Sub-cycle pulses in a two-level medium: The solitary wave
propagation
Xunming Cai
a,b
, Zhaoying Wang
a
, Jingyun Zhao
a
, Qiang Lin
a,c,
n
a
Institute of Optics, Department of Physics, Zhejiang University, Hangzhou 310027, China
b
College of Information Engineering, Guizhou Minzu University, Guiyang 550025, China
c
Center for Optics & Optoelectronics Research, Department of Applied Physics, College of Science, Zhejiang University of Technology, Hangzhou 310023,
China
article info
Article history:
Received 12 September 2014
Received in revised form
16 December 2014
Accepted 19 December 2014
Available online 23 December 2014
Keywords:
Sub-cycle pulse
Self-induced transparency
Intrinsic chirp
Complete population return
abstract
The expressions of hyperbolic secant sub-cycle pulse are derived from the oscillating dipole model
through the complex source point method. Using the expressions of sub-cycle pulse, the intrinsic
properties of the sub-cycle pulse as the intrinsic chirp and self-induced blueshift of the center frequency
are found to have an important effect on the propagation of sub-cycle pulse in a two-level medium. The
sub-cycle solitary waves are obtained in a two-level absorbing medium. The complete population in-
version cannot be obtained by an odd multiple
π
sub-cycle pulse. As the intrinsic properties of sub-cycle
pulse, the complete population return is obtained for both cases that the carrier frequency is resonant
and non-resonant with atom. Accordingly, the solitary wave regime and the self-induced transparency of
sub-cycle pulse are obtained in both the resonant and non-resonant cases.
& 2014 Elsevier B.V. All rights reserved.
1. Introduction
The propagation of laser pulse in the two-level or three-level
atomic media has attracted a lot of attentions for many years
[1–6]. Untill today, it is still difficult to produce the sub-cycle pulse
in laboratory directly. But the sub-cycle pulse can be produced by
some indirect methods [7–10]. For example, a single cycle laser
pulse is synthesized in frequency domain using sidebands [9]. The
pulse compression method can be used to produce the sub-cycle
pulse [11]. By the use of femtosecond pulse shapers, the hyperbolic
secant single cycle and sub-cycle pulses can be achieved through
the elaborate adjustment of waveform and spectral phase. For the
long pulse, the Maxwell–Bloch (M–B) equations can be solved
under the slowly varying envelop approximation (SVEA) and the
rotating-wave approximation (RWA). Some important results, for
example, the area theorem [1,12], which can predict and explain
many interesting phenomena such as self-induced transparency
(SIT) and pulse compression, has been found. However, for the
few-cycle and sub-cycle laser pulses, the RWA is not applicable any
more. The analytic solution of M–B equations without RWA is
difficult to be obtained for the few-cycle and sub-cycle pulses.
An iterative predictor-corrector finite-difference time-domain
method is used to solve the semiclassical Maxwell–Bloch system
numerically without RWA [13]. It was reported by Song that the
standard area theorem breaks down even for the small-area sub-
cycle attosecond pulses in a dense two-level medium [14]. But so
far, all these researches of the propagation of few-cycle and sub-
cycle pulses are based on the carrier-envelope model. The carrier-
envelope model is just an approximation model and can cause
non-physical results for the few-cycle and sub-cycle pulses.
Especially for the sub-cycle pulse, there is obvious direct current
component and the frequency spectrum depends heavily on the
carrier-envelope phase in the carrier-envelope model [15–17].
Such pulse cannot propagate in the space [18]. Recently, the ana-
lytical expressions of few-cycle, single-cycle, and sub-cycle pulsed
beams with arbitrary polarization have been found, which are the
exact solutions of Maxwell's equations [15,19]. Some new prop-
erties of sub-cycle pulse including the intrinsic chirp and self-in-
duced blueshift of the center frequency are found. So the effect of
the intrinsic properties of the sub-cycle pulse on the propagation
of pulse is worth to be investigated.
In this letter, the propagation of the hyperbolic secant sub-cycle
laser pulse in a two-level absorbing medium is investigated. The
intrinsic chirp and self-induced blueshift of the center frequency
exist in Gaussian type, hyperbolic secant type, Lorentz type, or
others type sub-cycle pulses. But for the different type sub-cycle
pulses, the degrees of these properties are different. For the
Lorentz sub-cycle pulse, there is a long fall time. Compared with
the Guassian pulse, the hyperbolic secant sub-cycle pulse has a
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journal homepage: www.elsevier.com/locate/optcom
Optics Communications
http://dx.doi.org/10.1016/j.optcom.2014.12.055
0030-4018/& 2014 Elsevier B.V. All rights reserved.
n
Corresponding author at: Institute of Optics, Department of Physics, Zhejiang
University, Hangzhou 310027, China.
E-mail address: qlin@zju.edu.cn (Q. Lin).
Optics Communications 342 (2015) 90–94