Knotted structures have been part of physical models since 1869 [1] and are solutions to several field theories [2]. In particular, the stability of knotted solutions [3] is important for many applications in topological quantum computing [4], superfluids [5], quantum matter [6,7], liquid crystals [8], matter control [9], information transmission and storage [10], and microfabrication [11]. 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 [12] 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 [7] at spatial scales of a few hundreds of micrometers in very brief time intervals on the order of 500 𝜇⁢s.
结绳结构自 1869 年以来一直是物理模型的一部分 [1] ，并且是几种场论的解 [2] 。特别是，结绳解的稳定性 [3] 对于拓扑量子计算 [4] 、超流体 [5] 、量子物质 [6, 7] 、液晶 [8] 、物质控制 [9] 、信息传输和存储 [10] 以及微加工 [11] 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 [12] ，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 [10,13,14] and phase singularities [15] in the paraxial regime. In particular, optical-vortex knots can be generated with two-dimensional phase masks. In the experimental implementation of Ref. [15], these structures were generated with axial lengths of a few centimeters. A similar approach has been used to generate acoustic vortex knots [16] with an axial length of about 40 cm. So far, the smallest optical-vortex knots (termed “ultrasmall”) have axial spatial scales of about 240⁢𝜆 [17], spanning a volume of 106 𝜇⁢m3 ∼3.94 ×106⁢𝜆3, 6 orders of magnitude smaller than the previous ones [15]. Several applications with optical-vortex knots have been implemented at macroscopic scales, such as the entanglement between topological features in a vortex link with single photons [4] and information transmission and storage [10]. Hence, these and other applications can be miniaturized to the fundamental optical-wavelength scale 𝜆 provided that the knotted vortex fields reach those dimensions. Wavelength-sized optical-vortex knots could be even more important for future experiments or applications that require interaction between these fields and a few atoms [18].
稳定的光学结结构已经在旁轴区域中通过极化 [10, 13, 14] 和相位奇点 [15] 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 [15] 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 [16] 。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 240⁢𝜆 [17] ，其体积为 106 𝜇⁢m3 ∼3.94 ×106⁢𝜆3 ，比之前的结构小 6 个数量级 [15] 。已经在宏观尺度上实现了几个光涡旋结的应用，例如在涡旋链中单光子之间的拓扑特征纠缠 [4] 以及信息传输和存储 [10] 。因此，只要结涡旋场达到这些尺寸，这些及其他应用就可以缩小到基本的光波长尺度 𝜆 。 波长级光涡结在未来需要这些领域与少数原子相互作用的实验或应用中可能更为重要 [18] 。

Theoretically, it has been shown that optical-vortex knots with spatial scales on the order of 𝜆 should be achievable by nonparaxial propagation of a tightly focused structured beam [19–21]. The main problem for experimental realization has been generation of these beams, because measurement of the position of the singularities can be done by locating the intensity minima as in Ref. [17,22]. However, a quantitative comparison with the calculated fields requires measurement of the phase and amplitude, which is not simple for tightly focused fields. Furthermore, the problem is exacerbated by the fact that there are no measurements of tightly focused light fields in small volumes (measurements across multiple spatial transverse planes).
理论上，已经证明具有 𝜆 数量级空间尺度的光涡旋结可以通过非平面传播的紧聚焦结构光束 [19–21] 来实现。实验实现的主要问题是生成这些光束，因为通过定位强度最小值可以测量奇点的位置，如参考文献 [17, 22] 所示。然而，与计算场的定量比较需要测量相位和幅度，这对于紧聚焦场来说并不简单。此外，问题还因在小体积内（跨多个空间横向平面）没有紧聚焦光场的测量而加剧。

