Field strength scaling in quasi-phase-matching of high-order harmonic generation by low-intensity assisting fields

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by  Emeric Balogh

Field strength scaling in quasi-phase-matching of high-order harmonic generation by
low-intensity assisting fields
Emeric Balogh1,2,3 and Katalin Varju´1,2
1Department of Optics and Quantum Electronics, University of Szeged, 6720 Szeged, Hungary
2ELI-ALPS, ELI-HU Nkft, 6720 Szeged, Hungary
3Center for Relativistic Laser Science, Institute for Basic Science (IBS), Gwangju 500-712, Republic of Korea∗
High-order harmonic generation in gas targets is a widespread scheme used to produce extreme
ultraviolet radiation, however, it has a limited microscopic efficiency. Macroscopic enhancement
of the produced radiation relies on phase-matching, often only achievable in quasi-phase-matching
6
arrangements. In the present work we numerically study quasi-phase-matching induced by low-
1
intensity assisting fields. We investigate the required assisting field strength dependence on the
0
wavelength and intensity of the driving field, harmonic order, trajectory class and period of the
2
assisting field. We comment on theoptimal spatial beam profile of theassisting field.

n
a
J I. INTRODUCTION focusingandlowionization),allowingtheconstructionof

8 a simple analytical model. This type of phase-matching
2 has already been discussed extensively [10–14], and now
The most promising way of coherent extreme ultravi-

olet and soft x-ray short pulse generation is high-order it is known that, depending on the characteristics of the
]
s harmonic generation (HHG) of near-infrared and mid- target gas, there is a limit on the achievable photon en-
c ergy (phase-matching cutoff). This limit is imposed by
infrared laser pulses in gases and solids. HHG in gases
i
t has the advantage of having less constraints on the laser the intensity of the driving field that produces a critical
p
ionizationrateabovewhichconventionalphase-matching
o parameters, however, it also has a moderate conversion
. efficiency (10−4 – 10−6) [1–3], which decreases further is not possible [13].
s
c with the wavelength of the generating laser pulse (λ−5.5
i – λ−6.5) [4, 5]. The increase in laser intensity to pro-
s
y ducehighphoton-fluxfromharmonicradiationislimited Above this phase-matching limit, quasi-phase-
h by the ionization it produces: through depletion of the matching (QPM) schemes are often used to increase
p medium and distortions of the laser pulse in the plasma. harmonicyield[13]. AsingasHHGthetraditionalQPM
[
The use of long gas cells also raises problems of spatial schemes based on birefringence are not possible other

1 phase-matching (PM) of the generated harmonic radia- methods have been proposed. These are based on some
v tion [6–8]. type of periodic modulation along the propagation axis,
0 Inamacroscopicmediumphasemismatchariseswhen which includes atomic density (modulated by acoustic
5
the phase velocity of the polarization induced by the waves [15], or using a multijet configuration [16–20]),
6
propagatinglaser field is different from that of the prop- driving field intensity (in modulated waveguides[21–24],
7
agating harmonic field [7, 9]. The exact description of and by multimode beating in capillaries [25, 26]), or
0
. phase-mismatch is a complex task mainly because gas modulation caused by a secondary periodic field, which
1
HHG is non-instantaneous: during the process the va- is either static [27], or propagating in another direction
0
lence electron leaves the core, gains energy and recom- than the driving field [28–34].
6
bineswiththeionreleasingitsenergyinformofaphoton.
1
: Along its trajectory the electron accumulates a phase
v
which is inherited by the harmonic field, making the
i In this paper we discuss QPMschemes employing sec-
X harmonic’s phase dependent on the phase, intensity and
ondaryassistingfields,andpresentnumericalresultsthat
r shape of the generating laser field as well as the length can give an indication to experiments on the intensities
a of the electron trajectory [8]. The complex relation be-
of the secondary field required to induce QPM, depend-
tween harmonic phase and generating laser field proper-
ing on the parameters of the driving field. This paper is
ties makes PM a complicated process, and there is no
organized as follows: In section II we briefly review the
known general formula for optimal PM when harmonics
known QPM schemes, and summarize their main fea-
are generated in a gas cell or gas jet by a laser pulse
tures, including the optimum phase-shift employed, and
producing considerable ionization rate [8].
themaximumachievableefficiency. Startingwithsection
Thedescriptionofphase-matchingisgreatlysimplified
III we describe our new results. There the magnitude of
whentheintensitydependenceoftheharmonicphasecan
the phase-shift produced by the assisting field in terms
be ignored,(for ex. inHHG in waveguides,or withloose
of the driving field strength, wavelengthratio of the two
fields and trajectory class responsible for HHG is pre-
sented. The presentation of the spatial profile of the op-
timum assisting field is discussed in section IV, and we
∗ [email protected] conclude in Section V.
2
II. METHODS OF QUASI-PHASE-MATCHING tion the gas cell is continuous and the secondary field is
EMPLOYING LOW-INTENSITY ASSISTING usedtoshiftthe phaseofthe generatedharmonicsinthe
FIELDS destructivezones,turningtheseinto(partially)construc-
tive zones.
