Fig. 1. Schematics of the proposed optical knot shaping method. (a) Illustration of a trefoil knot in the braid representation, consisting of two intertwined strands. These strands are defined by the polynomial given by Eq. (2), where N=2, s=s2=1.5, and φ1=0, φ2=π, and h ranges from −π to π. The color scheme highlights the three distinct lobes of the trefoil knot, as shown in panels (b)–(e). (b) The trefoil knot formed through a direct inverse stereographic projection given by Eq. (3) and described by the Milnor polynomial in the new coordinates (x,y,z). The optical knot appears on a torus, corresponding to the stereographic projection of a cylinder shown in panel (a). The upper part of panels (b)–(e) displays the field amplitude and phase patterns at the z=0 plane for the corresponding trefoil knots. (c) An example of a reshaped trefoil knot, represented by a modified Milnor polynomial. The dotted lines trace the original knot from panel (b). Here, each lobe of the knot is shifted along the z-axis, with the red-colored lobe also rotated counterclockwise. The intensity and phase distribution plots show discontinuities along the lobes. (d) An optical trefoil knot created using the adjusted Milnor polynomial shown in panel (c). Further shape refinements of the optical knot may require additional modifications to the polynomial shown by the dark blue arrow. Note that despite discontinuities in the Milnor space, all discontinuities in field intensity and phase are resolved in the optical domain. Once the optical field distribution displays the desired characteristics, it is ready for experimental application. (e) The experimental realization of the modified trefoil knot.

Fig. 2. An example of the new degrees of freedom for an optical Hopf link reshaping through the subject to varying rotation directions. The first column presents the phase distribution at z=0 plane and the corresponding field amplitude distributions shown in the insets. The second column shows a three-dimensional view of the corresponding Hopf links with various modifications of their loops. The last two columns depict the XY and ZX projections, respectively. (a) The original optical Hopf link with the Gaussian envelope waist parameter w=1.6. The angle between the two loop orientations shown by dashed lines is π (gray arrow). (b) A modified Hopf link, where one of the loops (blue) has undergone a counterclockwise rotation. As a result, the angle between the lobes has changed to 5π/6. (c) Both loops of the original Hopf link are rotated around the x-axis. The red loop has been rotated by 2π/9, while the blue loop is rotated by −2π/9. Dashed lines in the last column highlight the change in length along the z-direction, which is now only 27% of the length of the original optical knot L.

Fig. 3. Comparison of the original optical trefoil knot with its reshaped version. (a) The original optical trefoil knot with parameter w=1.3. (b) The reshaped trefoil knot. The reshaping involves a counterclockwise rotation of the right lobe, indicated in red. The first column shows the field amplitudes for both the original (a) and reshaped (b) trefoil knots in the z=0 plane. The second column shows the phase distributions at z=0. Here, dashed lines show the orientation of the lobes. These lines are drawn by connecting the singularity furthest from the center of each lobe (the rightmost lobe marked by a red arrow) to the midpoint between the other two singularities of the same lobe within the z=0 plane (indicated by gray arrows). The rotation of the right lobe is illustrated, with its angle shifting from 2π/3 to 5π/6. The third column offers a three-dimensional perspective of the trefoil knots, highlighting the rotated orientation of the red lobe while showing the unchanged positions of the other two lobes. The last column displays the top view (XY projection) of the trefoil knots. Black dashed lines mark the lobe orientations at z=0 planes, consistent with the orientation indicators in the second column.

Fig. 4. Trefoil knot reshaping aiming at enhanced intensity contrast between optical singularity lines and their alignment along the z-direction. The first and second columns depict the field amplitude and phase distributions, respectively. The third column provides a top view (XY projection) of the trefoils as well as XZ knot projections in the insets, while the fourth column displays their intensity distribution along the y=0 line in the z=0 plane. Purple dashed arrows indicate the positions of the singularities, while red double-edged arrows show the regions with the highest field intensity, separating the two singularities from each other. The maximum intensity values are noted on the panels. Row (a) shows the results of reshaping applied to the trefoil knot, where the three lobes are aligned along the propagation direction z by being rotated around the corresponding black dashed lines. The coefficients of the constituent LG modes for this trefoil knot are given in the following format (l,p): coefficient: (0, 0): 1.20, (0, 1): −2.85, (0, 2): 7.48, (0, 3): −3.83, (3, 0): −4.38, (−3, 1): −0.82. Row (b) shows the trefoil knot configuration obtained using the approach described in Ref. [23] with LG mode coefficients (0, 0): 1.51, (0, 1): −5.06, (0, 2): 7.23, (0, 3): −2.04, (3, 0): −3.97. Comparing the results in (a) and (b) indicates that the contrast between singularities has increased from 0.11 and 0.13 to 0.22 and 0.49, respectively. The XY projections show that the overall spread of singularity lines in case (a) is larger, resulting in greater distances between the singularity lines over the whole knot volume. Moreover, as seen in the insets with ZX-projections, the singularity lines of the trefoil in (a) are more aligned with the z-direction, with most of the line bends occurring at the very ends of the trefoil knot.