Here, we use the angular spectrum calculated in Ref. [19] to create a wavelength-sized optical-vortex knot, combined with a classical interferometric method to measure tightly focused fields with no approximations [23]. We show that it is possible to create a trefoil optical-vortex knot enclosed by a volume ( 𝑥 ×𝑦 ×𝑧) of 1.90×2.19×1.99⁢𝜆3. Our work represents a volume that is almost 6 orders of magnitude smaller compared with the ultrasmall knots [17] and about 12–13 orders of magnitude smaller with respect to the other paraxial realizations [10,15].
在这里，我们使用参考文献 [19] 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 [23] 。我们展示了可以创建一个被体积（ 𝑥 ×𝑦 ×𝑧 ）包围的三叶光涡结（ 1.90×2.19×1.99⁢𝜆3 ）。我们的工作代表了一个体积，几乎比超小结 [17] 小 6 个数量级，并且与其他旁轴实现 [10, 15] 相比小约 12-13 个数量级。

The angular-spectrum representation for a nonparaxial trefoil knot is described in Ref. [19] (another approach in Ref. [20,21]), where the authors have calculated the Fourier transform of their previous [15] 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 [24]. The complex angular-spectrum representation of the nonparaxial trefoil knot in cylindrical coordinates is [19]

where 𝜌𝑘 =(NA⁢𝑘0)−1⁢√𝑘2𝑥+𝑘2𝑦 is the dimensionless radial coordinate, 𝑘0 is the wave number in vacuum, and NA is the numerical aperture of the system. With this definition, 𝜌𝑘 =1.0 at the aperture, 𝜑𝑘 =tan−1⁡(𝑘𝑦/𝑘𝑥), 𝐸0 is the amplitude, 𝑆 is a scale parameter that determines the knot size and 𝜔0 is the dimensionless Gaussian waist.

The simulation considers linearly polarized light (horizontal) imprinted with the complex angular-spectrum representation [Eq. (1)] with 𝑤0 =0.75 and 𝑆 in a range between 0.8 and 1.2, which is tightly focused by a high-NA lens ( NA =1.11, refractive index 𝑛 =1.518). The focused field acquires measurable polarization components in the other directions ( 𝑦 and 𝑧). However, the vortex lines that can exhibit a knotted geometry only appear in the 𝑥-polarization component of the tightly focused field.

Figure 1 shows the calculated optical-vortex lines with their angular spectrum (amplitude and phase) for the cases of 𝑆 =0.8, 1.0, and 1.2 (constant 𝑤0 =0.75). The first case [Fig. 1(a)] with 𝑆 =0.8 results in an isolated trefoil knot with a size ( 𝑥 ×𝑦 ×𝑧) of 2.07 ×2.05 ×2.11⁢𝜆3 in the dominant 𝑥 component, which has about 83.5% of the power ( 16.0% in the 𝑧 component and 0.5% in 𝑦). Changing the scale parameter to 𝑆 =1.0 [Fig. 1(b)] results in a reconnection of vortex lines at one vertex, with about 84.4% of the power ( 15.1% in the 𝑧 component and 0.5% in 𝑦) with a size of 2.04 ×2.02 ×1.75⁢𝜆3. A further increase in the value of 𝑆 to 1.2 [Fig. 1(c)] results in breaking the knot into two separated loops that span a volume of 2.06 ×2.09 ×1.17⁢𝜆3 in the 𝑥 component, with about 85.4% of the power ( 14.2% in the 𝑧 component and 0.4% in 𝑦). Note that the transition from a knot to separated loops when increasing 𝑆 is similar to what has been observed in transient systems [12].

FIG. 1.

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. 2(a). The laser beam ( , waist of ) has a linear polarization state controlled with a half-wave plate (HWP, not shown) and overfills the screen of the phase-only spatial light modulator (SLM, Hamamatsu X-10468-07); in this way, the output from the SLM is a combination of horizontally polarized modulated light (structured beam) and vertically polarized unmodulated light (reference beam). The polarization of both beams is rotated by a second HWP and then both are projected into the horizontal state by a polarizing-beam-splitter (PBS) cube. A telescope (focal lengths and ) reduces the size of both beams, before both are reflected into the back aperture of the microscope objective (MO, Olympus Plan Fluorite) by a beam splitter (BS). The effective NA of the system (controlled with the aperture of the SLM) is 1.11.

