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Experimental Realization of a Wavelength-Sized Optical-Vortex Knot
波长级光涡结的实验实现
Phys. Rev. Applied 17, 064026 – Published 13 June, 2022
物理评论应用 17, 064026 – 发布于 2022 年 6 月 13 日
Abstract 摘要
In the past decade, optical-vortex knots with axial dimensions on the order of
在过去的十年中,具有轴向尺寸在
Physics Subject Headings (PhySH)
物理学主题词(PhySH)
Article Text 文章文本
Knotted structures have been part of physical models since 1869 and are solutions to several field theories . In particular, the stability of knotted solutions is important for many applications in topological quantum computing , superfluids , quantum matter , liquid crystals , matter control , information transmission and storage , and microfabrication . Just recently, it has been possible to generate fields exhibiting isolated knots in the laboratory but most of those experimental realizations are short lived and all have macroscopic spatial scales in the range between a few hundreds of micrometers and several centimeters. For example, in liquid, vortex knots have been created by the sudden immersion of a hydrofoil that has the desired shape (trefoil) with a size of a few centimeters. The knots appear for a few milliseconds and later (due to viscosity) the vortex lines reconnect and separate into vortex loops before disappearing. Transient knots have also been observed in Bose-Einstein condensates at spatial scales of a few hundreds of micrometers in very brief time intervals on the order of 500
结绳结构自 1869 年以来一直是物理模型的一部分 ,并且是几种场论的解 。特别是,结绳解的稳定性 对于拓扑量子计算 、超流体 、量子物质 、液晶 、物质控制 、信息传输和存储 以及微加工 等许多应用都很重要。最近,已经可以在实验室中生成表现出孤立结的场,但大多数实验实现都是短暂的,且所有这些都具有几百微米到几厘米范围的宏观空间尺度。例如,在液体中,通过突然浸入具有所需形状(如三叶形)的水翼,创造了涡旋结 ,其大小为几厘米。这些结出现几毫秒,随后(由于粘度)涡旋线重新连接并分离成涡旋环,然后消失。 在玻色-爱因斯坦凝聚态中也观察到了瞬态结,空间尺度为几百微米,时间间隔非常短,大约为 500。
Stable optical knotted structures have been created with polarization and phase singularities in the paraxial regime. In particular, optical-vortex knots can be generated with two-dimensional phase masks. In the experimental implementation of Ref. , these structures were generated with axial lengths of a few centimeters. A similar approach has been used to generate acoustic vortex knots with an axial length of about 40 cm. So far, the smallest optical-vortex knots (termed “ultrasmall”) have axial spatial scales of about
稳定的光学结结构已经在旁轴区域中通过极化 和相位奇点 被创建。特别是,光涡旋结可以通过二维相位掩模生成。在参考文献 的实验实现中,这些结构的轴向长度为几厘米。类似的方法已被用于生成轴向长度约为 40 厘米的声涡旋结 。到目前为止,最小的光涡旋结(称为“超小”)的轴向空间尺度约为
Theoretically, it has been shown that optical-vortex knots with spatial scales on the order of
理论上,已经证明具有
Here, we use the angular spectrum calculated in Ref. to create a wavelength-sized optical-vortex knot, combined with a classical interferometric method to measure tightly focused fields with no approximations . We show that it is possible to create a trefoil optical-vortex knot enclosed by a volume (
在这里,我们使用参考文献 中计算的角谱来创建一个波长大小的光涡结,结合经典干涉方法来测量没有近似的紧聚焦场 。我们展示了可以创建一个被体积(
The angular-spectrum representation for a nonparaxial trefoil knot is described in Ref. (another approach in Ref. ), where the authors have calculated the Fourier transform of their previous paraxial polynomial representation and then selected the terms that can give rise to a knot when propagating the beam in the nonparaxial regime with the Richards-Wolf integral . The complex angular-spectrum representation of the nonparaxial trefoil knot in cylindrical coordinates is
where
The simulation considers linearly polarized light (horizontal) imprinted with the complex angular-spectrum representation [Eq. ] with
Figure shows the calculated optical-vortex lines with their angular spectrum (amplitude and phase) for the cases of
The experimental setup is an holographic optical-tweezers apparatus with a few modifications that enable the measurement of the field with no approximations with step interferometry. A schematic of the simplified experimental setup is shown in Fig. . The laser beam (
The reference beam focuses at the imaging plane of the microscope objective [Fig. ], while the focused modulated beam is displaced axially
Both beams are reflected back into the objective by a mirror mounted on a piezo stage (Thorlabs, MAX311D) that controls the plane that is imaged by the objective [Fig. ].
The polarization of the imaged interferograms
The holograms are calculated with the algorithm proposed by E. Bolduc et al. to modulate both the amplitude and the phase of the angular spectrum of the beams. From Eq. in Cartesian coordinates, the required amplitude and phase are
where
We use the same parameters of the simulations (
Figure illustrates the process to encode the fields, the first column is for
The field is measured at 80 transverse planes axially separated by 40 nm (piezo stage). The axial scan starts in a position of
The phase and amplitude of the
and
where the extracted electric field is
Figure shows the four interferograms
The extracted phase and amplitude (simulated and measured) of a trefoil knot with
Figure shows a view from the top of the optical vortices that make up the isolated trefoil knot and some other surrounding vortex lines that are not connected to the knot. The color bar represents the height with respect to the bottom of the knot. The singularities that surround the isolated knot also appear in the calculations in regions of very low light intensity (a few percent of the maximum intensity at each frame), just as in the experiment. The isolated knot is contained in a volume of
To compare the simulated and the experimental beams across different planes, we have to take into account the error in the position of the experimental waist
where
Figures and show the comparison between the experiment and simulation for
Figures and shows the comparison between the experiment and simulation for
Figures – show the isolated vortex loops and knots but there are more vortices in the periphery, just as in the calculation. In order to show these singularities, we take the case of Fig. , which corresponds to the knot with
The isolated knot of Fig. is bounded on the top and bottom by a phase that has a triangular shape. Figure shows the measured phases that bound the knots and loops for
Figure shows the measured transition from an isolated knot to separated loops as a function of the scale parameter
We demonstrate an experimental realization of optical-vortex knots at the scale of a wavelength. The trefoil knot generated with parameters of
This work was partially funded by Dirección General de Asuntos del Personal Académico (DGAPA) Universidad Nacional Autónoma de México (UNAM) Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica (PAPIIT) Grant No. IN107719, Coordinación de la Investigación Científica (CTIC) - Laboratorio Nacional de Materia Ultrafría en Información Cuántica (LANMAC) 2021, and Consejo Nacional de Ciencia y Tecnología (CONACYT) Grant No. LN-299057. I.A.H.H. thanks PAPIIT for a scholarship. We thank José Rangel Gutiérrez for the fabrication of some optomechanical components.
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