01 Introduction
Recently, Professor Zeng Heping's team at the State Key Laboratory of Precision Spectroscopy Science and Technology at East China Normal University observed for the first time in theory and experiment the hierarchical structure of Farey trees and devil's ladders in breather lasers, revealing the structure of breather ultrafast lasers. fractal dynamics.
The relevant research results were published in Nature Communications on October 2, 2022 under the title "Farey tree and devil’s staircase of frequency-locked breathers in ultrafast lasers".
This figure graphically illustrates that the breathers produced in femtosecond fiber lasers are actually a fractal structure - a Farey tree. Further research revealed that the dimensions of the Farey Tree were consistent with the Devil's Ladder.
02 Research background
In 1975, mathematician Benoit Mandelbrot first proposed the term "fractal", and thus a new subject - fractal geometry was born. Mandelbrot defines a fractal as "A fractal is a shape in which the part and the whole are similar in some way." A typical example of fractals in nature is broccoli. The shapes of each branch and the whole of broccoli are similar; although fractals were originally just a mathematical concept, they have been observed in many systems, such as materials science and biology. , neurology, circuits, networks, geography and stock price fluctuations, etc.
In 2000, the connection between optical solitons and fractals was theoretically established (Phys. Rev. Lett. 84, 1902, 2000). Since optical solitons are relatively stable, "fractal" them into multiple branches requires multiple changes to the physical properties of the transmission medium (such as optical fiber), and requires the detection of optical solitons at multiple locations. These conditions are difficult to achieve experimentally, and related experiments are almost impossible to carry out.
03 Research innovation points
Breathers are localized wave packets whose parameters change periodically. Because they are closely related to nonlinear phenomena such as turbulence, chaos, extreme events, and modulation instability, they have attracted widespread attention from the academic community. For the first time, the team has experimentally and theoretically linked breathers and fractals. Their research found that by controlling the relevant parameters of the laser, the laser output light field can be switched between two types of breathers: frequency-locked and non-locked breathers. The breathing frequency of the frequency-locked breather is spontaneously locked to the laser repetition frequency and remains unchanged under external disturbances. Its frequency stability is about 3,000 times higher than that of non-locked breathers (Figure 1). Since manually searching for frequency-locked breathers requires a certain amount of experience and is time-consuming and labor-intensive, the team also demonstrated that the intelligent control system can be used to quickly search for frequency-locked breathers.
Figure 1 Comparison of the frequency stability of locked breathers and unlocked breathers. The stability of the former is about 3000 times that of the latter.
where SD (Standard Deviation) is the standard deviation
In addition, the team also found that the ratio of the breathing frequency (the reciprocal of the breather evolution period) to the laser repetition frequency (determined by the length of the laser) (i.e., the number of surrounds), will appear as a series of blue steps as the pump current changes. platform (Figure 2). The number of surrounds corresponding to the platform is very stable and does not change with the pump current. The values corresponding to these platforms exhibit a classic fractal structure - the Farey tree (inset of Figure 2). This fractal structure is called the Devil's Ladder because the value of its fractal dimension (0.87) is consistent with the Devil's Ladder.
Figure 2 A series of platform structures appear as the breathing frequency/winding number changes with the pump current. The corresponding number of wraps around the platform forms the farey tree, as shown in the illustration. The wrap number refers to the ratio of the breathing frequency (the reciprocal of the respiratory subevolution period) to the laser repetition frequency (determined by the laser length); the wrap number equals 1/5 means that the breathing frequency is 1/5 of the laser repetition frequency.
Another feature of fractal breathers is that they can generate dense frequency combs that are not limited by the laser cavity length. This research work shows that the comb tooth density of the laser emitted by this laser is significantly increased by 41 times compared with lasers with the same cavity length. It can be seen that fractal-based breather lasers are very attractive as an alternative to sub-MHz optical frequency combs.
04 Summary and Outlook
This work demonstrates that the breather laser can be used as a simple nonlinear system to study fractal dynamics. Due to the universality of breathers (microcavities, BEC, fluid mechanics, etc.), this work is expected to inspire the study of fractals in related physical systems, where fractals provide a new perspective for understanding the complex dynamics in these systems. More broadly, the physical model that describes femtosecond lasers is the nonlinear Schrödinger equation, which has strong universality. The fractal solution of the nonlinear Schrödinger equation discovered through experiments and theory in this work will also gain widespread attention in related physical systems.
The first author of the paper is Wu Xiuqi, a doctoral candidate at the State Key Laboratory of Precision Spectroscopy Science and Technology at East China Normal University, and the second author is doctoral candidate Zhang Ying. Professor Zeng Heping and researcher Peng Junsong are the co-corresponding authors of the paper. In recent years, Professor Zeng Heping’s team has done systematic work in the field of breather ultrafast lasers. Proposed and confirmed breather ultrafast lasers, including continuous breathers (Sci.Adv.5 eaax1110 2019) and discrete breathers (Phys.Rev.Appl.11 044068 2019); AI technology was used to realize its intelligent control (Laser & Photon.Rev.2100191 2021); revealed the breather explosion dynamics (Phys.Rev.Appl.12 034052 2019) and the breather multi-body interaction (Laser & Photon.Rev.2000132 2021). This work was funded by the Ministry of Science and Technology, National Natural Science Foundation of China, Shanghai Municipal Science and Technology Commission, and Chongqing Municipal Science and Technology Commission.
Source：https://mp.weixin.qq.com/s/hyFWpcv9J_jT-FfSuj__zw
Paper link：https://www.nature.com/articles/s41467-022-33525-0
