High capacity topological coding based on nested vortex knots and links

Optical knots and links have attracted great attention because of their exotic topological characteristics. Recent investigations have shown that the information encoding based on optical knots could possess robust features against external perturbations. However, as a superior coding scheme, it is also necessary to achieve a high capacity, which is hard to be fulfilled by existing knot-carriers owing to the limit number of associated topological invariants. Thus, how to realize the knot-based information coding with a high capacity is a key problem to be solved. Here, we create a type of nested vortex knot, and show that it can be used to fulfill the robust information coding with a high capacity assisted by a large number of intrinsic topological invariants. In experiments, we design and fabricate metasurface holograms to generate light fields sustaining different kinds of nested vortex links. Furthermore, we verify the feasibility of the high-capacity coding scheme based on those topological optical knots. Our work opens another way to realize the robust and high-capacity optical coding, which may have useful impacts on the field of information transfer and storage.

In this part, we give the detailed derivations on the complex polynomials of the nested vortex knots and links.
In a general nested knot or link, setting that each braid in the i-th generation contains strands. Each strand is numbered with the array = ( 1 , 2 , … , ) and recorded as S , which can be described mathematically in the ( ′ , ′ , ℎ) coordinate system as: ′ (ℎ) = ∑ ℓ ℓ=1 cos� ℓ ℎ + ℓ �, and ′ (ℎ) = ∑ ℓ ℓ=1 cos� ℓ ℎ + ℓ �. Here ℓ represents the initial rotation angle. ℓ is named winding number, which represents the number of turns of the strand around the braid centerline to which it belongs. By introducing the complex coordinates (u, v) as = ′ + ′ and = ℎ , the strands can be expressed as roots of a complex polynomials: . With the stereographic projection = 2 + ( + ) 2 2 + 2 + 1 and = 2 2 + 2 + 1 , the polynomial ( , ) can be converted into a complex field in the ( , , ) coordinate system as Here = � 2 + 2 , represents rotational coordinate in ( , , ) coordinate system. The ( , , ) is called the Milnor polynomial, which contains the nested knotted and linked zero lines. For the case where there are three generations ( = 3) and each generation contain three strands ( = 3), the zero line with a knotted and linked structure contained in ( , , ) is shown in Fig. 1c in main text.
Supplementary Note 2: Milnor polynomial and complex amplitude of the light field for nested knots and links with two generations.
In this part, we give the detailed derivations on the complex polynomials and complex amplitudes of the light field related to the nested vortex knots and links shown in Fig. 1 in main text.
When considering two generations and setting 1 = 3 and 2 = 2 , S 2 can be described mathematically in the ( ′ , ′ , ℎ) coordinate system as With complex coordinates (u, v), the strands can be expressed as roots of a complex polynomials Accordingly, the Milnor polynomial will degenerate into ( , , ) = � � 2 + ( + ) 2 The zero line with a linked structure contained in ( , , ) is shown in Fig. 1e in main text. Because Milnor polynomial divergence as , → ∞, it can not be used to represent the complex amplitude of the light field directly. The correct knot occurs by evolving the field ( , ) = ( , , = 0)( 2 + 1) − 2 2 ⁄ . The process of multiplying ( , , = 0) by sufficiently large powers of ( 2 + 1) is called 'overhomogenization' [1]. It is difficult to obtain the nested knotted and linked structure only by using the degree of freedom of phase singularity distribution. Here, we introduce the degree of freedom of frequency (or wavelength) and decompose the link into three sub-links as shown in Supplementary Figure 1a, which will be generated by three light fields with different wavelengths (λ 1 , λ 2 and λ 3 ), respectively. From Eq. (S5), the complex amplitudes of the light fields are: When only considering one generation and setting 1 = 3 and 1 = 1 , S 1 can be described mathematically in the ( ′ , ′ , ℎ) coordinate system as With complex coordinates (u, v), the strands can be expressed as roots of a complex polynomials: Accordingly, the Milnor polynomial will degenerate into: which contains the link 6 3 3 shown in Fig. 1g in the main text. With the process 'overhomogenization'. The complex amplitude of the light fields should be ( , ) = ( , , = 0)( 2 + 1) − 2 2 ⁄ . We introduce the degree of freedom of frequency (or wavelength) and decompose the linked structure into three rings as shown in Supplementary Figure 2a, which will be generated by three light fields with different wavelengths (λ 1 , λ 2 and λ 3 ), respectively. From Eq. (S9), the complex amplitudes of the light fields are:

