximum kinetic energies E of the
k β = 0.005 with E = 0.12 a.u., it reaches values close
directand the rescatteredelectronsare Ed =2U and 0
max p to ≈ 700 eV [Fig. 3(d)] against the ≈ 300 eV shown
Er = 10U , respectively (see below for more details).
max p by the homogeneous case. Another new feature present
In a quantum mechanical approach,however,it is possi-
forallthenonhomogeneouscasesisthestrongsensitivity
ble to find electrons with energies beyond the 10Up cut-
of the P(E) to the carrier envelope phase (CEP). This
off,althoughtheiryielddropsseveralordersofmagnitude
behavior can be clearly noticed by comparing the panels
depending strongly on the atomic species studied [1].
(a) of Figs. 2 and 3 (i.e. the homogeneous case) with
therestofthe plots. Itisclearthatforthehomogeneous
2(a) −2 2(b) −2 caseonlytwocurvesarepresent,due tothefactthatthe
−3 −3
k (a.u.)r1 −−54k (a.u.)r1 −−54 Pan(dE()φfo=r 3φπ/=2)0, raensdpe(cφtiv=elyπ./O2)natrheeiodtehnetrichaalntdo,φfo=r aπll
−6 −6
−02 −1 0 1 2 −7 −02 −1 0 1 2 −7 the nonhomogeneous cases it is possible to clearly dis-
kz (a.u.) kz (a.u.) tinguish the 4 cases, i.e. φ = 0, φ = π/2, φ = π and
2(c) −2 2(d) −2 φ = 3π/2. Indeed, this particular characteristic of the
−3 −3
k (a.u.)r1 −−54k (a.u.)r1 −−54 aPn(dEb)efottrenroCnEhoPmcohgaernaecoteursizfiaetlidosnctoouolld. makethemanew
−6 −6
It should be noted, however, that other well-known
−02 −1 0 1 2 −7 −02 −1 0 1 2 −7
kz (a.u.) kz (a.u.) and established CEP characterization tools, such as,
for instance, the forward-backward asymmetry or two-
FIG. 4. (Color online) Two-dimensional electron momen-
dimensional electron momentum distributions should
tum distributions (logarithmic scale) in cylindrical coordi-
complement the P(E) measurements [1]. Furthermore,
nates (kz,kr) using the exact 3D-TDSE calculation for an
hydrogen atom. The laser parameters are I = 1.140×1014 the utilization of nonhomogeneous fields would open the
W/cm2 (E0 =0.057 a.u.) and λ=800 nm. We have used a avenuefortheproductionofhigh-energyelectrons,reach-
sin-squaredshapedpulsewithatotaldurationoffouroptical ing the keV regime, if a reliable control of the spatial
cycles (10 fs) with φ=0. (a) β =0 (homogeneous case), (b) and temporal shape of the laser electric field is attained.
β =0.002, (c) β=0.003 and (d) β =0.005. Investigations in such direction has already started (see
e.g. [29] and references therein).
Experimentally speaking, both the direct and rescat- Adeepanalysisoftheelectrondistributionsforatomic
tered electrons contribute to the energy-resolved photo- ionizationproducedbyanexternallaserfieldcanbeper-
5
2(a) −2 2(b) −2 2(a) −1 2(b) −1
−3 −3 −2 −2
k (a.u.)r1 −−54k (a.u.)r1 −−54 k (a.u.)r1 −−−543k (a.u.)r1 −−−543
−6 −6
−02 −1 kz (0a.u.) 1 2 −7 −02 −1 kz (0a.u.) 1 2 −7 −02 −1 kz (0a.u.) 1 2 −6 −02 −1 kz (0a.u.) 1 2 −6
2(c) −1 2(d) −1
2(c) −2 2(d) −2 −2 −2
k (a.u.)r1 −−−543k (a.u.)r1 −−−543 k (a.u.)r1 −−−543k (a.u.)r1 −−−543
−6 −6 −02 −1 0 1 2 −6 −02 −1 0 1 2 −6
−02 −1 0 1 2 −7 −02 −1 0 1 2 −7 kz (a.u.) kz (a.u.)
kz (a.u.) kz (a.u.)
