-
Terahertz field induced near-cutoff even-order harmonics in
femtosecond laser
Bing-Yu Li,1, 2 Yizhu Zhang,3, ∗ Tian-Min Yan,2, † and Y. H.
Jiang2, 4, 5, ‡1Shanghai University, Shanghai 200444, China
2Shanghai Advanced Research Institute, Chinese Academy of
Sciences, Shanghai 201210, China3Center for Terahertz waves and
College of Precision Instrument and Optoelectronics
Engineering,
Key Laboratory of Opto-electronics Information and Technical
Science,Ministry of Education, Tianjin University, Tianjin 300350,
China
4University of Chinese Academy of Sciences, Beijing 100049,
China5ShanghaiTech University, Shanghai 201210, China
High-order harmonic generation by femtosecond laser pulse in the
presence of a moderately strongterahertz (THz) field is studied
under the strong field approximation, showing a simple
proportional-ity of near-cutoff even-order harmonic (NCEH)
amplitude to the THz electric field. The formationof the THz
induced-NCEHs is analytically shown for both continuous wave and
Gaussian pulse.The perturbation analysis with regard to the
frequency ratio of the THz field to the femtosecondpulse shows the
THz-induced NCEHs originates from its first-order correction, and
the availableparametric conditions for the phenomenon is also
clarified. As the complete characterization of thetime-domain
waveform of broadband THz field is essential for a wide variety of
applications, thework provides an alternative time-resolved
field-detection technique, allowing for a robust
broadbandcharacterization of pulses in THz spectral range.
I. INTRODUCTION
The development of terahertz (THz) technology hasmotivated a
broad range of scientific studies and appli-cations in material
science, chemistry and biology. TheTHz light is especially featured
by the availability to ac-cess low-energy excitations, providing a
fine tool to probeand control quasi-particles and collective
excitations insolids, to drive phase transitions and associated
changesin material properties, and to study rotations and
vibra-tions in molecular systems [1].
In THz science, the capabilities of ultra-broadbanddetection are
essential to the diagnostics of THz fieldof a wide spectral range.
The most common detec-tion schemes are based on the photoconductive
switches(PCSs) [2, 3] or the electro-optic sampling (EOS)
[4].Especially the EOS technique, which uses part of thelaser pulse
generating the THz field to sample the latter,has been widely
applied in the THz time-domain spec-troscopy, pump-probe
experiments and dynamic mattermanipulation [5]. Nevertheless, both
PCSs and EOS re-quire particular mediums— photoconductive antenna
forthe former and electro-optic crystal for the latter. Dueto
inherent limitations of the detection media, includingdispersion,
absorption, long carrier lifetime, and latticeresonances [6], the
typical accessible bandwidth of de-tected THz is limited below 7
THz [7–11].
On the other hand, gas-based schemes, including air-breakdown
coherent detection [12], air-biased coherentdetection (ABCD) [13],
optically biased coherent detec-tion [14], THz radiation enhanced
emission of fluores-cence [15] et al., allow for ultra-broadband
detections
∗ [email protected]† [email protected]‡ [email protected]
since gases being continuously renewable show no ap-preciable
dispersion or phonon absorption [16], thus ef-fectively extending
the accessible spectral range beyond10 THz. In particular, the ABCD
utilizes the THz-field-induced second harmonics (TFISH) [17] to
sense the THztransient through a third-order nonlinear process.
TheTHz field mixed with a bias electric field breaks the sym-metry
of air and induces the frequency doubling of thepropagating probe
beam. Such a nonlinear mixing re-sults in a signal of the intensity
proportional to the THzelectric field, allowing for the coherent
detection by whichboth amplitude and phase of the THz transient can
bereconstructed.
The air-based broadband THz detection utilizing thelaser induced
air plasma as the sensor medium is essen-tially an inverse process
of THz wave generation (TWG)in femtosecond laser gas breakdown
plasma. Both involverather complicated processes dominated by the
strongfield photoionization. From the perspective of single-atom
based strong field theory, the TWG has been in-terpreted as the
near-zero-frequency radiation due to thecontinuum-continuum
transition of photoelectron [18–20], complementary to the widely
acknowledged mech-anism for high-order harmonic generation (HHG) —
thecontinuum-bound transition that emits high energy pho-tons when
the released electron after the ionization rec-ollides with the
parent ion [21].
Under the same nonperturbative theoretical frameworkdepicting
the strong-field induced radiation, the TWGand the HHG within one
atom, however, present differ-ent facets of dynamics of electron
wave packet, providingpotential strategies to either characterize
the system orto profile external light fields. For example, the
synchro-nized measurements on angle-resolved TWG and HHGfrom
aligned molecules allow for reliable descriptions ofmolecular
structures [22]. On the other hand, the pres-ence of THz or static
fields may drastically affect the pho-
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toionization dynamics and reshape electron
trajectories,eventually altering photoelectron spectra and HHG
ra-diations. Applying the widely used streaking technique,the
influence on photoelectron spectra from an additionalTHz transient
can help characterize the time structure ofan attosecond pulse
train, e.g., pulse duration of individ-ual harmonic [23]. The
altered HHG spectral featuresby THz or static fields include the
increased odd-orderharmonic intensity in the low end of the
plateau, thesignificant production of even-order harmonics [24],
thedouble-plateau structure [25] and high-frequency exten-sion [26,
27]. The substantially extended HHG cutoff incombined fields
creates ultrabroad supercontinuum spec-trum, allowing for the
generation of single attosecondpulses. For example, the combination
of a chirped laserand static electric field has been proposed to
obtain iso-lated pulse as short as 10 attoseconds [28].