QPM is a powerful tool when conventional phase- In our discussion we assume ∆k to be constant over
matching is not possible, i.e. when the phase veloc- thelengthofthemedium,andweneglecttheeffectofab-
ity of the nonlinear polarization created by the driving sorption. These assumptions are justified in cases when
laser cannot be matched with the phase velocity of the the coherence length is much shorter than the absorp-
harmonic field, thus phase-mismatch (PMM) arises. In tion length and the laser intensity is constant along the
the forward propagation direction the magnitude of the propagation axis (for example in guided generation, or
wavevector of the propagating harmonic field (k ) and under loose focusing conditions). To describe the pro-
h
the wavevector of the high-harmonic polarization gener- cess it is convenient to use a coordinate frame mov-
ated by the laser (k ) are different (∆k = k − k ). ing with the phase velocity of the harmonic in question
H H h
′ ′
PMM in gases HHG has four different sources [14]: (z = z,t = t − z/vq). In the following we drop the
prime symbol for simplicity. The phase of the generated
∆k =∆kg+∆kn+∆ke+∆ka. (1) harmonic field in the moving frame can be expressed as
ϕ (z) = −∆kz = −2πz/L where L is the coherence
q c c
where ∆k arises from the Gouy phase-shift around the
g length.
focusand∆k fromfreeelectrondispersion. Theseterms
e An assisting electric field periodic in space induces a
are always negative. On the other hand the wavevector
periodicmodulationofthepolarizationphase,sothisbe-
mismatch from neutral dispersion (∆k ) is always posi-
n comesϕ (z)=−∆kz+Af(z),where Ais the amplitude
q
tive. The last term (∆k ) arises from the intensity de-
a of the phase-modulation induced by the assisting field
pendent atomic phase and it is negative before the focus
and f(z) is a normalized function with Λ spatial peri-
andpositive after the focus. InHHG the sameharmonic
odicity in the moving frame. QPM methods employing
canbe generatedbyelectronsperforminga shortorlong
low-intensityassistingfieldsarebasedonthefactthatthe
trajectory before recombination. In case of short tra-
phase-shift induced by the assisting electric field scales
jectoriesthe atomic phase is negligiblein manypractical
linearlywithitsamplitude (E )inthelimitwhenthatis
a
cases,andaslowlyvaryinglaserintensity(likeinloosefo-
much weaker than the amplitude of the generating field
cusing geometries) can make this contribution negligible
(E ≪ E , see [30] for details). As a result, the shape
a 0
forthelongtrajectoriesaswell. Byneglectingtheatomic
of the phase-modulation resembles that of the assisting
phase, the above equation shows that phase-matching
field, and its amplitude can be expressed as
can only be achieved when the total dispersion contri-
bution ∆k +∆k is positive, and balances the effect of A=ζE . (2)
n e a
focusing (∆k ). This relationcreates an upper bound to
g
The calculation of ζ is presented in section III.
the ionization rate, and limits the maximum laser inten-
Assuminganormalizedemissionrateindependentofz,
sity usable for phase-matched harmonic generation. At
the near fieldat the end ofcellwith length Lcanbe cal-
higher ionization rates PMM is unavoidable [12].
L ′ ′
culated as H (L) ∝ exp(−i∆kz )dz , the phase then
The consequence of PMM is that the intensity of q 0
is given by Φ (L)=Arg[H (L)], while the harmonic in-
the harmonic field periodically increases and decreases q q
tensitywillbeI (L)∝R |H (L)|2. Wedefinetheefficiency
along the propagation axis (Figure 1.a), which in non- q q
linear optics is known to be responsible for the appear- of the phase-matching method as
ance of Maker fringes [35]. The harmonic field builds η =I (L)/I0(L), (3)
q q
up until the phase difference between the locally gen-
erated and the propagated harmonic fields is smaller where I0(L) is the intensity of the propagated field pro-
q
than π/2, then, due to destructive interference the har- duced with perfect phase-matching (when ∆k =0).
monic intensity decreases. Zones where harmonic in- As seen in Figure 1.b-e, in QPM the intensity of the
tensity increases/decreases are called zones of construc- generated harmonic increases approximately quadrati-
tive/destructive interference. The basic idea of QPM is cally with the length of the cell as η · (L/L )2, with
c
to eliminate harmonic emission in destructive zones, or only slight sub-coherence-length oscillations around the
switch these into constructive zones, increasing the har- parabola. The intensity in optimal QPM conditions
monic yield over longer propagation distances. might increase until it reaches the absorption limit.
In multi-jet configuration QPM the elimination of Whereas in the case of PMM Figure 1.a, the peak inten-
emission in destructive zones is achieved by tuning the sity is reached at half of the coherence length, severely
gas pressure (and thus the value of ∆k and ∆k ) so limiting the achievable photon number in macroscopic
n e
the length of the constructive zone matches the length media.
of a single gas jet and the individual jets are placed at PeriodicassistingfieldsthatcaninduceQPMcanbeof
a distance along which vacuum propagation (now only manytypes: todateperiodicstaticelectricfields,perpen-
containing ∆k and ∆k ) continues over the destructive dicularly propagating THz fields, and counterpropagat-
g a
part [18]. By contrast, in the field-assisted configura- ing(totheIR)quasi-cwlaserandsawtooth-shapedfields
3
FIG. 1. (a) Illustration of phase-mismatch in harmonic generation via phase and intensity variation along the propagation
direction z. (b)-(e) Schematic presentation of QPM methods employing periodic assisting fields. Top row: illustration of the
assisting field distribution in units of A = πEa. Second row: Effect of the assisting field on the generated harmonic’s phase.