FIG. 8. (Color online) Two-dimensional electron momen-
FIG. 6. (Color online) Idem Fig. 4 but for φ=π tum distributions (logarithmic scale) in cylindrical coordi-
nates (kz,kr) using the exact 3D-TDSE calculation for an
hydrogen atom. The laser parameters are I = 5.0544×1014
formedintermsofthe two-dimensionalelectronmomen- W/cm2 (E0 = 0.12 a.u.) and λ = 800 nm. We have used a
sin-squaredshapedpulsewithatotaldurationoffouroptical
tumdistribution. The exactsolutionofthe threedimen-
cycles (10 fs) with φ=0. (a) β =0 (homogeneous case), (b)
sional Schr¨odinger equation (3D-TDSE) provide us with
β=0.002, (c) β=0.003 and (d) β =0.005.
anexcellenttooltoanalyzeindetailhowthetwocompet-
ingfields,namelythelaserelectricfieldandtheCoulomb
potential,modifytheelectronwavepacketofthereleased creasing the laser field intensity to I = 5.0544× 1014
electron. In Figs. 4-7 we calculate two-dimensional elec- W/cm2 (E = 0.12 a.u). The results are depicted in
0
tron momentum distribution for a laser field with an in-
Figs. 8-10 for φ = 0, φ = π/2, φ = π and φ = 3π/2,
tensity of I = 1.140× 1014 W/cm2 (E = 0.057 a.u),
0 respectively. Here, panel (a), (b), (c) and (d) represent
λ=800 nm and different values of the the β parameter:
the cases with β = 0 (homogeneous case), β = 0.002,
panel(a)β =0(homogeneouscase);panel(b)β =0.002;
β =0.003 and β =0.005, respectively.
panel(c)β =0.003andpanel(d) β =0.005. We employ
afew-cyclelaserpulsewith4totalcycles(10fs)andvari-
2(a) −1 2(b) −1
ousvaluesofthecarrierenvelopephase(CEP)parameter −2 −2
φ. Figs. 4-7 show the cases with φ = 0, φ= π/2, φ= π k (a.u.)r1 −−43k (a.u.)r1 −−43
and φ=3π/2, respectively. −5 −5
−02 −1 0 1 2 −6 −02 −1 0 1 2 −6
kz (a.u.) kz (a.u.)
2(a) −2 2(b) −2 2(c) −−21 2(d) −−21
k (a.u.)r1 −−−543k (a.u.)r1 −−−543 k (a.u.)r1 −−−543k (a.u.)r1 −−−543
−6 −6 −02 −1 0 1 2 −6 −02 −1 0 1 2 −6
−02 −1 0 1 2 −7 −02 −1 0 1 2 −7 kz (a.u.) kz (a.u.)
kz (a.u.) kz (a.u.)
2(c) −2 2(d) −2 FIG. 9. (Color online) Idem Fig. 8 but for φ=π/2
−3 −3
k (a.u.)r1 −−54k (a.u.)r1 −−54
−6 −6 2(a) −1 2(b) −1
−02 −1 0 1 2 −7 −02 −1 0 1 2 −7 −2 −2
kz (a.u.) kz (a.u.) k (a.u.)r1 −−43k (a.u.)r1 −−43
FIG. 7. (Color online) Idem Fig. 4 but for φ=3π/2 −5 −5
−02 −1 0 1 2 −6 −02 −1 0 1 2 −6
kz (a.u.) kz (a.u.)
Here, we concentrate our analysis on the low energy 2(c) −1 2(d) −1
−2 −2
rheogmioongeonfetihtieesdoisfttrhibeultaisoenrseilnecotrrdicerfietlodsatffuedcyththoewatnhgeuilnar- k (a.u.)r1 −−−543k (a.u.)r1 −−−543
electron yield. This regionshows the usual bouquet-type −02 −1 0 1 2 −6 −02 −1 0 1 2 −6
kz (a.u.) kz (a.u.)
structures (see [11] for details) with noticeable modifica-
tions for the nonhomogeneous cases. FIG. 10. (Color online) Idem Fig. 8 but for φ=π
Furthermore, the low-energy electrons appear to be
strongly influenced by the spatial inhomogeneity of the Following the trend observed in the previous studied
laser electric field (see panels (b)-(d) of Figs. 4-7). We case, we see strong modifications produced by the spa-
also can observe how the bouquet structures present in tial inhomogeneities in both the angular and low-energy
the homogeneous case disappear for particular values of structures. Here,themodificationseemstobeevenmore
φ (see e.g. Fig.6(d)). pronounced. For instance, the yield for electrons with
In order to complete our investigations, we calculate k < −1 a.u. for φ = 0 (Fig. 8(a)) and φ = π/2
z
two-dimensionalelectronmomentumdistributionsbyin- (Fig. 9(a)) drops several orders of magnitude. The sig-
6
nificantdropoftheyieldoftheelectronemissionbetween
k < −1 a.u and k > 0 a.u. opens a new approach to
z z
characterize the CEP.