In this work, we study the influence of an additionalmoderately
strong THz field on the HHG by femtosecondlaser pulse. By
"moderately" we mean the THz field iseasily attainable in nowadays
laboratories — its inten-sity is not as high as to modify global
harmonic featuresas considered in previous theoretical works.
Instead, wefocus on the more subtle THz-induced even-order
har-monics. The dependence of all-order HHG on time de-lay between
the THz field and the femtosecond pulseis studied, showing the
near-cutoff even-order harmon-ics (NCEHs) are of particular
synchronicity with the ex-ternal THz field. Accordingly, the
measurement of THzinduced NCEHs, similar to the widely used TFISH,
issupposed to provide an alternative all-optical
ultrabroadbandwidth method to characterize the time-domain
THztransient. The influence of the THz field on NCEHs
istheoretically investigated under the strong field approxi-mation,
showing a direct link inbetween, further confirm-ing the
availability of the detection scheme.
The paper is organized as follows: In Sec. II, wepresent
time-delay dependent all-order HHG calculationsand show the
significant correlation between NCEHsand the external THz field,
illustrating the possibleschematics to reconstruct the time-domain
THz field us-ing NCEHs. In Sec. III, the detailed analysis of
theNCEHs is presented. A brief retrospect of the
analyticalderivation for odd-order harmonics is given in Sec. III
A,followed by the discussion in Sec. III B on the simplestcase, the
NCEHs in fields of continuous wave. When afemtosecond laser has a
finite pulse width, the influencefrom pulse envelope is shown in
Sec. III C, where the ac-tion for a Gaussian envelope is explicitly
derived. In Sec.IIID, a complete description of the THz-induced
NCEHsin femtosecond pulse is presented, showing the relationbetween
the NCEH and the THz field at the center ofthe femtosecond laser
pulse. In the end, exemplary re-construction schematics are shown
with parametric con-ditions discussed for the applicability.
II. SCHEME OF COHERENT DETECTIONAND NUMERICAL SIMULATIONS
We consider an atom subject to combined fields in-cluding a
linearly polarized femtosecond laser pulse E0(t)and a THz field
E1(t). With both polarizations along thesame direction, the total
fields read E(t) = E0(t)+E1(t).Denoting the associated vector
potentials by A(t) =A0(t) + A1(t), we evaluate the harmonics using
theLewenstein model [21] under the strong field approxima-tion
[29–31], which is usually applicable in the tunnelingregimes,
providing a reasonable and intuitive descriptionof harmonic
radiation from highly energetic recollidingelectrons. The
time-dependent dipole moment d(t) inRef. [21] as an integration
over the intermediate mo-mentum can be dramatically simplified by
applying thestationary phase approximation, yielding
d(t) = iˆ ∞0
dτ
(π
�+ iτ/2
)3/2µ∗[pst(t, τ) +A(t)]
×µ[pst(t, τ) +A(t− τ)]E(t− τ)e−iS(t,τ) + c.c.,(1)
where integration variable τ is the return time of theelectron,
i.e., the interval between the instants of ion-ization and
rescattering. In this work, atomic unitsare used unless noted
otherwise. The dipole matrixelement µ(k) = 〈k|x̂|Ψ0〉 between bound
state |Ψ0〉and continuum state |k〉 of momentum k is given
byi∂k〈k|Ψ0〉 along the polarization direction. Taking |Ψ0〉the 1s
state of a hydrogen-like atom for example, µ(k)
=−i27/2(2Ip)5/4k/[π(k2 +2Ip)3], where Ip is the
ionizationpotential. In Eq. (1), the action reads
S(t, τ) =
ˆ tt−τ
dt′(
1
2[pst(t, τ)−A(t′)]2 + Ip
), (2)
and the stationary momentum pst(t, τ) = −[α(t)−α(t−τ)]/τ is
determined by the electron excursion α(t) =´ t
dt′A(t′) after the electron is released by external lightfields.
The subsequent spread of the continuum electronwave packet is
depicted by [π/ (�+ iτ/2)]3/2 in the inte-gral of Eq. (1) with an
infinitesimal �.
Evaluating d(t) of Eq. (1) and |d̃(ω)| from its
Fouriertransform, we demonstrate in Fig. 1 the all-order har-monics
as a function of the time delay between a near-infrared pulse and a
THz field. The near-infrared pulsehas the vector potential of a
Gaussian envelope, A0(t) =(E0/ω0)e
−t2/(2σ2) sin(ω0t), with ω0 = 0.0353 (1.3 µm),E0 = 0.06
(intensity I = 1.3 × 1014 W/cm2), σ =2106 (FWHM of 120 fs). The THz
field is modeled byA1(t) ∝ −(E1/ω1)(ω1t)−10/[exp(a/ω1t)− 1]
sin(ω1t+ φ)with a = 50, E1 = 2× 10−5 (100 kV/cm), ω1 = 2× 10−4(1
THz) and φ = 0.3π, and the corresponding electricfield E1(t) =
−∂tA1(t) is shown in Fig. 1(a).
When the THz field is absent, Fig. 1(b) shows |d̃(ω)|in the
logarithmic scale with a typical plateau structurebetween the 20th-
and 80th-order. The harmonic yield
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3
0
2
0
20
40
60
80
100 -2.0
-2.5
-3.0
-3.5
-4.04
2
0
-2
-4
0 1 2 3 4 5
Figure 1. Schematics of the reconstruction of THz field
withnear-cutoff even-order harmonics, which uses a femtosecondlaser
pulse to scan over a THz field and to record the gen-erated
harmonics as a function of the time delay betweenthe fields. Panel
(a) shows an exemplary THz field withω1 = 2 × 10−4 (1 THz), and E1
= 2 × 10−5 (100 kV/cm).Without the THz field, panel (b) shows the
harmonics gen-erated by a femtosecond pulse with ω0 = 0.0353 (1.3
µm),E0 = 0.06 (I = 1.3 × 1014 W/cm2) and σ = 2106 (FWHMof 120 fs).
