Curved singular beams for three-dimensional particle manipulation

For decades, singular beams carrying angular momentum have been a topic of considerable interest. Their intriguing applications are ubiquitous in a variety of fields, ranging from optical manipulation to photon entanglement, and from microscopy and coronagraphy to free-space communications, detection of rotating black holes, and even relativistic electrons and strong-field physics. In most applications, however, singular beams travel naturally along a straight line, expanding during linear propagation or breaking up in nonlinear media. Here, we design and demonstrate diffraction-resisting singular beams that travel along arbitrary trajectories in space. These curved beams not only maintain an invariant dark “hole” in the center but also preserve their angular momentum, exhibiting combined features of optical vortex, Bessel, and Airy beams. Furthermore, we observe three-dimensional spiraling of microparticles driven by such fine-shaped dynamical beams. Our findings may open up new avenues for shaped light in various applications.


S2: Detailed theoretical anaylsis of accelerating Bessel-like singular beams
In the following we describe in detail the analysis leading to the design of accelerating singular beams of the higher-order Bessel type with arbitrary trajectories.

Formulation of the problem
We begin with the Fresnel integral of diffraction that describes the paraxial propagation of an optical wave Error! Reference source not found. The main task of the analysis is to determine the phase ( ) , Q ξ η that is required to produce an accelerating vortex beam with a given transverse width and a topological charge . m As a measure of the transverse width of the beam, we use the diameter of the inner low-intensity disk which is defined in Fig. S1.
We now employ ray optics. The equations of the rays follow from the condition of first-order stationarity of the function Figure S1: Real part of a vortex Bessel beam with order m = 6. Indicated is the inner low-intensity disk which is defined at the radius where the argument of the Bessel function is equal to its order.
which are the equations of a ray from the input to the field point. Subsequently, we require that, at an arbitrary transverse plane z , the rays emitted at skew angles with respect to the z axis from a (yet unknown) locus ( ) L z on the input aperture pass from a circle with center ( ) L z is mapped to the distance z, then a two-variable function ( ) , z ξ η is obtained. The latter is critical to finding ( ) , Q ξ η but is yet unknown.

Determination of the phase Q
There are two key steps needed to proceed. The first is the requirement that the phase ( ) , Q ξ η and its first two derivatives are continuous. A necessary requirement is that its mixed second-order partial derivatives should be equal, i.e. Q Q ξη ηξ = , or using Eqs.
where the subscripts , ξ η imply the partial derivatives of the corresponding functions and Equation (6) is a differential one for the unknown functions ( ) where θ is the azimuth coordinate of the point ( ) cos ,sin | cos ,sin cos ,sin where the prime denotes the derivative / . d dz From the obvious relation The gradient , ξ η ϕ ∇ can also be obtained if we require that angles θ and ϕ satisfy ( ) where ( ) w z is another function that is unknown for the moment. From the above equation we Subsequently, we substitute Eqs.
To solve this system, the radius ( ) R z is required. This is found from the field profile in the neighbourhood of the center ( ) which reveals that the beam behaves locally around ( )  We now proceed to solve the system (19), which, due to Eq. (31), simplifies to However, the initial condition that is required to solve this system cannot be determined from we are able to compute the input phase, noting that the only numerical part of the procedure is the solution of Eq. (8) for z through the Newton-Raphson method. An example is shown in Fig. S3 in terms of the input phase and intensity snapshots of the beam. The ray structure for this example is shown in Fig. S2.

A final note
The condition Eq. (33) is a prerequisite for computing the phase through the presented method, ensuring that the circles of Eq. (8) are expanding but never intersecting each other.
However, for trajectories whose acceleration does not approach to zero as z → ∞ , this condition is satisfied only for distances below a certain bound, or max z z ≤ . Beyond this distance, a new trajectory must be defined in order to satisfy the condition, as for example a straight line. The procedure is then similar to that of the zero-order Bessel beams discussed previously in Ref 2.