Third row: Phase difference (modulo 2π) between the generated and propagated fields and local efficiencies shown in color
scale. Bottom row: Harmonic intensities, whose values at Lc also show the overall efficiency of the process. ϕ and Φ denote
phases of thegenerated and the propagated harmonic fields.
and pulse trains have been proposed or used. Although ever only for odd n orders, and these produce the same
the experimental realizationof the different schemes dif- efficiency as first order QPM. In conclusion, using this
ferwidely,thebasicphysicsbehindthephenomenaisthe scheme 40.5% efficiency can be obtained by A = π rad
same in all cases, and in Section 3 we will show that the phase-shift with Λ=L periodicity.
c
optimalamplitudeoftheassistingfieldcanbecalculated
with a general formula.
A. Periodic static electric fields
B. Sinusoidal electric fields matching the coherence
QPMinHHGbyusingperiodicstaticelectricfieldhas length
been proposed by Biegert et al. [27, 36]. In this scheme
high-order harmonics are periodically generated with no
assisting field over half a coherence length, then, just QPM is also achievable with sinusoidal phase-
before destructive interference would occur a DC field modulation as illustrated in Figure 1.c. Such schemes
shifts the phase of the selected harmonic by A=π, and were proposed where the phase-modulation is achieved
constructive interference continues over the other half of by counterpropagating quasi-cw fields [30, 38], or per-
the coherence length (see Figure 1.b). Alternating zones pendicularly propagating THz pulses [33]. In both cases
withandwithoutstaticelectricfieldcreatethecondition the optimalphase-shiftinducedby theassistingfieldhas
for QPM. Under the approximations presented earlier, tobeA=1.85radian(thepositionofthefirstextremum
this scheme produces the same efficiency asconventional of the first order Bessel function of the first kind J (A))
1
QPM (2/(mπ))2 in case of second-harmonic generation, [30]. Higher order QPMs can be achieved when the
whichhas beendiscussedextensivelyby Fejer etal. [37]. spatial period or amplitude of the phase-modulation is
This also means that higher order spatial QPM is pos- higher than required for first order. QPM of mth or-
sible, where the periodicity is mΛ, m being a positive derinspaceandnthorderinamplitude occurswhenthe
integer number. For odd m orders the length of 0 and phase-modulationperiod is mL and the amplitude is at
c
π phase-shift zones should be mL /2, however for even the position of the nth maximum of (J (A))2. The effi-
c m
orders (m−1)L /2 and (m+1)L /2 long zones should ciencies in these cases can be calculated by the values of
c c
alternate [37]. It also follows that higher order QPM in (J (A))2, the highest being 33.7% for first order QPM
m
theamplitudeofphase-modulation(nπ)ispossible,how- [30].
4
C. Counterpropagating pulse trains of the field is determined by the coherence length of the
high order harmonic generation process which, in some
Another method of QPM is to scramble the phase cases,canbecalculated[10]orevenmeasured[28,32,45].
of the generated harmonics at zones of destructive in- The calculation of the optimal electric field amplitude is
terference by a counterpropagating pulse or pulse train, presented in the next section.
thatsuppressesemissionintheseregions(seeFigure 1.d).
These schemes have been extensively discussed already
III. MAGNITUDE OF PHASE-MODULATION
[28, 29, 31, 32, 39–41], and it has been found that the
INDUCED BY ASSISTING FIELDS
harmonic emission can be eliminated by counterprop-
agating light pulse [29, 42] and this can induce QPM
A. Assisting field wavelength identical with driver
[28]. The intensity of counterpropagating pulse interfer-
wavelength
ingwiththe forward-propagatingdrivingpulsehastobe
onlyasmallfractionofthe drivingintensity toeliminate
emission [29]. With this method the coherence length Using assisting fields of the same wavelength as the
shouldmatchthewidthofthecounterpropagatingpulse, driver (λa = λ0) the phase-modulation induced by the
not its wavelength, therefore it is easiest to implement weak assisting field can be expressed analytically. The
whenthe coherencelengthismuchlargerthanthewave- phase of a harmonic q (ϕq), generated by a quasi-
length of the assisting pulse (λ ≪L ) [31]. monochromatic field (apart from a constant) can be ex-
a c
For this type of QPM, flat-top laser pulses have been pressed as [7]:
generated and applied experimentally [34, 40], and the
αU
effect of sech2-shaped pulses has been analyzed theoret- ϕ =qϕ − p, (4)
q 0 ~ω
ically [31]. Complete elimination of emission in destruc- 0
tive zones can achieveanefficiency of 10.1%(1/π2) with
whereϕ andU arethephaseandponderomotiveenergy
0 p
flat-top pulses [31]. However, destructive zones can also
of the generating field, the latter is proportional to the
be switched into partially constructive zones, increas-
intensity. The α coefficient depends on the length of
ing efficiency [28, 31, 43]. The phase-shift induced by
theelectrontrajectoryinvolvedingeneratingharmonicq,
the counterpropagating light yielding the best efficiency
and its value can be obtained from classical or quantum
for sech2-shaped pulses is A = 4.5rad (case shown in
mechanicalHHG models [8], anditis shownin Figure 2.
Figure 1.d), increasing the overall efficiency to 14% [31]
usingtheoptimallengthofthecounterpropagatingpulse
of 0.23 L (intensity FWHM) [31]. In case of square-
c
shaped pulses the best efficiency of 20% is produced by
6 long
a phase-shift of A = 3.83 rad, (the global minimum of
short
J0(A) [31]). 5
The obvious advantage of this scheme is, that any
phase-modulation comparable or larger than π would d) 4
a
produce partial extinction of harmonic yield, therefore (r
3
this method is not very sensitive to the parameters of
the assisting field [29].