With the THz field, panel (c) shows the all-orderharmonics versus
the time delay. The difference ∆|d̃(ω)| be-tween the harmonics with
THz field, as given in (c), and theone without THz field, as shown
in (b), is presented in panel(d) in the linear scale. All
near-cutoff even-order harmonics,as indicated by the box around
Ecutoff = Ip+3.17Up, exhibitssynchronous change with the intensity
of the THz field.
dramatically decreases beyond the cutoff around 80th-order, as
depicted by the maximum kinetic energy of arecolliding electron,
Ecutoff = Ip + 3.17Up, where Up =E20/4ω
20 is the ponderomotive energy of the electron.
When the THz field is present, the full scope of all-order
|d̃(ω)| in the logarithmic scale versus the time de-lay is
presented in Fig. 1(c). To better observe the in-fluence from the
accompanied THz field, the differencebetween |d̃(ω)| with and
without THz field, ∆|d̃(ω)|, isshown in Fig. 1(d) in the linear
scale, with positiveand negative values indicated by distinct
colors. A closescrutiny reveals most of the nonvanishing ∆|d̃(ω)|
shownin (d) appear at even orders. In the low-order region,the
second-order harmonic follows the change of THz in-tensity similar
to the phenomenon utilized by TFISH,though the applicability of the
model in this low-orderregion is dubious. As the harmonic order
increases, thedelay dependence of harmonics loses the regularity
andthe distribution seems rather chaotic. When the har-monic order
increases up to the near-cutoff region, thedelay-dependent harmonic
yields, however, again follow
the time profile of |E1(t)|. Such a concurrence is notewor-thy,
since it provides an alternative detection strategy tocharacterize
an arbitrary time-domain THz waveform.
III. ANALYSIS OF NEAR-CUTOFFEVEN-ORDER HARMONICS
A. Monochromatic light field
Before the analysis of THz induced modulation onNCEHs, we
retrospect the simplest case where the HHGis induced by a
monochromatic field of continuous waveE(t) = E0 cos(ω0t) and the
vector potential A(t) =A0 sin(ω0t) with E0 = −A0ω0 [21]. For
concision, defin-ing phases ϕt = ω0t and ϕτ = ω0τ , we have pst(ϕt,
ϕτ ) =A0[cosϕt − cos(ϕt − ϕτ )]/ϕτ , and the action in Eq.
(2)reads
S0(ϕt, ϕτ ) = F0(ϕτ )−(Upω0
)C0(ϕτ ) cos(2ϕt − ϕτ )(3)
with the ponderomotive potential Up = E20/4ω20 = A20/4,F0(ϕτ ) =
[(Ip + Up)/ω0]ϕτ − (2Up/ω0)(1 − cosϕτ )/ϕτ ,and
C0(ϕτ ) = sinϕτ −4 sin2(ϕτ/2)
ϕτ. (4)
Here, subscript "0" in S0 is used to label the actionwithout the
influence from the additional accompa-nied field, i.e., the THz
field as will be discussedin the subsequent sections. Applying the
Anger-Jacobi expansion, the exponential part exp(−iS0)with S0 of
Eq. (3) reads exp[−iS0(ϕt, ϕτ )] =exp[−iF0(ϕτ )]
∑∞M=−∞ i
MJM (UpC0(ϕτ )/ω0) exp[iM(ϕτ−2ϕt)].
In Eq. (1), the part including dipole matrix el-ements
µ∗[pst(ϕt, ϕτ ) + A(ϕt)]µ[pst(ϕt, ϕτ ) + A(ϕt −ϕτ )]E(ϕt − ϕτ ) can
be represented by Fourier series∑n bn(ϕτ ) exp [−i(2n+ 1)ϕt] with
respect to ϕt. For
simplicity, we assume the dipole moment of the formµ(p) ∼ ip,
leading to mostly vanishing bn except for oneswith 2n+ 1 = ±1 and
±3.
Applying the above expansions and substituting n =K −M , Eq. (1)
eventually has the form
d0(t) =
∞∑K=−∞
d̃0,2K+1e−i(2K+1)ϕt . (5)
For K > 0, the coefficients for odd-order harmonics aregiven
by
d̃0,2K+1 = i
ˆ ∞0
dτ
(π
�+ iτ/2
)3/2e−iF0(ϕτ )
×∞∑M=0
iMeiMϕτJM
(Upω0C0(ϕτ )
)bK−M (ϕτ )
+c.c. (6)
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4
while coefficients of all even-order harmonics vanish dueto the
restricted value range of n as long as the atomicpotential is
spherically symmetric.