We now employ a classical model in order to explain
and characterize the extension of the energy-resolved
photoelectron spectra. According to the simple-man’s
model [53] the physical mechanism behind ATI process
willbeasfollows: atagiventime,thatwecallionization
time t , an atomic electronis releasedor born at the po-
i
sition z = 0 with zero velocity, i.e., z˙(t )= 0. This elec-
i
tron now moves only under the influence of the oscillat-
ing laser electric field (this model neglects the Coulomb
interaction with the remaining ion) and will reach the
detector either directly or through the process known as
rescattering. By using the classical equation of motion
of an electron moving in an oscillating electric field, it is
possibletocalculatethemaximumenergyoftheelectron
for both the direct and rescattered processes.
2(a) −1 2(b) −1
−2 −2
k (a.u.)r1 −−43k (a.u.)r1 −−43
−5 −5
−02 −1 0 1 2 −6 −02 −1 0 1 2 −6
kz (a.u.) kz (a.u.) FIG. 12. (Color online) Numerical solutions of the Newton
2(c) −1 2(d) −1 equation [Eq. (3)] plotted in terms of the direct and rescat-
−2 −2
k (a.u.)r1 −−−543k (a.u.)r1 −−−543 tlaesreerdpealercatmroentekrsinaerteicIen=er1g.y1,4E0kd×a1n0d14EWkr,/rcemsp2e(cEti0ve=ly.0.T05h7e
−02 −1 0 1 2 −6 −02 −1 0 1 2 −6 a.u), λ = 800 nm and φ = 0. We employ a few-cycle laser
kz (a.u.) kz (a.u.) pulse with 4 total cycles (10 fs). Different panels correspond
to various values of β (see labels). Green filled circles: direct
FIG. 11. (Color online) Idem Fig. 8 but for φ=3π/2
electrons; red filled circles: rescattered electrons.
The Newton equation of motion for the atomic elec-
tron in the laser electric field can be written, using the
withβ =0, itcanbe shownthat the maximumvalue for
functional form of Eq. (1), as follows
Ed is 2U while for Er it is 10U [1]. These two values
k p k p
z¨(t)=−∇ V (r,t) (3) appear as cutoffs in the energy resolved photoelectron
z l
spectrum as can be observedin panels (a) of Figs. 2 and
=E(t)[1+2βz(t)] (4)
3.
where we have collected the time-dependent part of the In Figs. 12-19, we present the numerical solutions of
electric field in E(t), i.e., E(t)=E0f(t)sin(ωt+φ). Eq. (3), which is plotted in terms of the kinetic energy
In the limit where β = 0 in Eq. (3), we recover the (in eV) of the direct and rescattered electrons.
homogeneous case. For the direct laser ionization, the Figures 12-15 are for a laser intensity of I = 1.140×
kinetic energy of an electron released or born at time ti 1014 W/cm2 (E =0.057 a.u.) meanwhile in Figs. 16-19
0
is the laserintensity isI =5.404×1014 W/cm2 (E =0.12
0
[z˙(t )−z˙(t )]2 a.u.). Figures12(16),13(17),14(18)and15(19)arefor
Ed = i f , (5) φ = 0, φ = π/2, φ = π, φ = 3π/2, respectively and for
k 2
differentvaluesofthe β parameter(β =0(homogeneous
wheretf istheendtimeofthelaserpulse. Fortherescat- case), β = 0.002, β = 0.003 and β = 0.005, from top to
teredlaser-ionizedelectron,inwhichtheelectronreturns bottom).
to the coreat a time tr andreversesits direction,the ki- From the curves for β 6= 0 we can observe the strong
netic energy of the electron yields modifications that the nonhomogeneous character of the
laser electric field produces in the kinetic energy of the
[z˙(t )+z˙(t )−2z˙(t )]2
Er = i f r . (6) laser-ionized electron. Furthermore, it is possible to ob-
k 2
servehowthekineticenergyismoresensitivetotheCEP.