2
1
D. Sawtooth-shaped fields
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
In theory, perfect elimination of the PMM can be ob-
( -1.32 Ip) / Up
tained, if a sawtooth-shaped field is applied as proposed
in[44]. Therefore,thisinnotatraditionalQPMmethod, FIG.2. Coefficientofintensitydependentharmonicphasefor
wementionitduetothefactthatitalsousesanassisting different trajectories. For more details see [8].
field and, in theory, this can achieve 100% efficiency.
The two terms in the rhs. of Equation 4 show that
at the interference of two fields the modulation of the
E. Summary harmonic’s phase has two sources: the modulation of
the driver’s phase, and the modulation of the driver’s
We conclude that the implementation of all QPM intensity. Let us call these two the direct and indirect
schemeswhichemployaweak,periodicorquasi-periodic phase modulations, following the works [39] and [29]. In
electricfieldrequiresaprecisedeterminationoftheassist- the limit of E ≪ E both of these contributions will
a 0
ingfield’s parametersto achievesignificantenhancement cause the harmonic’s phase to change sinusoidally with
of the macroscopic radiation. The assisting field can be thephasedifferencebetweenthedriverandassistingfield
described by its period and amplitude. The periodicity (ϕ −ϕ ). Theamplitudeofthedirectphase-modulation
a 0
5
for harmonic q can be expressed as (see Appendix for For cutoff harmonics it is known that q = (1.32I +
p
more details): 3.2U )/~ω and α ≈ π (where I > 0 is the ionization
p 0 p
energy), therefore in cases when I ≪ U , the scaling
p p
Ea factor becomes
∆ϕ ≈q (5)
p
E
0 7U
ζcutoff ≈ p ∝E ·λ3. (9)
0 ~ω E 0 0
The indirect phase-modulation (arising from the sec- 0 0
ond term in Equation 4) is linked to the intensity-
Thisshowsthatforcutoffharmonics,whereQPMmeth-
dependence of the harmonic’s phase (i.e. the atomic
ods are found to be most effective, the required field
phase mentioned in Sec.2.). The amplitude of this mod-
strength scales inversely with the driving field strength
ulation can be approximated by
and the third power of its wavelength:
∆ϕ ≈ −αe2E0Ea = −α2UpEa, (6) Ecutoff = A ∝E−1λ−3. (10)
I 2m ~ω3 ~ω E a ζcutoff 0 0
e 0 0 0
where e and me denote the electron charge and mass As the cutoff energy in HHG scales as E02·λ20, the same
respectively and ω is the angular frequency of the gen- energy photons still require weaker assisting fields when
0
erating laser field (see the Appendix for more details). generatedbyweaker,butlongerwavelengthdriverfields.
The maximaofthe two –directandindirect–compo-
nentsofthephase-shiftoccursshiftedbyπ/2inphasedif-
B. Assisting IR field with different wavelength
ferencebetweenthetwointerferingfields,asillustratedin
Figure 3. Due to this delay, the total phase-modulation
With anassistingfieldofarbitrarywavelengthwe rely
onnumericalcalculationstoobtainthesameinformation.

We use the nonadiabatic saddle-point approximation to
0.3
p calculate the harmonic phases [46, 47].
Saddle-points of the Lewenstein integral are known to
0.2 I
) reproducewellthephase-derivativesofthegeneratedhar-
d
(ra 0.1 p+ I monics. Using this method the phase of a selected har-
hift moniccanbeexpressedasϕq =qω0tr−S(ti,tr)/~,where
-s 0.0 ti and tr are the solutions of the saddle-point equations
e
s representing the ionization and return times of the most
a
h
P -0.1 relevant electron trajectories, and S(ti,tr) is the quasi-
classical action. The solved equations read as:
-0.2
1 tr
p = A(t)dt (11)
-0.3 s t −t
0.0 0.5 1.0 1.5 2.0 r i Zti
[p +A(t )]2
a- 0 ( rad) s i −Ip =0 (12)
2
FIG. 3. Direct and indirect harmonic phase-modulation [p +A(t )]2
caused by interfering driver and assisting waves, shown as s r +Ip =q~ω0 (13)
2
a function of the phase difference between the two fields.
PhasescalculatedusingEquation 4,withparameters: α=π, wherep isthestationaryvalueofthecanonicalmomen-
q=93,adriverandassistingfieldwithintensitiesof6×1014 tum andsA(t) is the vectorpotential whichhasno direct
W/cm2 and1.32×109 W/cm2 respectively,andawavelength
relation to the scalar A used in the other equations.
of λ0 =800 nm. More details in [29, 39] and Appendix. To calculate the effect of the assisting field, we cal-
culate electron trajectories in the two-color field while
can be calculated simply as changing the phase-difference between the two fields.
Fromthistheoscillatingharmonicphaseϕ liketheblue
q
A = ∆ϕ2+∆ϕ2. (7) solid line in Fig3 can be obtained, revealing the ampli-
0 I p
tude of the total phase-modulation.
q
From the above equation, the scaling factor ζ between We observe that for the obtained phase-modulation
the assisting field strength and the phase-shifting effect amplitude the same scaling rules apply than in the pre-
(of Equation 2) for cases when the assisting and driver vious case. In fact the obtained phase-modulation effect
fields have the same wavelength can be expressed as (ζ) can be related to the previous case where the two
fieldshadthesamewavelength(characterizedbyζ )and
0
a simple correction factor can be introduced:
2 2
q α2U
p
ζ = + . (8)
0 s(cid:18)E0(cid:19) (cid:18)~ω0E0(cid:19) ζ =ζ0β(λa/λ0,τ) (14)
6
where τ distinguishes trajectories with different travel static during the electron’s travelin the continuum), the
times, and λ /λ is the ratio of the two wavelengths. value of the β correction factor goes to 1.45 for cutoff
a 0
Thevalueofβ(λ /λ ,tr)obtainedfromthesaddle-point harmonics. For other trajectories this factor varies as
a 0
solutions is shown in Figure 4. shown in Figure 5, being very close to one in case of the
shortest trajectories and going slightly higher than two
for trajectories with a return time of one optical cycle.