B. THz field induced NCEHs
An extra THz field of the same polarization direction,A1(t),
induces even-order harmonics. In the followings,the detailed
analysis of NCEHs is presented to show theirrelations with the THz
field. Defining pst,i the stationarymomentum for a single field i
of vector potential Ai(t)with i = 0 and 1, the stationary momentum
when bothfields are present satisfies pst = pst,0 + pst,1. Also,
letSi(t, τ) =
12
´ tt−τ [pst,i +Ai(t)]
2+Ipτ be the single action
for the ith field and note that´ tt−τ Ai(t
′)dt′ = pst,iτ , thetotal action S(t, τ) relates to partial
action Si(t, τ) by
S(t, τ) = S0(t, τ) + [S1(t, τ)− Ipτ ]
−pst,0pst,1τ +ˆ tt−τ
A0(t′)A1(t
′)dt′. (7)
Given the vector potential of the additional THz fieldA1(t) = A1
cos(ω1t + φ) with ω1 � ω0 and φ an ar-bitrary initial phase, we
define the frequency ratio ε =ω1/ω0 � 1. Substituting A0(t) and
A1(t) into Eq. (7),it is shown that the first-order correction with
regard toε arises completely from the cross term in Eq. (7)
(i.e.,the two terms on the second line), while S1(t, τ) − Ipτmerely
has the contribution of O(ε2). Retaining terms ofS(t, τ) only up to
the first order of ε, we have the yield-ing action S(ϕt, ϕτ ) '
S0(ϕt, ϕτ ) + ∆S(ϕt, ϕτ ) with theTHz-induced correction
∆S(ϕt, ϕτ ) = −pst,0pst,1τ +ˆ tt−τ
A0(t′)A1(t
′)dt′
' εA0A1ω0
sinφC1(ϕτ ) cos(ϕt −
ϕτ2
)(8)
and C1(ϕτ ) = ϕτ cos(ϕτ/2)− 2 sin(ϕτ/2).The action correction
∆S(ϕt, ϕτ ) can also be treated
with Anger-Jacobi expansion, exp[−i∆S(ϕt, ϕτ )] =∑∞N=−∞ i
NJN (−εA0A1 sinφC1(ϕτ )/ω0) exp[iN(ϕτ/2 −ϕt)], resulting in the
expansion for the full action,
e−iS(ϕt,ϕτ ) = e−iF0(ϕτ )∞∑
M,N=−∞iM+NJM
(Upω0C0(ϕτ )
)
×JN(−εA0A1
ω0sinφC1(ϕτ )
)×ei(M+
N2 )ϕτ e−i(2M+N)ϕt . (9)
Considering that the relatively small THz electric com-ponent
E1(t) is negligible, the similar Fourier series ex-pansion,
µ∗[pst(ϕt, ϕτ ) + A(ϕt)]µ[pst(ϕt, ϕτ ) + A(ϕt −ϕτ )]E(ϕt − ϕτ )
=
∑M bM (ϕτ ) exp [−i(2M + 1)ϕt], can
still be performed as in Sec. III A with E(ϕt − ϕτ ) 'E0(ϕt − ϕτ
).
0 5 10 15 200
2
4
6
8
10
Figure 2. The function 2|C0(ϕτ )| (solid line) and |C1(ϕτ
)|(dashed line) versus ϕτ of the return time. The maximumof the
2|C0(ϕτ )| is associated to the maximum kinetic en-ergy gain that
corresponds to the cutoff energy of the HHG,as indicated by the
black line for 3.17Up. The shaded area(light blue) highlights ϕτ
contributing to the near-cutoff har-monic spectrum of the the
energy between 2.4Up (red line)and 3.17Up (black line).
Substituting the above expansion and Eq. (9) into Eq.(1), we
obtain
d(t) = i∞∑
n=−∞
ˆ ∞0
dτ
(π
�+ iτ/2
)3/2bn(ϕτ )e
−iF0(ϕτ )
×∞∑
M,N=−∞iM+NJM
(Upω0C0(ϕτ )
)
×JN(−εA0A1
ω0sinφC1(ϕτ )
)×ei(M+
N2 )ϕτ e−i[2(M+n)+N+1]ϕt + c.c.. (10)
The near-cutoff harmonics are usually featured with re-stricted
range of ϕτ . As shown in Fig. 2, the photonenergy within the range
2.4Up < 2|C0(ϕτ )| < 3.17Up re-quires ϕτ ∈ [3, 5],
corresponding to the first return of thereleased electron (see also
Fig. 1 of Ref. [21]). Withinthe range of ϕτ contributing to the
near-cutoff harmon-ics, as indicated by the shaded area in Fig. 2,
the globallyincreasing function C1(ϕτ ) also remains low, resulting
inthe whole argument of JN being small. Taking a typical800 nm, 1×
1014 W/cm2 femtosecond pulse and 1 THz,1 MV/cm THz field for
example, the argument in JN (z)is roughly |z| ≈ 0.3. Since JN (z)
becomes exponentiallysmall whenN > |z|, only whenN = 0,±1 does
JN (z) sig-nificantly contribute, since we have the typical value
|z| <1. Using JN (z) ' (z/2)N/Γ(N + 1) when z → 0 [32], wefind
from Eq. (10) that d(t) = d0(t)+d1(t)+O(ε2), whered0(t), simply
given by Eq. (5) for odd-order harmonicsin a monochromatic field,
stems from the contribution ofN = 0 as J0(z) = 1. The odd-order
harmonics are barelyinfluenced by the additional low-frequency
field aroundthe near-cutoff region as long as the THz field
remainsrelatively low to fulfill |z| < 1 for JN (z). It is
noteworthy
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5
that an extra correction to the dipole moment, d1(t), isinduced
by the THz field, corresponding to N = ±1 asJ±1(z) = ±z/2. In
contrast to Eq. (5), we have d1(t) inthe frequency domain,
d1(t) =
∞∑K=−∞
d̃1,2Ke−i(2K)ϕt ,
containing only even-order harmonics, whose coefficientsare
given by
d̃1,2K = εA0A12ω0
sinφ
ˆ ∞0
dτ
(π
�+ iτ/2
)3/2×e−iF0(ϕτ )C1(ϕτ )
∞∑M=0
iMeiMϕτJM
(Upω0C0(ϕτ )
)×[bK−M (ϕτ )e
− i2ϕτ + bK−M−1(ϕτ )ei2ϕτ]
+c.c.. (11)
The prefactor of d̃1,2K thus suggests the proportionalityto E1,
showing a simple relation between NCEHs withthe THz field. In
addition, the initial phase φ betweenthe pair of continuous waves
also tunes the amplitudes ofeven-order harmonics, showing a
sinusoidal dependenceof NCEHs on φ.