FIG. 2.

The reference beam focuses at the imaging plane of the microscope objective [Fig. 2(b)], while the focused modulated beam is displaced axially (above the imaging plane of the microscope objective) by adding the phase of , ( , with and ) at the SLM as described in Ref. [23].

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. 2(b)].

The polarization of the imaged interferograms is selected with a third HWP and a second PBS. A lens projects the image into a USB camera (Thorlabs DCC1545M-GL). We choose to image the dominant -polarization component of the field where the vortex knot can appear.

A. Holograms

The holograms are calculated with the algorithm proposed by E. Bolduc et al. [25] to modulate both the amplitude and the phase of the angular spectrum of the beams. From Eq. (1) in Cartesian coordinates, the required amplitude and phase are and , respectively. Taking into account the amplitude of the Gaussian beam at the SLM, we use the corrected amplitude for the complex modulation. The following expressions are used to encode the amplitude and the phase:

(2)

where displaces the beam axially and replaces the commonly used transverse diffraction grating. Then, we calculate the hologram that is projected at the SLM:

(3)

We use the same parameters of the simulations ( and ) with an NA of , which is controlled by the diameter of the hologram at the SLM.

Figure 3 illustrates the process to encode the fields, the first column is for , the second for , and the last column contains the case . The first two rows (top to bottom) are the amplitude and phase of the angular-spectrum representation of Eq. (1). We observe that the amplitude range decreases with increasing : , , and for , and , respectively. The range in phase for the same cases is , and . The third row depicts the holograms that encode the complex modulation without the axial displacement . We observe that the holograms have very small variations in phase, which makes them very sensitive to perturbations. The holograms in the final row are the ones projected by the SLM and contain the axial displacement . We observe subtle differences for the different values: has a triangular symmetry, which is eroded for the larger of and .

FIG. 3.

B. Measurement

The field is measured at 80 transverse planes axially separated by 40 nm (piezo stage). The axial scan starts in a position of ( ) relative to the waist ( ) and increases until it reaches . The amplitude and the phase of the focused field are extracted with four-phase-shift interferometry at a given axial position with a diverging reference beam [23]. We only measure the component of the electric field that contains the knotted field: , where and represent the amplitude and the phase at the transverse plane at the axial position. The reference beam is . For a monochromatic wave, the intensity ( component) is proportional to , , where is the permittivity and , the speed of light in the medium (units of power per unit area). The proportionality constants are not important, as the intensity is measured at the camera in arbitrary units. In particular, we measure the intensity of four interferograms ( ) that are represented by . Removing the spatial dependence to simplify the notation, , where is the intensity of the reference beam and , , , and are the phase shifts added to each hologram.

The phase and amplitude of the -polarization component of the tightly focused field are directly extracted from the raw data (interferograms ) using the relations

(4)

and

(5)

where the extracted electric field is (the prime indicates that the difference with is essentially a complex constant): and . Close to the optical axis, both and are almost constant due to the large radius of curvature of the diverging reference: , . In this way, and . To obtain the position of the phase singularities the value of the constants is not important but when we compare the measured phases to the calculated ones, we choose to maximize a normalized cross-correlation (C) [26]. The amplitudes (calculated and measured) are normalized and also compared using the C.

Figure 4(a) shows the four interferograms at a height ( ) for , while the reconstructed phase and amplitude are shown in Fig. 4(b). The extracted real and imaginary parts of the -component of the field, and , are shown in Fig. 4(c). Figure 4(d) depicts the zero contours for the real and imaginary parts of the field and the intersections locate the singularities.

FIG. 4.