For assisting fields with λ > 5λ the values of β cal-
a 0
culated for static fields (shown in Figure 5) are already
reasonably accurate.
The figure also indicates, that for λ < λ the de-
a 0
pendence of the correction factor on trajectory length is
reversed; the longest the trajectory, the less the effect
– which can be understood as the perturbation caused
by the assisting field can averageout through the longer
traveling time of the electron. This means that in this
regime the relative effect of the assisting field on shorter
trajectories becomes more and more pronounced. We
wouldliketopointoutthepracticalityofthislimit: since
the assisting field’s wavelength is determined by the co-
herence length, and L scales inversely with harmonic
c
order[15],itmightreachverysmallvalueswhenincreas-
FIG. 4. Phase-modulation coefficients for arbitrary wave- ing driver wavelengths are applied to generate very high
length assisting fields. Trajectory lengths (τ) are shown in harmonics in the x-ray region [48, 49]. In this scenario
color scale represented by their final kinetic energy in units
under very unfavorable PM conditions short wavelength
of Up. assisting fields might be useful in achieving QPM.
Weperformedcalculationswithdifferentlaserfieldand
ionization potential parameters, all yielding very similar
results to what is shown in Figure 4, only finding small
deviations from it. The results are found to be more
2.0 long accurate in the high-intensity regime, where U >I .
p p
short Finally, combining equations 2, 8 and 14, the formula
1.8 forthestrengthoftheassistingfieldcausingtherequired
) A phase-modulation for harmonic order q can be ex-
r
¥,t 1.6 pressed as
(
AE
1.4 0
E = (15)
a
2
β q2+ 2αUp
1.2 ~ω0
r
(cid:16) (cid:17)
the value of β depending on the ratio of the driver and
1.0
assisting fields wavelength, and shown in Figure 4.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
We note here, that in some QPM schemes the wave-
( -1.32Ip) / Up
length of the assisting field, λ is not a free choice, it is
a
FIG. 5. Coefficient for calculating phase-modulation caused determined by the coherence length. This implies, that
bystatic electric field. Trajectories represented by theirfinal the correction factor β has only an indirect dependence
kinetic energy in unitsof Up. onthe generatinglaserpulse parametersthroughthe co-
herence length, and depends directly only on the chosen
Bydefinitionoftheparameters,atλ =λ allthetra- trajectory,thus the scalinglawexpressedinEquation 10
a 0
jectory dependence of the phase shift is included in ζ . for cutoff harmonics holds generally.
0
Itis interestingto seehowthe correctionfactorforshort In many practical cases, especially in free focusing ge-
and long trajectory components cross at this point. For ometries,the intensity ofthe beam canchangealongthe
all values of λ > λ we observe that the value of the propagation axis, moreover the pulse shape and struc-
a 0
correction factor is almost constantly 1 for the shortest ture is also affected by dispersion, self-phase modulation
trajectories,andforthelongesttrajectoriesthedeviation anddiffractioncausedbynon-lineareffects. Theseeffects
from1 has the largestmagnitude. This finding is consis- also change the coherence length along the propagation
tent with the simple view, that the longer the electron axis. To compensate this effect the assisting field’s pe-
stays in the continuum, the more sensitive it becomes to riodicity has to match the changing L along the whole
c
the effect of the assisting field [30]. In the limit when gas medium for efficient QPM. To this end, the use of
λ ≫λ (i.e. when the assisting field can be considered chirped assisting fields [30, 33], or counter-propagating
a 0
7
pulse-trains with variable separation has been proposed
[40]. When using chirped assisting pulses, however, the 8 short trajectories q=35
wavelength ratios become ill-defined. In these cases our q=75 50 2 m)
(a)
results remain indicative. 6 q=110 40 /c
W
Anotherimportantaspecttomentionisthatthecoher-
ence lengthisalsotime-dependent: the ionizationrate is 2 m) 4 x 6 30 11 0
increasingwithinthedurationofthedriverpulse. Forthe W/c 20 y (1
boresQtPoMveriasllacehffiiecvieendcayroitunisddtehseirpedeatkhoafttphheapsue-lmsea,twchhienrge 14 0 2 x 15 10 nsit
tacphanhebFdoltiemnwoa-ithiclooelrynrot,eishzwcteaohetpceinoiachonstei,ggeserhtenwehpeshalrtaeatchntoiianoutrgnrhmemteohffinaoesidccsriesieoslnatlieicsrnyeooggbfisfievtneiuuolendsurunasaaeltyllelioldnynn.goehtiicogaanhpuiepzsslaetis--, er intensity (1 680 (blo)ng trajectories qqq===3715510 010..92 sting field inte
tion in the three-step model of HHG. Las 4 0.6 Assi
2 0.3
IV. ASSISTING BEAM PROFILE
0 0.0
In order to induce QPM in a macroscopic media the -1.0 -0.5 0.0 0.5 1.0
phase-shifting effect of the assistingfield for a givenhar- Radial coordinate (units of w0)
monic hastobe the sameatdifferentspatialcoordinates
FIG. 6. Numerically calculated radial intensity profiles
across the beam. However, due to the intensity profile
needed to induce π phase-shift for different plateau harmon-
across the beam the same harmonic order falls at differ-
ics, when generated by a Gaussian beam having a beam ra-
ent parts of the plateau and thus has an α and β value diusof w0,andpeak intensity8×1014 W/cm2. Calculations
varying with the radial coordinate. Both of these affect done for λ0 = 800 nm driving field, and long wavelength as-
the phase-shifting effect of the assisting field. To com- sisting fields λa ≫ λ0. At the peak intensity the values of
pensatethis,theassistingfieldmusthaveanappropriate (~ω−1.32Ip)/Up for harmonics 110, 75 and 35 are 2.96, 1.8
spatial profile. and 0.54 respectively.