0 20 40 60 801.0
0.5
0.0
0.5
1.0
1.5
2.0
Figure 3. Emission of all-order harmonics under the laserfield
of continuous wave accompanied by a THz field. Thelaser parameters
are given by ω0 = 0.0353 (1.3 µm) and E0 =0.06 (intensity 1.3× 1014
W/cm2) for the infrared laser, andω1 = 3.53× 10−4 (2.3 THz) and E1
= 2× 10−5 (100 kV/cm)for the THz field. The |d̃(ω)| (blue line),
evaluated from theFourier transform of d(t) as the direct numerical
integrationof Eq. (1), is compared with analytical formula, Eq. (6)
andEq. (11), for odd- (green) and even-order (orange)
harmonics,respectively. The even-order harmonics are maximized
withinitial phase φ = π/2.
Fig. 3 presents the comparison of harmonics |d̃(ω)|from
numerical integration of Eq. (1) with analytical
formula, Eq. (6) and Eq. (11), for odd- and even-orderharmonics,
respectively, when initial phase φ = π/2 ischosen to maximize the
yield of even-order harmonics.The agreement between odd-order
|d̃(ω)| evaluated byEq. (1) with the analytical solution of Eq. (6)
confirmsthe conclusion that the additional THz field with the
cur-rent laser parameters imposes no influences on
odd-orderharmonic generation. On the other hand, the yields
ofeven-order harmonics are typically lower than their odd-order
counterparts due to the small ratio of frequencies εin the
prefactor of Eq. (11). The derived solution of Eq.(11) is also in
excellent agreement with the numericalresult for NCEHs, showing the
validity of the assumedconditions that only JN (z) of N = ±1
contribute. Eq.(11) even works in a broader parametric range than
ex-pected — it correctly describes all even-order harmonicsabove
40th-order, which corresponds to a much lower en-ergy than that of
the cutoff.
The above analysis establishes the basis for the gener-ation of
NCEHs. In the following, the envelope effect fora more realistic
laser pulse will be presented to accountfor the time-resolving
capacity of the femtosecond laserpulse in THz detection.
C. Effect of pulse envelope
The envelope of a femtosecond laser pulse should beconsidered in
practice. It is expected that the temporallocality specified by the
envelope plays an essential rolein determining the waveform of the
THz field at exactlythe time of pulse center. In this section, we
first discussthe envelope effect on harmonics in the absence of
theTHz field.
Assuming the vector potential has a Gaussian-envelope, A(t) = A0
exp[−t2/(2σ2)] sin(ω0t), withthe time center at t = 0 and the pulse
width σ,the excursion of the electron is given by α(t) =−A0
exp[−(ω0σ)2/2]
√π/2σIm
[erf(t/√
2σ + iω0σ/√
2)]
with the error function erf(z) = (2/√π)´ z0
exp(−t2)dt.Substituting into Eq. (2), we find the action
S0(t, τ) = Ipτ + Upe−( tσ )
2√π
2σG
(t√2σ,ω0σ√
2,τ√2σ
)(12)
where
G(x, y, η) = Re[ei4xyg(x− iy, η)− g(x, η)]
−√
2π
η
[Imei2xyg
(x− iy√
2,η√2
)]2,
and
g(z, η) = w(
i√
2z)− e4η(z−η/2)w
(i√
2(z − η))(13)
with w(z) the Faddeeva function defined by w(z) =exp(−z2)[1−
erf(−iz)].
In comparison with the action for a continu-ous wave, Eq. (3),
which can be recast as
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6
100 50 0 50 1000
1
2
3
4
5
6
7
Figure 4. Comparison of contributing phases in actions asa
function of ϕt between fields of continuous wave and ofGaussian
envelope. Parameters ϕσ = 50 and ϕτ = 4 are usedwhich are typical
for the generation of NCEHs.
S0(ϕt, ϕτ ) = (Ip/ω0)ϕτ + (Up/ω0)Φcw with phaseΦcw = ϕτ − 2(1 −
cosϕτ )/ϕτ − C0(ϕτ ) cos(2ϕt −ϕτ ), action (12) for a
Gaussian-enveloped pulsetakes the similar form, S0(ϕt, ϕτ ) =
(Ip/ω0)ϕτ +(Up/ω0) exp[−(ϕt/ϕσ)2]Φgauss, with Φcw replaced bya
Gaussian-windowed one exp[−(ϕt/ϕσ)2]Φgauss andΦgauss = (
√π/2)ϕσG
(ϕt/√
2ϕσ, ϕσ/√
2, ϕτ/√
2ϕσ).
Here, all notions of phases ϕt = ω0t and ϕτ = ω0τ arestill used
for consistency. Besides, we have introducedϕσ = ω0σ. A common
factor of Gaussian exp[−(ϕt/ϕσ)2]in Eq. (12) specifies a filtering
window whose center co-incides with that of the femtosecond pulse.
A pictorialanalysis on the difference of actions between the
continu-ous wave and the Gaussian-enveloped pulse is presentedin
Fig. 4. The curves of both Φcw and Φgauss contain thesame dominant
oscillating components versus ϕt, propor-tional to cos(2ϕt−ϕτ ).
Within the region of interest, i.e.,around the center of the
Gaussian window, the differencebetween Φgauss and Φcw is that the
oscillating amplitudeof Φgauss increases with ϕt while that of Φcw
remains con-stant. Such an increase becomes more significant
withdecreasing ϕσ. On the contrary, when ϕσ is sufficientlylarge,
the amplitude of Φgauss approaches Φcw. That is,when the pulse is
infinitely long, i.e., ϕσ →∞, Eq. (12)for a Gaussian envelope
degenerates to Eq. (3), the ac-tion for a monochromatic continuous
wave laser.