For homogeneous fields, Equations (5) and (6) be- If we analyze, for instance, the top panels of Figs. 12-15
comethe usualexpressionsEd = [A(ti)−A(tf)]2 andEr = (i.e the homogeneous case) we conclude that the shape
[A(ti)+A(tf2)−2A(tr)]2, with A(tk) being the2 laservectorkpo- oelfecbtortohntahreekiidneenttiiccaelneforgryφo=f th0e(dφir=ectπa/n2d) arnesdcaφtt=ereπd
tential A(t) = − tE(t′)dt′, respectively. For the case (φ = 3π/2). On the other hand, for the different values
R
7
FIG. 13. (Color online) Idem Fig. 12 for φ=π/2. FIG. 15. (Color online) Idem Fig. 12 for φ=3π/2.
this way it spends more time in the continuum acquir-
ing energy from the laser electric field. Consequently,
higher values of the kinetic energy are attained. This
distinct behavior is more evident for E =0.12 a.u. and
0
β =0.005,butitappearstosomedegreeforallthestud-
ied cases.
A similar behavior was observed recently in above
threshold photoemission (ATP) using metal nanotips.
Accordingto the modeldevelopedinRef.[31]the strong
localizedfields modify the electronmotionin suchaway
to allow sub-cycle dynamics.
In our approach, however, we include the full picture
of the ATI phenomenon, namely both direct and rescat-
tered electrons are considered (in Ref. [31] only direct
electrons are taken into account) and consequently the
characterizationofthe dynamics ofthe photoelectronsis
more complex. Nevertheless, the higher kinetic energy
of the rescattered electrons is a clear consequence of the
strong modifications the laser electric field produces in
theregionwheretheelectrondynamicstakesplace,asin
the above mentioned case of ATP.
FIG. 14. (Color online) Idem Fig. 12 for φ=π.
B. Multiphoton regime
of β the kinetic energy has a unique shape for a given Forthemultiphotoncase,weconsiderafew-cyclelaser
value of φ. pulse with 6 complete optical cycles and E = 0.05 a.u.
0
Theparticularfeaturespresentforβ 6=0arerelatedto (I =8.775×1013 W/cm2) and ω =0.25 a.u. (λ=182.5
the changes in the laser-ionizedelectron trajectories (for nm). Inhere,theKeldyshparameterisγ =5,indicating
detailsseee.g.[37–39]). Insummary,theelectrontrajec- the predominance of the multiphoton process [11]. We
tories are modified in such a way that now the electron have computed the P(E), two-dimensional electron dis-
ionizes at an earlier time and recombines later, and in tributionsandtheclassicalelectronenergiesforalltheset
8
FIG. 17. (Color online) Idem Fig. 16 for φ=π/2.
FIG. 16. (Color online) Numerical solutions of the Newton
equation[Eq. 3]plottedintermsofthedirectandrescattered
electron kinetic energy, Ed and Er, respectively. The laser
k k
parameters are I = 5.0544×1014 W/cm2 (E0 = 0.12 a.u),
λ = 800 nm and φ = 0. We employ a few-cycle laser pulse
with 4 total cycles (10 fs). Different panels correspond to
various values of β (see labels). Green filled circles: direct
electrons; red filled circles: rescattered electrons.
of cases presented in Sec. III.A. In this paper, however,
we just present the most extreme case with an inhomo-
geneity factor of β = 0.005 and CEP of φ= π/2. These
results are presented in Figs. 20, 21 and 22 for P(E),
two-dimensional electron distributions and the classical
electron energies, respectively. The P(E) exhibits the
usual multiphoton peaks [2, 48] and the inhomogeneity
of the field does not play any significant role. In the
whole range, the values of the yields have a difference of
less than 5% and in a logarithmic scale this is hard to
discern.
Thetwo-dimensionalelectrondistributionsarealsothe
same interms ofshape andmagnitude forboth homoge-
neous and inhomogeneous cases, as shown in Fig. 21. It FIG. 18. (Color online) Idem Fig. 16 for φ=π.
means the differences introduced by the spatial nonho-
mogeneous character are practically imperceptible. We
should note that our calculation is practically identical
these observable quantities for both variations in the
to the one presented in [11].
CEP and the strength of the inhomogeneity parameter
ThenumericalsolutionsofEq.(3)asfunctionoftheki- β. As a result, we conclude that in the multiphoton
neticenergy(ineV)ofthedirectandrescatteredelectron regime the modifications introduced by the spatial in-
is depicted in Fig. 22. In here, we could also observe, in homogeneitiesfielddonotproduceappreciablemodifica-
supporttoourquantummechanicalcalculations,thatthe tions in the electron dynamics and consequently in the
inhomogeneity of the field does not change the electron measurablequantities. Inaddition,thelaser-ionizedelec-
energies in both the direct and rescattered processes. tron, in both the direct and rescattered processes, has a
In general, we do not find noticeable differences in very small kinetic energy in the multiphoton regime due
9
2(a)
−2
k (a.u.)r1 −4
−6
−02 −1 0 1 2 −8
kz (a.u.)