Assuming that the generating laser beam has a Gaus-
sian spatial profile, the intensity profile of the assisting
fieldcanbedeterminedusingnumericalcalculationspre- However, in case of counter-propagating pulse trains,
sentedintheprevioussection. InFigure 6thecalculated the only constraint for (partial) elimination of harmonic
intensity profile is shown for different harmonics gener- emission from destructive zones is that the phase-shift
ated by 800 nm driving field for the case when A=π ra- shouldbelargerthanπ. Inthisrespectshorttrajectories
dian, and λ ≫λ . dominate the selection of the field strength, since those
a 0
In case of short trajectories a lower IR intensity (off- alwaysrequirehigherintensityassistingfieldforthesame
axis) means that the same harmonic is closer to the cut- phase-shift. This is also consistent with the findings of
off, has both higher α and higher β values, therefore the Landreman et al. [29].
required assisting field strength is lower. The opposite Itshouldbenotedthatacrossthegeneratingbeam,not
standsforlongtrajectories,whereαandβ decreaseswith only the driver intensity, but also the coherence length
intensity (see Figure 2). This issue is not risenfor cutoff can vary, which can limit the efficiency of QPM, even
harmonics which are only generated close to the axis. when using an assisting field with optimal beam profile.
The intensity profile required by QPM (Figure 6) for
short trajectories closely resembles a Gaussian suggest-
ing that counter-propagating fields may be used to in- V. CONCLUSIONS
duce QPM in the whole cross section of the gas cell. As
for long trajectories the required field intensity is higher In this paper we first reviewed proposed QPM tech-
off-axis,which couldbe anexplanationwhy the mostef- niques employing periodic assisting fields. In ideal sit-
ficient QPM was found for harmonics close to the cutoff uations these methods can yield an enhancement of the
[27,33]. AnotherimportantaspectofFigure 6isthatfor harmonic signalalong the cell in contrastto the oscillat-
long trajectoriesthe requiredfield strengthis two orders ingoutputfromthephase-mismatchedsituation. Theen-
ofmagnitudelowerandalmostconstantfordifferenthar- hancementachievableisoftheorderof(0.14−0.4)·( L )2
Lc
monics (close to the axis), while for short trajectories it in contrast to the oscillation between 0 to 0.1 for the
showshighvariationwithharmonicorder(aresultconsis- PMM case. We discussed the required field strength
tentwiththefindingsofZhangetal. [28]). Thusspectral of the assisting field for the implementation of efficient
selectionmight be easierto achievefor shorttrajectories QPM. The presented formula – Eq. 15 together with
by varying the strength of the assisting field. Figure 4 – can be used to determine its value for experi-
8
mental realization. This term has its first maximum at δ = π/2, and the
Inconclusionwehaveanalyzedhowthe phaseofhigh- magnitude of the phase-modulation is given by E /E .
a 0
order harmonic radiation generated by an infrared laser
In HHG the phase of the generatedharmonic depends
field can be manipulated by low-intensity assisting fields
on the phase of the generating wave multiplied by the
in order to achieve quasi-phase-matching. A generalfor-
harmonicorderq[7]. Asaresultthephase-modulationof
mula was presented that allows the calculation of the
thegeneratingfielddescribedbyEquation A2willtrans-
optimal assisting field strength in terms of the gener- late to a direct modulation of the harmonic phase with
ating laser pulse intensity, on the two fields’ relative
amplitude
wavelength and the length of the trajectory in ques-
tion. We discussedtherelationshipbetweenthe simplest
caseofcounter-propagatingassistingfieldswiththesame Ea
∆ϕ ≈q . (A3)
wavelength (that is analytically treatable), to the case p E
0
when a different wavelength assisting field is used, and
showedthatthetwocanberelatedthroughawavelength-
dependent correction factor. The optimal field profile of The harmonic’sphase also depends on the intensity of
assisting fields for short and long trajectory components the generatingwave,this contributionis usually referred
requiredfor efficientQPMwasalsodiscussed,andfound as the atomic (or dipole) phase, because it is inherited
that short trajectories have the advantage of requiring from the electron which accumulates it during its travel
the same profile for the driver and the assisting beam. from ionization to recombination. This is well approxi-
Funding. mated as ϕI =−αUp/(~ω0), where Up =e2E2/(4meω02)
Hungarian Scientific Research Fund (OTKA is the ponderomotive energy in the driver field [8], and
NN107235). K.V. acknowledges the support of the the value of α is shown Figure 2. The amplitude of the
Bolyai Foundation. resulting wave in Equation A1 is given by
Acknowledgments. WethanktheNIIFInstitutefor
computation time.