Fig. 5 shows the comparison of harmonics generationsbetween
using a continuous wave and using Gaussian-enveloped femtosecond
pulse. With the same femtosec-ond laser parameters as considered
for the continuouswave (ω0 = 0.0353, E0 = 0.06), the result in Fig.
5(a)is actually a zoom-in spectrum of Fig. 3 around thenear-cutoff
energy, while Fig. 5(b) shows the one with aGaussian envelope of σ
= 100 fs. The |d̃(ω)| at each odd-order, for either without or with
envelope effect, is high-lighted by orange curve, showing the
similar distributionsof odd-order harmonics. Action (12) modified
by the fi-
2
1
0
60 65 70 75 80 85 90
4
3
2
Figure 5. Effect of pulse envelope on harmonic generation.The
panels show the near-cutoff harmonics under (a) a contin-uous
monochromatic laser of the same parameters used in Fig.3, and (b) a
Gaussian-enveloped pulse of σ = 1755 (FWHMof 100 fs). The red marks
label all harmonics of odd-orders,whose distribution is highlighted
by orange curves. Blackdashed lines indicate Ecutoff.
nite pulse width, which has also been numerically ex-amined,
however, contains extra frequency components,leading to multiple
sidebands in Fig. 5(b) around theoriginal odd-order harmonics. In
the next section theTHz field combined with Gaussian-enveloped
femtosec-ond pulse is analyzed to show the theory behind the
THzfield reconstruction.
D. THz induced NCEHs under Gaussian-envelopedpulse
The above discussions allow for a straightfor-ward extension to
consider the harmonic genera-tion under a Gaussian-enveloped
femtosecond laserpulse accompanied by a THz field. Let A(t) =A0
exp(−t2/2σ2) sin(ωt) + A1 cos(ω1t+ φ) be the vectorpotential of the
combined fields. As presented in Sec.III B, the corresponding
action Eq. (7) eventually takesthe form S(t, τ) ' S0(t, τ) + ∆S(t,
τ), where S0(t, τ) isgiven by Eq. (12) for a Gaussian pulse as
presented inSec. III C, while the correction ∆S(t, τ) derives
fromthe cross term −pst,0pst,1τ +
´ tt−τ A0(t
′)A1(t′)dt′. Simi-
lar to Sec. III B, denoting the ratio between frequenciesε =
ω1/ω0, solving ∆S(t, τ) yields
∆S(t, τ) = −A0A12
√π
2σe−(
t√2σ
)2Im[eiϕtK(ϕt, ϕσ, ϕτ )],
where
K(ϕt, ϕσ, ϕτ ) = 2 cos[ε(ϕt −
ϕτ2
)+ φ
]sinc
(εϕt2
)g0
−e−i(εϕt+φ)g+ − e+i(εϕt+φ)g−
with g0 ≡ g(z/√
2, η/√
2)
and g± ≡g(z/√
2± iεϕσ/2, η/√
2)defined by function g(z, η) of
-
7
Eq. (13). Here, arguments z and η are dimensionlesscompositions
of time variables, z = (ϕt/ϕσ − iϕσ)/
√2
and η = ϕτ/√
2ϕσ. With a small ε, the series expansionof K(ϕt, ϕσ, ϕτ ) with
respect to ε up to the first orderresults in
K(ϕt, ϕσ, ϕτ ) ' −2√
2εeiϕtϕσ sinφΞ(z, η),
where
Ξ(z, η) =(z − η
2
)g
(z√2,η√2
)− 1− e
2η(z− η2 )√π
.
Hence the action is given by
∆S(t, τ) '√πA0E1e
−(
ϕt√2ϕσ
)2sinφσ2Im[eiϕtΞ(z, η)].(14)
With a typically large value of ϕσ when the femtosec-ond laser
of several tens of optical cycles is used, theFaddeeva function
w(z) ' iz/
√π(z2 − 1/2) if |z|2 > 256
[33]. Using the approximation and explicitly expandingthe
imaginary part in ∆S(t, τ), the full expression canbe rearranged by
trigonometric functions, whose coeffi-cients, each as a polynomial
of time variables, can befurther simplified by retaining only the
highest order ofϕσ. Eventually, we find
∆S(t, τ) ' A0E12ω20
e−(
ϕt√2ϕσ
)2sinφ{−ϕτ cosϕt + 2 sinϕt
−eϕtϕτϕ2σ [ϕτ cos(ϕt − ϕτ ) + 2 sin(ϕt − ϕτ )}.
When the pulse duration is sufficiently long,exp(ϕtϕτ/ϕ
2σ) ' 1, ∆S(t, τ) approaches
∆S(t, τ) ' −A0E1ω20
e− ϕ
2t
2ϕ2σ sinφC1(ϕτ ) cos(ϕt −
ϕτ2
),(15)
recovering action (8) under continuous waves as discussedin Sec.
III B, except for the presence of an extra prefactorexp(−t2/2σ2)
that serves as a temporal window.
Fig. 6 shows the comparison of near-cutoff harmon-ics evaluated
by direct numerical integration of Eq. (1)with that obtained by
applying the action of analyti-cal form, S(t, τ) = S0(t, τ) + ∆S(t,
τ), with S0(t, τ) and∆S(t, τ) given by Eqs. (12) and (15),
respectively. Thecomparison presents a rather good agreement,
justify-ing the analytically derived action with the assumed
ap-proximations. Comparing with harmonics in Fig. 5(b)without THz
field, it is shown that the odd-order har-monics are dominantly
determined by S0(t, τ), as thoseharmonics in both Fig. 5(b) and
Fig. 6 are almost thesame, though odd-order harmonic peaks in the
latter areslightly sharper due to the use of longer pulse width of
thefemtosecond laser. In Fig. 6, however, NCEHs emerge,clearly
indicating that even-order harmonics originatefrom the THz-induced
correction ∆S(t, τ).
