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Optimization of temporal gate by two-color

chirped lasers for the generation of isolated

attosecond pulse in soft X rays

LIWEI HE,

GUANGLU YUAN,

KAN WANG,

WEIJIE HUA,

CHAO YU,

1,3

AND CHENG JIN

1,2,

Department of Applied Physics, Nanjing University of Science and Technology, Nanjing 210094, China

State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics,

Chinese Academy of Sciences, Xi’an 710119, China

e-mail: chaoyu@njust.edu.cn

*Corresponding author: cjin@njust.edu.cn

Received 17 July 2019; revised 9 September 2019; accepted 22 September 2019; posted 24 September 2019 (Doc. ID 372966);

published 15 November 2019

We propose a simple and efficient method to optimize two-color chirped laser pulses by forming a “temporal

gate” for the generation of isolated attosecond pulses (IAPs) in soft X rays. We show that the generation process

for higher and cutoff harmonics can be effectively limited within the temporal gate, and the harmonic emission

interval can be further reduced with the help of phase-matching by only selecting the contribution from short-

trajectory electrons. This two-color gating mechanism is verified by increasing the pulse duration, raising the gas

pressure, and extending the target cutoff. Compared to the five-color waveform in Phys. Rev. Lett. 102, 063003

(2009), our waveform can be used to generate the IAP in the long-duration laser pulse while the cutoff energy is

higher without the reduction of harmonic yields. Our work provides an alternative temporal gating scheme for

the generation of IAPs by simultaneously improving the harmonic conversion efficiency, thus making the atto-

second soft X rays an intens e and highly time-resolved tabletop light source for future applications.

Chinese Laser Press

https://doi.org/10.1364/PRJ.7.001407

1. INTRODUCTION

The high harmonic generation (HHG) by the interaction of an

ultrafast strong laser with a gaseous medium, resulting in light

with wavelength covering the extreme ultraviolet (XUV) to X

rays [1], has been used as a tabletop light source and has ini-

tiated attosecond science [2–5]. Compared to the bulky, costly,

and inconvenient larger light sources, e.g., synchrotron radia-

tion accelerators and free electron lasers, the temporal charac-

teristics of HHG have made it a unique tool for studying time-

resolved spectroscopy and for probing electron dynamics. One

of the most important applications of HHG is to provide iso-

lated attosecond pulses (IAPs), often with the idea of limiting

the efficient generation process only in a fraction of the driving

laser period, i.e., the temporal gating.

Since the first ge neration of IAP in 2001 [6], several differ-

ent methods have been developed to produce the IAP in the

XUV based on the traditional 800-nm Ti:sapphire lasers, in-

cluding the polarization gating for an isolated 130-as pulse

[7,8], the amplitude gating [9] for an 80-as pulse with carrier-

envelope phase (CEP)-stabilized few-cycle pulses, and the dou-

ble optical gating (DOG) for a 67-as IAP [10]. Until recently,

the soft X-ray IAPs have emerged with the development of laser

technology in the mid-infrared [11,12]. With a 2-μm and mul-

ticycle laser, Chen et al. [13] showed the IAP at photon energies

up to 180 eV via the transient phase-matching gating [14]. By

rotating the wavefront of the driving laser, the continuous

harmonic spectra up to the carbon K-edge of 284 eV were pre-

sented by Silva et al. [15], indicating the existence of a pulse as

short as 400 as. Later on, the same group demonstrated the

generation of 0.5-keV supercontinuum spectra to support a

13-as IAP [16]. In 2017, the new record of the shortest pulse

with the duration of 53 as in the soft X rays was reported by

Li et al. [17] through the mid-infrared polarization gating tech-

nique. In the same year, a 43-as pulse was also claimed with a

passively CEP-stable mid-infrared laser by Gaumnitz et al. [18].

Very recently, Johnson et al. [19] applied a 1.8-μm laser in the

overdriven regime to rapidly modify the beam’s spatiotemporal

shape, and high harmonics with photon energies up to 600 eV

were generated to support soft X-ray IAPs. However, all the

above generation methods have their own limitations. As soft

X-ray pulses are highly in demand for attosecond experiments

[20], two serious issues need to be resolved. One is how to im-

prove the emission efficiency, since the HHG yields from each

atom scale unfavorably with laser wavelength λ, typically as

Research Article

Vol. 7, No. 12 / December 2019 / Photonics Research 1407

−4

− λ

−6

[21–23]. The other is how to develop efficient gating

schemes for the IAP, which can be applied in the soft X rays.

In recent years, advances in laser technology have made it

possible to generate any optical waveforms by accurately con-

trolling and adjusting the synthesis of two-color or even multi-

color laser pulses [24–29]. They have been used to control the

conversion efficiency and spectral region of HHG by optimiz-

ing subcycle laser waveforms (see review in Ref. [30]). For in-

stance, in 2009, Chipperfield et al. [31] presented the so-called

“ideal waveform” to dramatically increase the harmonic cutoff

energy, which can be roughly accomplished through the syn -

thesis of five-color laser pulses. With only 30% of pulse energy

from the longest wavelength component, the five-color wave-

form enables the harmonic cutoff energy in the XUV with only

an 800-nm laser to be efficiently increased into the soft X rays

by a factor of 2.5, while the harmonic yields are maintained. In

2014, Jin et al. [32] proposed an optimization scheme to con-

struct the multicolor laser waveform in the single-atom level by

considering the macroscopic effects of HHG propagation in the

nonlinear medium. They were able to show the enhancement

of soft X-ray harmonic yields by 1 to 2 orders of magnitude

compared to the single-color field. Therefore, the optimization

of multicolor laser pulses has been established as a meaningful

way to boost the HHG conversion efficiency in soft X rays.