Fig. 5. Experimental setup. M stands for the mirror, L for the lens, I for the iris diaphragm, BS for the beam splitter, SLM for the spatial light modulator, and CMOS for the complementary metal oxide semiconductor camera.

Fig. 6. Experimental realization of optical knots. Panels (a) and (b) display the original optical Hopf link (with the Gaussian beam waist parameter w=1.6) and trefoil knot (w=1.3), respectively. Panels (c) and (d) show reshaped versions of these knots, with one lobe rotated by 4.8π/6 and 5.2π/6 radians counterclockwise. Panel (e) illustrates the reshaping applied to the trefoil knot for the case shown in Fig. 4(a). Panel (g) shows the corresponding field intensity along the y=z=0 line, with red arrows indicating positions of the highest intensity that separate the singularities. The values of these intensities are displayed above the arrows. Panels (f) and (h) present the optimized configuration of the trefoil knot, based on Ref. [23], analogous to Fig. 4(b). Note that the intensity contrast between the singularities in the trefoil knot (e), (g) is higher, consistent with the theoretical results shown in Fig. 4. For the panels showing the measured three-dimensional structures, colored lines were added to guide the dots and facilitate the visualization of the knots. This helps to observe the close-to-straight parallel singularity lines with respect to the measured cross-sections often seen at the top and bottom ends of the structures.

Fig. 7. The example of a Hopf link reshaping. Panel (a) displays the phase, amplitude (inset), and singularity line structure of an optical Hopf link with one of the loops being rotated (blue color) without any z-shifts. This transformation of the Hopf link is described by the corresponding Milnor representation Eq. (A6). Panel (b) illustrates the ZY-projection of the same rotated Hopf link. Panel (c) shows the ZY-projection of the unmodified Hopf link. Panel (d) presents the ZY-projection of the Hopf link with both lobes shifted along the z direction, as indicated by the green arrows. The expression for this Hopf link (d) is described by the Milnor polynomial Eq. (A11). Black dashed lines in panels (b)–(d) demonstrate the additional shifts of the loops along the propagation direction (z), highlighting that the position is almost identical for both the standard Hopf link (c) and the rotated one with shifts (d), while it is strongly different for the rotated link without shifts (b).

Fig. 8. Illustration of the effect of LG decomposition on the Milnor polynomial corresponding to the optical trefoil knot, with one of the lobes being rotated. Panel (a) represents the field amplitude and phase of the Milnor polynomial Eq. (D3), while panel (b) displays the optical field obtained from the LG decomposition Eq. (A6) of this Milnor polynomial. Panel (c) shows the amplitude of the modes resulting from the LG decomposition Eq. (A6).

Fig. 9. Demonstration of the z-shift in the trefoil knot lobes. Panel (a) displays the field of the optical trefoil knot with one of its rotated lobes (indicated in red) without additional z-shifts. This knot reshaping is described by the corresponding Milnor representation Eq. (D3). Panel (b) depicts ZY-projection of this trefoil knot’s 3D structure. Panel (c) illustrates the ZY-projection of the unmodified trefoil knot. Panel (d) presents the modified knot with extra z-shifts, as green arrows indicate. Black dashed lines demonstrate the additional shifts of the lobes along the propagation direction z, highlighting that the position is now identical for both the standard trefoil knot (c) and the rotated one with the z-shifts (d).

Fig. 10. An example of the reshaping of the optical trefoil knot, emphasizing the enhanced intensity contrast between optical singularity lines and their alignment along the z-direction. The first and second columns show the field amplitude and phase distributions. The third column offers a top view (XY-projection) of the trefoils, while the fourth column presents the intensity distribution along the y=0 line in the z=0 plane. Purple arrows pinpoint the positions of the singularities, and red double-edged arrows highlight areas with the highest intensity separation. The maximum intensity values are indicated on the panels. Row (a) demonstrates the modifications applied to the trefoil knot, wherein the three lobes are aligned along the propagation direction z and rotated around their axis, indicated by the black dashed lines. The coefficients of the LG modes (l, p) for this knot configuration are (0, 0): 1.20, (0, 1): –2.85, (0, 2): 7.48, (0, 3): –3.83, (3, 0): –4.38. A gray dashed box in this row accentuates the improved alignment of singularity lines along the z-direction. Row (b) shows the optimized trefoil knot configuration based on Ref. [23], with LG mode coefficients (0, 0): 1.51, (0, 1): –5.06, (0, 2): 7.23, (0, 3): –2.04, (3, 0): –3.97. This serves as a baseline for comparison. In our modified version, the contrast between singularities has increased from 0.11 and 0.13 in the original to 0.19 and 0.34, and the extension of singularity lines along the z-direction is more pronounced.