From Eq. (15), following the similar procedure of anal-ysis in
Sec. III B, the reasoning behind the generation ofNCEHs in an
enveloped laser pulse is straightforward.
60 65 70 75 80 85 904.5
4.0
3.5
3.0
2.5
2.0
Figure 6. Comparison of |d̃(ω)| evaluated by direct
numericalintegration of Eq. (1) (blue) and by using the derived
action,S = S0 + ∆S, with S0 and ∆S given by Eqs. (12) and
(15),respectively. The same parameters of femtosecond pulse as
inFig. 5 are used except for σ = 2106 (FWHM of 120 fs). TheTHz
field is parametrized by ω1 = 1 × 10−4, E1 = 2 × 10−5and φ =
π/2.
Within the temporal window specified by the Gaussianenvelope,
the strength of NCEHs is approximately pro-portional to THz field
strength exactly at the center ofthe envelope. In other words,
under the influence of theTHz field that induces even-order
harmonics, the fem-tosecond pulse with a filtering temporal window
maps theinstantaneous strength of THz field onto that of
NCEHs,allowing for a complete characterization of the THz
time-domain spectrum with the femtosecond pulse scanningover the
THz field.
4
3
2
60 65 70 75 80 85 90
4
3
2
fs pulseTHz
fs pulseTHz
Figure 7. Near cut-off harmonics without THz field (blue)
andwith THz field (orange) when (a) φ = π/2 and (b) φ = π. Theinset
of each panel shows the time center of femtosecond pulse(blue)
relative to that of the THz field (orange) for different φ.Marks
"◦" (“×”) label the even-order harmonics with (with-out) the THz
field. The position of Ecutoff is indicated by theblack dashed
line. The same parameters as in Fig. 6 are used.
-
8
Fig. 7 shows the dependence of NCEHs on initial phaseφ, or
equivalently, the pulse center of the femtosecondlaser relative to
the electric component of the THz field.When φ = π/2, factor sinφ =
1 in Eq. (15) maximizesthe coefficient of NCEHs as analyzed in Eq.
(11). Asshown in Fig. 7(a), the even-order harmonics under aTHz
field is significantly higher than its counterpart with-out the THz
field, and the amplitude relative to theiradjacent odd-order
harmonics becomes even more signif-icant when the order approaching
the cutoff. On the con-trary, in Fig. 7(b), when φ = 0, the
even-order harmonicsvanish and the harmonic distribution in the
presence ofTHz field is exactly the same as the one without the
THzfield. Except for even-order harmonics, the harmonics ofother
energies are almost identical between panels (a) and(b), showing
they are barely influenced by the THz field.As indicated by insets
of Fig. 7, at φ = π/2 (φ = 0), thecenter of the femtosecond pulse
is at the maximum (thezero-point) of the THz electric field. The
coincidence ofthe NCEHs yields with the THz electric field shows
thefeasibility to reconstruct the latter using the former.
0 20000 40000 60000 80000 1000000.0
0.5
1.0
1.5
2.01e 5
2000 0 2000 4000 6000 80000.0
0.5
1.0
1.5
2.01e 5
0 50000 100000
0 5000
fs pulse
THz
fs pulse
THz
Figure 8. Reconstruction of time-domain spectrum of THzwaves.
(a) The parameters are the same as used in Fig. 1.ω0 = 0.0353 (1.3
µm), E0 = 0.06 (I = 1.3 × 1014 W/cm2),σ = 2106 (FWHM of 120 fs), ω1
= 2 × 10−4 (1 THz), andE1 = 2× 10−5 (100 kV/cm). (b) The
reconstruction for THzwave of higher frequency with ω0 = 0.1139
(400 nm), E0 =0.06 (I = 1.3 × 1014 W/cm2), σ = 702 (FWHM of 40
fs),ω1 = 0.003 (20 THz), and E1 = 2×10−5 (100 kV/cm). Insetsshow
the temporal profiles E0(t) and E1(t) of the femtosecondpulse and
the THz field, respectively.
In Fig. 8, the reconstruction of THz field from theNCEHs is
demonstrated by examples. Changing the timedelay between the
femtosecond pulse and the THz field,the even-order harmonic nearest
to the cutoff on the lower
energy side is retrieved and compared with |E1(t)| ofthe THz
field. Using the same parameters of fields asmentioned above, we
present the reconstruction with the78th-order NCEH in Fig. 8(a),
which in general showsa good agreement of |d(ω)| with |E1(t)|.
Another exam-ple to detect the THz field of higher frequency is
pre-sented in (b) to show the universality of the scheme. Inorder
to resolve the THz field of 20 THz, a femtosec-ond pulse of higher
frequency is required to ensure alow ratio ε = ω1/ω0. Using a 400
nm laser pulse withε ' 0.03 and reducing the pulse width to 40 fs,
the gen-erated 6th-order harmonic can also be used to reveal
thetime-domain THz wave. The successful reconstruction ofTHz wave
of short-time scale suggests the possibility ofTHz broadband
detection under the aid of the NCEHsmeasurement.
