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

∗ zhangyz@sari.ac.cn† yantm@sari.ac.cn‡ jiangyh@sari.ac.cn

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

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)

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

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

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