The extracted phase and amplitude (simulated and measured) of a trefoil knot with and at different axial heights are shown in Figs. 5(a) and 5(b). Note that we label the heights starting from the bottom of the knot. In contrast, the waist of the simulated beam (where the polynomial is defined) is at . In the experiment, this plane is located close to the middle of the axial scan (at in Fig. 5).

FIG. 5.

Figure 5(c) 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 , which is smaller than the calculated one. The slight difference in dimensions might be caused by minor errors in the experiment (aberrations and in the implementation of the complex angular spectrum).

A. Comparing experiment with simulations

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 (after approximately displacements). The measurement is matched to the corresponding simulated cross section that is located at , with . With that reference, we compare the measurements with the simulations (in planes located at , with an integer) with the normalized cross-correlation [26], defined by

(6)

where is the dot product between the vectorized form of the squared array of measurements (A) and simulations (B). In the case of Fig. 5, there is good agreement between measurements and simulations with C mean values of for the phase and for the amplitude. We take the mean value of the selected planes as the representative C of the beam and the standard deviation as the error.

Figures 6(a) and 6(b) show the comparison between the experiment and simulation for ( ). The Cs have values of for the phase [Fig. 6(a)] and for the amplitude [Fig. 6(b)]. The reconstructed vortex lines are shown in Fig. 6(c), where a reconnection is indicated by the arrow. In the simulation, this reconnection happens at the crossing that is at the top left. The transition from a triangular to a circular symmetry in the holograms (Fig. 2) might explain the appearance of the reconnection at a different vertex when in combination with the high sensitivity to perturbations of the complex encoding of the angular spectrum of Eq. (1).

FIG. 6.

Figures 7(a) and 7(b) shows the comparison between the experiment and simulation for ( ), the normalized cross-correlations have values of for the phase [Fig. 7(a)] and for the amplitude [Fig. 7(b)]. The three-dimensional (3D) position of the vortex lines (view from the top) is in Fig. 7(c), which shows the isolated loops.

FIG. 7.

Figures 5–7 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. 5, which corresponds to the knot with and , and expand the transverse range (Fig. 8). Figures 8(a) and 8(b) show the amplitude and phase of the measured central plane in Figs. 5(a) and 5(b). The dots mark the position of the singularities and the arrows point at the singularities that do not belong to the isolated knotted structure. Figure 8(c) shows the 3D structure in the same axial range of (compared with Fig. 5) with expanded transverse sizes of .

FIG. 8.

The isolated knot of Fig. 5 is bounded on the top and bottom by a phase that has a triangular shape. Figure 8(d) shows the measured phases that bound the knots and loops for , , and , showing that the structures are indeed isolated.

Figure 9 shows the measured transition from an isolated knot to separated loops as a function of the scale parameter . Figure 9(a) depicts the trefoil knot (Fig. 5) in 3D with , while Fig. 9(b) shows the knot generated with with a reconnection (indicated by an arrow). A further increase of to 1.2 results in two separated loops [Fig. 9(c)]. All cases are contained in a similar transverse area (approximately 4.0–4.3 ) and have axial dimensions of 1.99, 1.50, and 0.75 for , 1.0, and 1.2, respectively. The measured vortex lines (knots and loops) are slightly smaller than the calculated ones (Fig. 1), which also span transverse areas with the same range of 4.0–4.3 and axial sizes of , 1.75, and 1.17 for , 1.0, and 1.2, respectively. Interestingly, the axial dimensions of the measured and simulated knots and loops are similar to the transverse sizes as calculated in Ref. [19], in contrast to the paraxial realizations [10,15].

FIG. 9.

We demonstrate an experimental realization of optical-vortex knots at the scale of a wavelength. The trefoil knot generated with parameters of and is contained in a volume of , which is and covers a spatial volume of . Furthermore, the reconnections hint at the fundamental size limit of these knotted structures. Future work will involve other polarization states (e.g., circular) and the exploration of other possible angular-spectrum representations [20,21].

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.