The applicability of the reconstruction scheme isclosely related
to the approximations applied for the anal-ysis in previous
sections. From the temporal perspec-tive, ε = ω1/ω0 � 1 is a must,
indicating that thecharacterization of THz field of high frequency
needshigh frequency femtosecond pulse. Moreover, the ap-proximation
of Faddeeva function to solve Eq. (14) re-quires (ϕ2t/ϕ2σ + ϕ2σ)/2
> 256, necessitating ϕσ > 23,suggesting a femtosecond pulse
should contain as least9 cycles within its FWHM. The range of ϕσ
also natu-rally satisfies both conditions that η = ϕτ/ϕσ � 1
andexp(ϕtϕτ/ϕ
2σ) ' 1 to derive (15). In general, the scheme
favors the use of large ϕσ, which also helps suppress
thesideband caused by the finite pulse width. Nevertheless,a
smaller ϕσ allows for a better time resolution of thewaveform
reconstruction. Therefore, a balance inbetweenshould be considered
for the choice of ϕσ, which also de-pends on the frequency range of
THz field to detect.
In addition, the choice of laser parameters, includ-ing field
amplitudes E0, E1 and frequency ω0, is criticalto the applicability
of the reconstruction scheme. Thesmall value of the argument of the
Bessel function inEq. (10) imposes the condition |ε(A0A1/ω0)C1(ϕτ
)| =|(E0E1/ω30)C1(ϕτ )| < 1. As shown in Fig. 4, the value
of|C1(ϕτ )| ∈ [1.7, 5.2] when ϕτ ∈ [3, 5] for near-cutoff
har-monics allows for an estimation of the loosely
restrictingcriterion, E0E1/ω30 < 0.2. Moreover, neglecting the
THzfield E1(t) in the prefactor of dipole matrix
elements,µ∗[pst(ϕt, ϕτ ) +A(ϕt)]µ[pst(ϕt, ϕτ ) +A(ϕt−ϕτ )]E(ϕt−ϕτ
), requires E1 � E0. Both conditions indicate an up-per limit for
E1. That is, the detected THz field in thiswork is not supposed to
be overwhelmingly intense, oth-erwise the approximation of the
Bessel function in Eq.(10) breaks down, resulting in nonvanishing
high-ordercomponents that contribute to other complicated
effectsaccompanied by a strong low-frequency field, e.g.,
thefield-induced multi-plateau structure. Since the theoryworks in
the tunneling regime, Ip 6 2Up, the femtosec-ond laser also
satisfies E20 > 2ω20Ip.
Besides the conditions required to justify the recon-struction
scheme, we also need to take the finite signal-to-noise ratio into
account. The even-order harmonics
-
9
should be large enough to observe. Assuming the am-plitude of
even-order harmonics to the adjacent odd-order one is no less than
one percent, we may imposean extra condition with the coefficients
of harmonics,εA0A1/2ω0 = E0E1/2ω
30 > 10
−2, yielding E0E1/ω30 >0.02. In contrast to the condition
specified by the ap-proximation of Bessel function, it indicates
the field am-plitudes should be large enough to generate
even-orderharmonics.
Figure 9. Available parametric range for THz reconstructionwith
NCEHs with regard to frequency ω0 and field amplitudesE0 and E1.
The detailed conditions are specified in the maintext.
All above conditions for the reconstruction scheme canbe
pictorially illustrated in the parametric space as shownin Fig. 9,
where the appropriate parametric range ishighlighted. When E1 of
the THz field is low, a fem-tosecond pulse of longer wavelength
avails the measure-ment; on the contrary, the THz field of
increasing E1 re-quires a femtosecond pulse of higher ω0, whose
optionalfrequency range also becomes broader. Concerning
therelation ε = ω1/ω0 � 1, the accessible frequency of theTHz field
for waveform reconstruction thus depends onfield amplitudes.
Especially both E0 and E1 being highfavors the use of higher ω0,
allowing for the detection of
THz field of higher ω1.
IV. SUMMARY AND CONCLUSION
The harmonic generation by a half-wave symmetricdriving laser
that interacts with isotropic media has longbeen known to yield
odd-order harmonics only [34]. Theemergence of even-order harmonics
usually attributes tocertain broken symmetries [35], e.g., the THz
field in-duced broken symmetry in this work. Here, the even-order
harmonic generation near the cutoff is found tohave a particularly
synchronous relation with the THzelectric field. The analytical
derivation with perturba-tive expansion shows the NCEHs originate
from the first-order correction with regard to the ratio between
frequen-cies of the THz field and that of the femtosecond
laserpulse. The linear relation between the NCEH amplitudeand the
THz electric field derives from an approximationof the Bessel
function, which can be fulfilled by the spe-cific range of return
time corresponding to the near-cutoffenergy region.
The direct mapping from the THz field to NCEHs thusprovides an
alternative conceptually simple approach toreconstruct time-domain
THz wave from NCEHs. Theproposal to measure NCEHs as a function of
the timedelay between the femtosecond laser pulse and the THzfield
has been numerically verified, showing the appli-cability of the
method for the broadband THz detection.The analytical derivations
also help identify the paramet-ric region for such applications,
indicating high-frequencyTHz field characterization should require
higher laser in-tensity. The encoding of the time-domain
information ofTHz wave into NCEHs may inspire new routes towardsthe
realization of coherent detection in broad spectralrange.
ACKNOWLEDGMENTS
This work is supported by the National Natural Sci-ence
Foundation of China (Grants No. 11874368, No.11827806 and No.
61675213).
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Terahertz field induced near-cutoff even-order harmonics in
femtosecond laserAbstractIntroductionScheme of Coherent Detection
and Numerical SimulationsAnalysis of near-cutoff even-order
harmonicsMonochromatic light fieldTHz field induced NCEHsEffect of
pulse envelopeTHz induced NCEHs under Gaussian-enveloped pulse
Summary and conclusionAcknowledgmentsReferences