Solitons and topological waves

doi:10.1126/science.abb5162

GSTDTAP > 气候变化

DOI	10.1126/science.abb5162
	Solitons and topological waves
	Mark J. Ablowitz; Justin T. Cole
	2020-05-22
发表期刊	Science
出版年	2020
英文摘要	The intense coherent emission from lasers enabled the study of light propagation in nonlinear media, which spurred many important applications. More recently, the study of electromagnetic wave propagation in periodic media, where linear band structures play an important role, has advanced in new directions. By breaking certain symmetries, such as time reversal, the medium can support so-called “topologically protected” modes that possess uncommon robustness to material defects. Theory has suggested that certain nonlinear waves can inherit the topology of associated linear waves. On page 856 of this issue, Mukherjee and Rechtsman ([ 1 ][1]) describe experiments where such nonlinear waves, called solitons, can now be observed in the bulk of photonic topological media. These localized waves exhibit cyclotronic motion as the light propagates down a specifically engineered waveguide. When a different mode is considered—one with trivial topology—the waves no longer circulate but remain essentially fixed in their initial spatial distribution. ![Figure][2] Tracking a topological soliton Five snapshots (left to right) show a topological soliton as it propagates down an array of waveguides laser-fabricated by Mukherjee and Rechtsman. The waveguide is fabricated in such a way that as adjacent waveguides come close together, the light transfers in a counterclockwise fashion from one waveguide to the next. GRAPHIC: JOSHUA BIRD/ SCIENCE Investigations of solitons trace their roots back to 1834, when naval architect John Scott Russell first recognized their remarkable character in the Union Canal near Edinburgh, Scotland ([ 2 ][3]). This wave was not oscillatory; it was a solitary surface wave that propagated over surprisingly long distances (2 to 3 km) with fixed form. Some years later, mathematicians described this solitary wave in terms of approximate equations derived from the governing water-wave equations. For nearly 70 years, this was essentially all that was known theoretically. The situation changed in 1965 ([ 3 ][4]) when it was found that two such solitary waves have extraordinary interaction properties. Their interaction is elastic in nature, and the two waves exit the interaction with the same amplitude and speed with which they entered. Such solitary waves were termed solitons. This paper motivated major research studies in both mathematics and physics. In mathematics, it gave rise to a new field of study: integrable nonlinear wave systems. These are nonlinear partial differential equations that are exactly solvable and possess an infinite number of symmetries and conservation laws. In physics, researchers have observed solitary waves and solitons not only in water waves and nonlinear optics but also in plasmas, electrical circuits, and Bose-Einstein condensates. These solitary waves do not necessarily have special interaction properties. In a different research direction, it was shown ([ 4 ][5]) that topological properties and invariants could be used to explain the integer quantum Hall effect. As an external magnetic field is gradually increased, the conductivity in materials such as gallium arsenide heterostructures decreases by quantized jumps. The field of topological insulators in electromagnetic media can be traced to 2008, when topologically protected modes were theoretically identified in suitably constructed material permittivity ([ 5 ][6]). By using media that break time-reversal symmetry, linear edge waves were found that propagate unidirectionally and possess nontrivial topological invariants (Chern numbers). The first experimental observation of a topologically protected mode in an electromagnetic system found that localized edge waves could propagate in suitable magnetic media that break time-reversal symmetry ([ 6 ][7]). These linear magneto-optical waves propagated unidirectionally without backscatter from defects. The topology here is spectral in nature and is different from spatial topology observed in dark solitons and vortices. ![Figure][2] Undeterred by defects A topological edge soliton localized along the boundary of a waveguide array cannot reflect backward and instead propagates around a defect. GRAPHIC: JOSHUA BIRD/ SCIENCE A few years later, a photonic system was constructed ([ 7 ][8]) that broke time-reversal symmetry by creating a helical rotation of the lattice waveguides in the propagation direction. The media vary periodically in the direction of propagation, and models of this system involve wave equations with coefficients that share this periodicity. The mathematician Gaston Floquet studied differential equations with periodic coefficients, so this system is referred to as a Floquet topological insulator. All of these systems are linear. A nonlinear waveguide system was proposed in ([ 8 ][9]) that exhibits a similar type of cyclotronic motion, as has now been observed by Mukherjee and Rechtsman (see the first figure). From a mathematical perspective, the model used to describe both systems consists of a discrete nonlinear Schrödinger (NLS) equation in two spatial dimensions, with periodic coefficients in the propagation variable. One-dimensional solitons in uniform waveguides, but without topology, were theoretically predicted in ([ 9 ][10]) and observed a decade later ([ 10 ][11]). They were subsequently observed in two-dimensional uniform waveguides ([ 11 ][12]). These uniform systems are modeled by one- and two-dimensional discrete NLS equations with constant coefficients. Researchers have theoretically predicted the existence of solitons ([ 12 ][13]) on the boundary edge of the helically varying waveguides used in the experiments of ([ 7 ][8]). The linear topology and the unidirectional propagation through defects appear to be naturally inherited by the nonlinear soliton modes (see the second figure). We anticipate future research that will continue to examine how the presence of topology affects the behavior of solitons. An aspect of the work of Mukherjee and Rechtsman indicates the extent to which waveguide fabrication has progressed. Early fabricated optical structures created waveguides with uniform features in the late 1990s. This approach was extended to waveguides with helical structure in the longitudinal direction with femtosecond laser writing techniques in ([ 7 ][8]). The waveguides used in the present study are engineered so that during one period, each waveguide couples to its nearest neighbors sequentially and one at a time. 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领域	气候变化 ; 资源环境
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文献类型	期刊论文
条目标识符	http://119.78.100.173/C666/handle/2XK7JSWQ/270482
专题	气候变化资源环境科学
推荐引用方式 GB/T 7714	Mark J. Ablowitz,Justin T. Cole. Solitons and topological waves[J]. Science,2020.
APA	Mark J. Ablowitz,&Justin T. Cole.(2020).Solitons and topological waves.Science.
MLA	Mark J. Ablowitz,et al."Solitons and topological waves".Science (2020).