E = E2+E2+2E E cos(δ), (A4)
tot 0 a 0 a
q
Appendix A: Harmonic phase modulation in
interfering laser fields therefore the generating field’s amplitude is also mod-
ulated with changing δ and it has its first maximum
at δ = 0. This amplitude modulation causes an indi-
Here we describe the harmonic phase modulation in-
rect modulation of the harmonic phase. In this case the
duced by the interference of the driver with a weak as-
atomic phase is expressed as
sisting field, resulting Equation 5 and Equation 6. Let
ustakeadriverlaserfieldintheHHGmediumalongthe
z axis, described as E0sin(ϕ0), where ϕ0 =ω0t+|~k0·~z| ϕ = −αe2 E2+E2+2E E cos(δ) , (A5)
is the phase of the laser field, which is a function of I 4m ~ω3 0 a 0 a
e 0
space and time. The assisting field can be described as
(cid:2) (cid:3)
E sin(ϕ +δ), where δ =ϕ −ϕ denotes the phase dif-
a 0 a 0
ference between the two fields, and ϕ = ω t+|~k ·~z| which is modulated with an amplitude given by
a 0 a
is the phase of the low-intensity assisting field. Thus δ
becomes dependent on z, when the wavevectors ~k0 and −αe2
~ka enclose a nonzero angle. The resulting wave is ∆ϕI = 2me~ω03E0Ea. (A6)
E sin(ϕ +θ )=E sin(ϕ )+E sin(ϕ +δ), (A1)
tot 0 tot 0 0 a 0
Both of these factors have been described in [29, 39],
with E denoting its amplitude and θ its phase.
tot tot for the most relevant case, when the phase difference is
In the following calculations we use that E ≪ E .
a 0 calculatedfortwocounterpropagatingwaves,whereϕ =
Thephaseoftheresultingwavethencanbeexpressedas 0
ωt−k z and ϕ =ωt+k z, giving δ =2k z.
0 a 0 0
E sin(δ) Easin(δ) These two different sources of phase-modulation are
θtot =arctan(cid:20)E0+aEacos(δ)(cid:21)=arctan"1+E0EEa0cos(δ)# cshomowpsartahbaltethineymaalwganyitsuodcec,ubruwtitthhea sdheisftcroipftπio/n2 ianbothvee
Ea Ea valueofδ. Thisrelationformsthebasisofourdiscussion
≈arctan sin(δ) ≈ sin(δ). (A2)
E E in section III.
(cid:20) 0 (cid:21) 0
[1] J.-F. Hergott, M. Kovacev, H. Merdji, C. Hubert, and P. Sali`eres, Phys.Rev. A 66, 021801 (2002).
Y. Mairesse, E. Jean, P. Breger, P. Agostini, B. Carr´e,
9
[2] K. Midorikawa, Y. Nabekawa, and A. Suda, Hooker, Phys. Rev.Lett. 99, 143901 (2007).
Progress in QuantumElectronics 32, 43 (2008). [26] B.Dromey,M.Zepf,M.Landreman, andS.M.Hooker,
[3] E. Skantzakis, P. Tzallas, J. Kruse, C. Kalpouzos, and Opt. Express15, 7894 (2007).
D.Charalambidis, Opt.Lett. 34, 1732 (2009). [27] C. Serrat and J. Biegert,
[4] J. Tate, T. Auguste, H. G. Muller, Phys. Rev.Lett. 104, 073901 (2010).
P. Salieres, P. Agostini, and L. F. DiMauro, [28] X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou,
Phys.Rev.Lett. 98, 013901 (2007). H. C. Kapteyn, M. M. Murnane, and O. Cohen,
[5] A. D. Shiner, C. Trallero-Herrero, N. Kajumba, H.- Nature Physics 3, 270 (2007).
C. Bandulet, D. Comtois, F. L´egar´e, M. Gigu`ere, J.- [29] M. Landreman, K. O’Keeffe, T. Robinson,
C. Kieffer, P. B. Corkum, and D. M. Villeneuve, M. Zepf, B. Dromey, and S. M. Hooker,
Phys.Rev.Lett. 103, 073902 (2009). J. Opt.Soc. Am. B 24, 2421 (2007).
[6] P. Sali`eres, A. L’Huillier, and M. Lewenstein, [30] O. Cohen, X. Zhang, A. L. Lytle, T. Pop-
Phys.Rev.Lett. 74, 3776 (1995). mintchev, M. M. Murnane, and H. C. Kapteyn,
[7] P.Balcou,P.Sali´eres,A.L’Huillier, andM.Lewenstein, Phys. Rev.Lett. 99, 053902 (2007).
Phys.Rev.A 55, 3204 (1997). [31] O. Cohen, A. L. Lytle, X. Zhang, M. M. Murnane, and
[8] M. B. Gaarde, J. L. Tate, and K. J. Schafer, H. C. Kapteyn,Opt.Lett. 32, 2975 (2007).
J.Phys. B: At, Mol. and Opt.Phys. 41, 132001 (2008). [32] A. L. Lytle, X. Zhang, R. L. Sandberg, O. Co-
[9] T. Brabec and F. Krausz, hen, H. C. Kapteyn, and M. M. Murnane,
Rev.Mod. Phys. 72, 545 (2000). Opt. Express16, 6544 (2008).