Supplementary Note 4: Prime factorization in coding scheme based on the nested knots and links
In this part, we give the detailed derivations on the prime factorization. According to the theoretical description in main text, the coding scheme relies on a pair of numbers ( , ), where is a positive integer, and is a number both related to and to the topological structure. Here the is given by represents the sum of all values of winding numbers.

Supplementary Note 5: Optimizing the geometry of a single pixel of the metasurface.
In this part, we give detailed numerical simulations of the conversion efficiency for the supercell of metasurface holograms with four components. Each multiplexed supercell contains four silicon nanopillars with three different types, where two smallest nanopillars are positioned diagonally and two larger nanopillars locate on the other diagonal. The dimension of the pixel is 500 nm × 500 nm. In addition, lengths and widths of three species equal to (160 nm, 130 nm), (140 nm, 100 nm) and (93 nm, 70 nm), respectively. The height of all nanopillars is 300nm. It is noted that each type of nanopillars responds to a specific wavelength upon the illumination of a light beam containing three different wavelengths ( 1 = 532nm, 2 = 645nm and 3 = 810nm), making the separately manipulation of transmitted phases of three wavelengths become feasible. Due to the fact that each nanopillar has different dimensions along two orthogonal axes, the locally transmitted phase can be engineered by the in-plane orientation angle ϕ of the nanopillar. In this case, when a CPL beam is normally incident on the structure, the transmitted light with the opposite handedness could possess a phase delay ±2ϕ with ±depending on the handedness of the incident beam. Using such a rotation-angle dependent geometric phase, we could design the three-wavelength hologram to achieve three-frequency vortex knots and links.
The numerical calculations for the conversion efficiency of our designed unit under the CPL illumination is shown in Supplementary Figure 3a. We can see that our designed supercell possesses strong responses on three target wavelengths, and the corresponding conversion efficiencies are over 80%, 60% and 40%, respectively. To further demonstrate the frequency-selective characteristics of the designed super-cell, we present the corresponding distributions of light near-field at three wavelengths. At 1 = 532nm, the incident light fields are totally located onto two smallest nanopillars of the supercell, as shown in Supplementary Figure   3b. This indicates that two smallest nanopillars could manipulate the electromagnetic field at 532nm. The spatial distributions of input fields at 2 = 645nm and 3 = 810nm are displayed in Supplementary Figure   3c and Supplementary Figure 3d, respectively. We can see that the associated light fields are mainly located around the second largest and largest nanopillars, respectively. In this case, we could change the rotation angle of these three-type nanopillars to control the transmitted field separately at 1 = 532nm, 2 = 645nm and 3 = 810nm. Therefore, these three types of nanopillars can be considered as three wavelength-dependent field filters. In such a case, each component of the designed metasurface hologram could possess a specific (wavelength-dependent) functionality. Hence, by suitably designing the spatial rotation of nanopillars with different sizes, linked optical vortex rings with three different frequencies could be fulfilled. To embed the nested optical knots into the waist of a Gaussian beam, the diffractive holographic scheme based on the phase-only metasurface hologram can be used. It is proved that the phase-only holograms can be modified to control not only the phase structure of the diffracted beams but their intensity [2]. The general process is summarized as follows. Firstly, the phase distribution of knotted vortex field at the z=0 plane should be added with a suitable blazed diffraction grating to construct the required phase distribution of the designed hologram. In this case, the first-order diffracted energy is angularly separated from the other orders. Then, the desired intensity of the knotted beam in the z=0 plane is applied as a multiplicative mask to the phase distribution of the hologram, acting as a selective beam attenuator imposing the necessary intensity distribution on the first-order diffracted beam.