However, it is not straightforward to employ this approach

for the generation of IAP [33,34]. So it is very necessary to

develop some innovative optimization methods.

Our main goal in this work is to establish a novel scheme to

optimize the multicolor laser wav eform for the generation of

intense soft X-ray IAPs. We su ggest that the waveform consist-

ing of two-color chirped laser pulses is optimized to form a

“temporal gate” only for selected higher harmonics, which in

turn creates an IAP in time. Note that our waveform with only

two color components has better performance than the five-

color one by Chipperfield et al. [31], i.e., it can produce even

longer cutoff without reducing the harmonic yields and gener-

ate the IAP in the long pulse. The paper is arranged as follows.

In Section 2, we will present the waveform optimization

method, HHG propagation equations, and the formulation

for the generation of attosecon d pulses; in Section 3, we will

show the optimized waveforms, and the resulting harmonic

spectra and IAPs; the conclusion of this paper is given in

Section 4.

2. THEORETICAL METHODS

A. Optimization of the Two-Color Chirped Laser

Waveform

In the optimization, we consider the synthesized waveform

consisting of two-color chirped laser pulses in the form of

EtE

tcosω

t  α

 ϕ



 E

tcosω

t  α

 ϕ

: (1)

Here E

, ω

, A

, α

, and ϕ

(i  1, 2) are the respective ampli-

tude, angular frequency, temporal envelope, linear chirp rate,

and phase of each pulse. Due to the addition of chirp, the syn -

thesized waveform does not have periodicity anymore. The

optimization is thus executed over the entire temporal pulse

instead of one optical cycle of the fundamental pulse, as in

the previous studies [32,35,36]. Note that the effects of chirp

on the HHG in single- or two-colo r laser pulses have been

substantially discussed [37–47]; however, systematically opti-

mizing the chirp for the generation of attosecond pulses has

not been achieved.

In order to simplify the optimization procedure, some param-

eters in Eq. (1) can be fixed as in the following: (i) the first laser’s

wavelength is λ

 800 nm, considered to be the fundamental,

and ϕ

 0; (ii) both colors have the Gaussian envelope with the

full width at half-maximum (FWHM) duration of 8 fs (three

optical cycles of the fundamental); (iii) the total pulse energy

of two colors is fixed, i.e., jE

jE

is a constant. We search

parameters fE

, ω

, α

, ϕ

g to maximize the single-atom

HHG yields at given cutoff energies.

We use the genetic algorithm (GA) [48] to carry out the

optimization, and tens of thousands of iterations are taken

to ensure that the converged results are obtained. At each iter-

ation, the quantitative rescattering (QRS) model [49,50] is ap-

plied to compute the single-atom HHG. In the QRS, the

induced dipole moment of an atomic target is expressed as

DωW ωdω, (2)

where d ω is the complex photo-recombination transition di-

pole matrix element, which can be accurately calculated

through solving the time-independent Schrödinger equation

(TISE) by including the Coulomb potential effect, and

W ω is the complex electron returning wave packet and

can be calculated by the strong field approximation (SFA) [51].

The fitness function in the optimization is written as

F 

max

min

jDωj

dω: (3)

Here ω

min

 ω

cutoff

− 10ω

, and ω

max

 ω

cutoff

 10ω

where ω

is the angular frequency of the fundamental, and

cutoff

is the preset cutoff frequency (or, equivalently, photon

energy). Since the electron ionization occurs in a very short

time interval, the enhancement of harmonics in a small speci-

fied energy r ange would automatically enhance a broad range of

harmonics. The ionization probability at the end of the laser

pulse is set as a constraint, which is less than 2%–5%, to avoid

the defocusing of laser by the electron plasma in the gas

medium.

B. Macroscopic Propagation in the Nonlinear

Medium

To calculate the soft X-ray high harmonics emitted in a gaseous

medium, both the single-atom-induced dipole and the macro-

scopic response are needed. The former is calculated by using

the QRS model, and the latter can be accounted for by solving

the propagation equations of driving laser and high-harmonic

field in the nonlinear medium. Then the near-field harmonics

are obtained at the exit face of the gas medium. The details of

these propagation equations have been given in Refs. [52–56].

Here we only outline the key equations.

The propagation equation of a driving laser in dense and

ionized media is affected by the refraction, nonlinear self-

focusing, ionization, and plasma defocusing. In the atomic gas

medium, its evolution is given by the three-dimensional (3D)

Maxwell’s wave equation [56],

1408 Vol. 7, No. 12 / December 2019 / Photonics Research

Research Article

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