复杂Swift-Hohenberg方程下的高能平面与复合脉冲发现

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本文主要探讨了在激光系统中，通过复杂Swift-Hohenberg方程模型来研究高能量平面波和复合脉冲的新发现。Swift-Hohenberg方程是一种常用的数学模型，用于模拟光在非线性介质中的传播行为，特别是在研究模式形成、稳定性以及自组织现象时。在这个研究中，作者S.C.V. Latas及其团队基于先前由Soto-Crespo和Akhmediev提出的复合脉冲，通过调整参数，成功地数值发现了这些高能量的纯态和复合态脉冲。首先，作者在参数空间中确定了这类高能量脉冲存在的区域，这对于理解和控制实际激光系统的性能至关重要。他们详细分析了这些脉冲的特性，包括其时间域和频域（即光谱）的形状，以及它们的色散特性（即脉冲的频移或频率变化，即所谓的色散效应）。结果显示，尽管脉冲的能量与光的色散参数有关，但对非线性项（即非线性光学效应）的依赖更为显著。特别值得注意的是，研究还揭示了一种极端幅度的脉冲，这可能具有特殊的物理意义或潜在的应用价值，比如在数据传输、光开关或光存储等高速光通信技术中。这样的脉冲由于其高强度和紧凑的时空结构，可能提供比传统脉冲更高效的信息处理能力。此外，该工作对于激光物理学和光纤通信领域具有重要意义，因为它扩展了我们对非线性光学现象的理解，特别是关于如何通过调控参数来控制和优化高能量脉冲的形成和传播。这对于设计新型高性能光纤激光器和超快光通信系统具有直接的指导作用。这篇文章通过复杂Swift-Hohenberg方程的数值模拟，深入研究了高能量激光脉冲的性质，并为未来的光子学研究和实际应用提供了有价值的数据和理论基础。

High-energy plain and composite pulses in a laser

modeled by the complex Swift – Hohenberg equation

S. C. V. Latas

I3N—Institute of Nanostructures, Nanomodelling and Nanofabrication, Department of Physics, University of Aveiro,

3810-193 Aveiro, Portugal (sofia.latas@ua.pt)

Received October 29, 2015; revised January 8, 2016; accepted January 10, 2016;

posted January 15, 2016 (Doc. ID 252953); published March 8, 2016

In this work, new plain and composite high-energy solitons of the cubic–quintic Swift–Hohenberg equation were

numerically found. Starting from a composite pulse found by Soto-Crespo and Akhmediev and changing some

parameter values allowed us to find these high energy pulses. We also found the region in the parameter space in

which these solutions exist. Some pulse characteristics, namely, temporal and spectral profiles and chirp, are

presented. The study of the pulse energy shows its independence of the dispersion parameter but its dependence

OCIS codes: (060.5530) Pulse propagation and temporal solitons; (140.3510) Lasers, fiber; (140.7090)

Ultrafast lasers.

http://dx.doi.org/10.1364/PRJ.4.000049

1. INTRODUCTION

Laser systems with passive mode locking can be described by

the cubic–quintic complex Ginzburg–Landau (CGLE) and

Swift–Hohenberg (CSHE) equations [1]. Both equations have

a wide diversity of analytical and numerical solutions, as can

be seen for example in [1–6], and references therein. Among

other characteristics, both models have in common two par-

ticular numerical solutions: the plain and the composite

solitons [6–9].

Recently, ultrashort high-energy pulses and extreme ampli-

tude spikes, solutions of the CGLE, have been found [10,11].

Such diversity and complexity of solutions is due, in part,

to the fact that the CGLE has several free parameters. A small

change of one of these parameters can have a huge impact on

the nature of the solution [11].

Some technological applications of high-energy pulses

are, for example, dissipative soliton fiber lasers in the

normal dispersion regime or with parameter management

[12–14].

One of the restrictions of the CGLE model concerns

spectral filtering, which is limited to a second-order term,

characterized by a single maximum spectral response. In gen-

eral, in experiments the gain spectrum is large and might

have several maxima. For more realistic modeling it is nec-

essary to add higher-order spectral filtering terms. The

addition of a fourth-order spectral filtering term into the

cubic–quintic CGLE transforms it into the CSHE [1,6].

The role of this higher-order term in spectral filtering has

been studied in [6]. In particular, narrow composite pulse

(NCP) and wider composite pulse (WCP) were numeri-

cally found.

In this work new plain and composite high-energy

solitons of the CSHE, and their region of existence in the plane

(ε, D), have been numerically found. Some pulse characteris-

tics are presented. Additionally, a plain pulse with extreme

amplitude peak was also found.

2. THEORY

The CSHE can be written in the following form [1,6]:



juj

u  νjuj

u  iδu  iβu

 iεjuj

 iμjuj

u  iγu

TTTT

; (1)

where Z is the propagation distance or the cavity round-trip

number, T is the retarded time in a frame of reference moving

with the group velocity, and u is the normalized complex

envelope of the optical field. On the left-hand side, D repre-

sents the cavity dispersion, with D>0 in the anomalous re-

gime and D<0 in the normal regime, and ν corresponds, if

negative, to the saturation of the nonlinear refractive index.

On the right-hand side, δ represents the difference between

linear gain and loss; β and γ account for spectral filtering

and higher order spectral filtering, respectively. The terms

with ε and μμ < 0, represent the nonlinear gain and the

saturation of the nonlinear gain, respectively.

The CSHE given by Eq. (1) has been numerically solved,

with a split-step Fourier symmetric method, with a time step

ΔT  0.03125 and a distance step ΔZ  0.0003, described for

example in [15]. Absorbing boundary conditions were used, as

presented in [16]. In order to guarantee the convergence to a

single pulse, an initial condition of the form 2 sech2T has

been considered. For initial conditions with a larger width

multiple pulses can be formed.

3. RESULTS

At this stage we refer that the starting point for this work was

the composite pulse (CP), discovered in [6] for the following

set of parameter values: β  −0.3, δ  −0.5, ε  1.6, μ  −0.1,

ν  0, and γ  0.05. We might ask, what is the impact of a

small change of the saturation of the nonlinear gain, μ,on

the nature of the solution? The pulse amplitude is gradu-

ally reduced as μ becomes slightly more negative. On the other

hand, as μ → 0 the pulse amplitude grows. If μ  0, blow-up

S. C. V. Latas Vol. 4, No. 2 / April 2016 / Photon. Res. 49

Corrected 22 March 2016

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