[10] C. G. Durfee, A. R. Rundquist, S. Backus, [33] K.Kova´cs,E.Balogh,J.Hebling,V.To¸sa, andK.Varju´,
C. Herne, M. M. Murnane, and H. C. Kapteyn, Phys. Rev.Lett. 108, 193903 (2012).
Phys.Rev.Lett. 83, 2187 (1999). [34] K. O’Keeffe, D. T. Lloyd, and S. M. Hooker,
[11] A.Paul,E.Gibson,X.Zhang,A.Lytle,T.Popmintchev, Opt. Express22, 7722 (2014).
X. Zhou, M. Murnane, I. Christov, and H. Kapteyn, [35] P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M.
QuantumElectronics, IEEE Journal of 42, 14 (2006). Savage, Phys.Rev.Lett. 8, 21 (1962).
[12] T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, [36] C. Serrat and J. Biegert, NIMA653, 134 (2011).
P.Sidorenko,O. Cohen, I.P.Christov, M. M. Murnane, [37] M. Fejer, G. Magel, D. Jundt, and R. Byer,
and H. C. Kapteyn,PNAS 106, 10516 (2009). Quantum Electronics, IEEE Journal of 28, 2631 (1992).
[13] T.Popmintchev,M.-C.Chen,P.Arpin,M.M.Murnane, [38] H. Ren, A. Nazarkin, J. Nold, and P. S. Russell,
and H. C. Kapteyn,NaturePhotonics 4, 822 (2010). Opt. Express16, 17052 (2008).
[14] P. Rudawski, C. M. Heyl, F. Brizuela, J. Schwenke, [39] S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons,
A. Persson, E. Mansten, R. Rakowski, L. Rading, N. Terry, J. Titensor, Q. Wang, and J. Peatross,
F. Campi, B. Kim, P. Johnsson, and A. L’Huillier, Phys. Rev.Lett. 87, 133902 (2001).
Reviewof ScientificInstruments84, 073103 (2013). [40] T. Robinson, K. O’Keeffe, M. Zepf, B. Dromey, and
[15] M. Geissler, G. Tempea, and T. Brabec, S. M. Hooker, J. Opt.Soc. Am. B 27, 763 (2010).
Phys.Rev.A 62, 033817 (2000). [41] K. O’Keeffe, T. Robinson, and S. M. Hooker,
[16] T. Auguste, B. Carr´e, and P. Sali`eres, Opt. Express20, 6236 (2012).
Phys.Rev.A 76, 011802 (2007). [42] J. Sutherland, E. Christensen, N. Powers,
[17] V. Tosa, V. S. Yakovlev, and F. Krausz, S. Rhynard, J. Painter, and J. Peatross,
NewJournal of Physics 10, 025016 (2008). Opt. Express12, 4430 (2004).
[18] J. Seres, V. S. Yakovlev, E. Seres, C. Streli, [43] J. Peatross, S. Voronov, and I. Prokopovich,
P. Wobrauschek, C. Spielmann, and F. Krausz, Opt. Express1, 114 (1997).
NaturePhysics 3, 878 (2007). [44] P. Sidorenko, M. Kozlov, A. Bahabad, T. Pop-
[19] A.Willner,F.Tavella,M.Yeung,T.Dzelzainis,C.Kam- mintchev, M. Murnane, H. Kapteyn, and O. Cohen,
peridis, M. Bakarezos, D. Adams, M. Schulz, R. Riedel, Opt. Express18, 22686 (2010).
M.C.Hoffmann,W.Hu,J.Rossbach,M.Drescher,N.A. [45] S.Kazamias,D.Douillet,C.Valentin,F.Weihe,F.Aug´e,
Papadogiannis, M. Tatarakis, B. Dromey, and M. Zepf, T. Lefrou, G. Grillon, S. Sebban, and P. Balcou,
Phys.Rev.Lett. 107, 175002 (2011). Phys. Rev.A 68, 033819 (2003).
[20] R. A. Ganeev, M. Suzuki, and H. Kuroda, [46] G. Sansone, C. Vozzi, S. Stagira, and M. Nisoli,
Phys.Rev.A 89, 033821 (2014). Phys. Rev.A 70, 013411 (2004).
[21] I. Christov, H. Kapteyn, and M. Murnane, [47] M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier,
Opt.Express 7, 362 (2000). and P. B. Corkum, Phys.Rev. A 49, 2117 (1994).
[22] A. Paul, R. Bartels, R. Tobey, H. Green, S. Weiman, [48] E. J. Takahashi, T. Kanai, K. L. Ishikawa,
I. Christov, M. Murnane, H. Kapteyn, and S. Backus, Y. Nabekawa, and K. Midorikawa,
Nature421, 51 (2003). Phys. Rev.Lett. 101, 253901 (2008).
[23] E. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gau- [49] T.Popmintchev,M.-C.Chen,D.Popmintchev,P.Arpin,
diosi, S. Backus, I. Christov, A. Aquila, E. Gullik- S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciu-
son, D. Attwood, M. Murnane, and H. Kapteyn, nas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim,
Science 302, 95 (2003). S.E.Schrauth,A.Gaeta,C.Hernandez-Garcia,L.Plaja,
[24] D.Faccio,C.Serrat,J.M.Cela,A.Farr´es,P.D.Trapani, A.Becker,A.Jaron-Becker,M.M.Murnane, andH.C.
and J. Biegert, Phys.Rev.A 81, 011803 (2010). Kapteyn,Science 336, 1287 (2012).
[25] M.Zepf,B.Dromey,M.Landreman,P.Foster, andS.M.