2(b)
−2
k (a.u.)r1 −4
−6
−02 −1 0 1 2 −8
kz (a.u.)
FIG. 21. (Color online) Two-dimensional electron momen-
tum distributions (logarithmic scale) in cylindrical coordi-
nates (kz,kr) using the exact 3D-TDSE calculation for an
hydrogen atom. The laser parameters are E0 = 0.05 a.u.
(I =8.775×1013 W/cm2),ω=0.25a.u. (λ=182.5nm)and
φ=π/2. We employ a laser pulse with 6 total cycles. Panel
(a) corresponds to the homogeneous case (β = 0) and panel
(b) is for β =0.005.
FIG. 19. (Color online) Idem Fig. 16 for φ=3π/2. IV. CONCLUSIONS AND OUTLOOK
We have extended our previous studies of ATI pro-
duced by nonhomogeneous fields using the three dimen-
sional solutions of the TDSE. We have modified the 3D-
TDSE to model the ATI phenomenon driven by spatial
nonhomogeneous fields by including an additional term
in the laser-atom coupling. In the tunneling regime we
predict an extension in the cutoff position and an in-
crease of the yield of the energy-resolved photoelectron
spectra in certain regions. In addition, both the pho-
toelectron spectra and the two-dimensionalelectron mo-
mentum distributions appearto be more sensitive to the
carrier envelope phase of the laser electric field. This
feature indicates that the ATI produced by spatial in-
homogeneous field could be a good candidate for few-
cycle laser pulse characterization. Furthermore, our pre-
dictions pave the way for the production of high-energy
photoelectrons, reaching the keV regime, using plasmon
FIG. 20. (Color online) Energy-resolved photoelectron spec- enhancedfields. Inthemultiphotonregime,weshowthat
traP(E)calculatedusingthe3D-TDSEforanhydrogenatom
boththeP(E)andthetwo-dimensionalelectrondistribu-
(Ip = −0.5 a.u.). The laser parameters are E0 = 0.05 a.u. tionsarehardlyaffectedbythespatialnonhomogeneities
(I =8.775×1013 W/cm2)andω=0.25a.u. (λ=182.5nm).
of the laser electric field. Our quantum mechanical cal-
We have used a sin-squared shaped pulse with a total dura-
culations are supported by the classical simulations. In
tionofsixopticalcyclesandφ=π/2. Redline: homogeneous
particular, the P(E) characteristics are reasonably well
case (β=0); blue line: β=0.005;.
reproduced by simulations based on classical physics.
ACKNOWLEDGMENTS
to low intensity field. Indeed, in this case the maximum
energy after the rescattering process has taken place is
≈3 eV. As a result, it is reasonableto have a very small We acknowledge the financial support of the
or almost no differences between the final kinetic ener- MINCIN/MINECO projects (FIS2008-00784TOQATA)
gies, when a spatial inhomogeneity of small strength is (M. F. C. and M.L.); ERC Advanced Grant QUA-
present. GATUA, Alexander von Humboldt Foundation (M. L.);
10
J. A. P.-H. and L. R. acknowledge support from Span-
ish MINECO through the Consolider Program SAUUL
(CSD2007-00013) and research project FIS2009-09522,
from Junta de Castilla y Le´on through the Program for
GroupsofExcellence(GR27)andfromtheERCSeventh
Framework Programme (LASERLAB-EUROPE, grant
agreement n 228334); L. R. acknowledges the Junta de
Castilla y Le´on through the project CLP421A12-1.;this
research has been partially supported by Fundaci´o Pri-
vada Cellex.
FIG. 22. (Color online) Numerical solutions of the Newton
equation [Eq. (3)] plotted in terms of the direct and rescat-
tered electron kinetic energy, Ed and Er, respectively. The
k k
laser parameters are E0 = 0.05 a.u. (I = 8.775 × 1013
W/cm2), ω = 0.25 a.u. (λ = 182.5 nm) and φ = π/2. We
employ a laser pulse with 6 total cycles. Panel (a) corre-
sponds to the homogeneous case (β =0) and panel (b) is for
β = 0.005. Green filled circles: direct electrons; red filled
circles: rescattered electrons.
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