Optical tweezers, pioneered by Arthur Ashkin in the mid-1980s [1], utilize a single tightly focused laser beam to create strong gradient force for trapping. Nowadays, optical tweezers have widely been used in various areas such as physics, chemistry and biology in particular. Based on the choice of different laser beam modes, optical tweezers can trap either high-index or low-index particles. For example, a conventional Gaussian beam traps a high-index particle with refractive index higher than that of the surroundings. The low-index particles can be trapped at the dark core of a vortex beam [2], which is also commonly known as a helical beam. Apart from the ability of trapping low-index particles, vortex beams usually carry orbital angular momentum which can be transferred to the trapped particles and therefore induces rotation [3, 4].

In recent years, along with the applications of using optical vortices for optical trapping, modified helical beams have been proposed and studied as novel beams for the generation of optical vortices in certain shapes rather than the conventional doughnut shape [5–7]. Curtis and Grier demonstrated an approach to generating modified helical beam with Lissajous pattern in the focal plane by adding a modulation part to the phase function [5]. It is noted that this work was implemented in an implicit way that the modified helical beams were only formed when the phase function was altered by changing the depth and orientation constants for the Lissajous pattern. Hence, the intensity patterns of the modified helical modes were only obtained ambiguously by varying the constants of the Lissajous pattern. However, a direct and deterministic technique for generation of helical beams with various specific intensity patterns explicitly will be more useful in practice. In our previous work in Ref. 6, it has been shown that by using a spatial light modulator (SLM) for a phase-only hologram a modified helical beam embedded with a dark core of optical vortex in a desired shape can be reconstructed at a fixed propagation distance when the Fresnel diffraction is taken into account. Furthermore, the modified helical beam was generated by a micro-fabricated spiral phase plate and it was then employed for optical trapping in Ref. 7 recently. It is noted, however, that in many applications it is required to focus the helical beam onto a sample in a microscope. Hence, the modified helical beams generated at a fixed distance could encounter some problems in terms of imaging distance and the shape of the desired dark core of optical vortices in the optical tweezers configuration. Therefore this paper introduces a deterministic method to show how to obtain a modified helical beam with specified dark core shape as a far-field pattern in the focal plane of a focusing lens.

A conventional helical beam can be approximated by the phase function exp(ilθ), where the integer l is the topological charge and θ is the azimuth angle in the polar coordinates. The phase structure represented by the phase function is a |l|-started helicoids and the sign of l determines the spiral direction. The phase function of a modified helical beam can be written in a generalized form of exp[iφ(θ)], where φ(θ) represents the phase values and φ(θ)=lθ in the case of conventional helical beam. According to Ref. [5], the radius of the maximum intensity of optical vortex at certain azimuth angle in the focal plane after the modified helical beam passing through a focusing lens is

where a_I and b_I are constants depending on the beam’s radial amplitude profile, λ is the wavelength of light and NA is the numerical aperture of the focusing lens. Obviously, R_I (θ) forms a close loop about the origin. Compared to the circumference of dark core of optical vortex, the maximum intensity loop R_I (θ) is not easy to be visualized in terms of an intensity pattern. So the main purpose of this paper is to realize optical vortex with a desired dark core shape. Apparently the perimeter of the dark core is not exactly the loop R_I (θ). For the sake of brevity, the perimeter of the dark core is referred to as the perimeter below. Since the experimental result in Ref. 5 indicated that the perimeter almost kept the same shape as the maximum intensity loop, it is reasonable to suggest that the perimeter of the dark core R(θ) is also dependent on the phase gradient d φ(θ)dθ linearly

where a and b are constants. For conventional helical beams, since the phase gradient is determined as a constant l, the perimeter is formed in a circular shape. For modified helical beams, since the phase values can be modified to increase by the azimuth angle in a nonlinear fashion, it results in different local radii at different angles. The term of aλ(b×NA) in Eq. (2) can be neglected during the calculation of the specific phase function because it does not affect the shape of the perimeter but only the size, whereas the size can be altered by choosing different focusing lens later.

Rewrite Eq. (2)

Given phase function φ(θ) in a range between 0 and l×2π, where l is the overall topological charge of the modified optical vortex, and in order to keep the overall topological charge as an integer l, we assume

Now, if a certain shape of the perimeter R(θ) is determined in Eq. (4), then the phase function φ(θ) can be expressed as

where we assume φ(0)=0.

In this paper, without losing generality, only polygonal perimeter is discussed because polygons approximate most of the useful shapes. Consider a generalized polygon with n vertices (P ₁, P ₂,…P _n-1, P_n ). For the sake of simplicity, vertex P ₁ is rotated onto the axis θ=0.

 

Fig. 1. Desired polygonal perimeter R(θ) of dark core

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The polygonal perimeter can be expressed as follows:

Insert Eq. (6) into Eq. (5), the phase function is determined by

where

$I\left(\theta \right)=\{\begin{array}{cc}{D}_{1,2}\left[\ln \mid \tan \left(\frac{\theta -\alpha }{2}\right)\mid -\ln \mid \tan \left(\frac{\alpha }{2}\right)\mid \right]& \mathrm{for}0\le \theta <{\theta }_2\\ D_{2,3}\ln \mid \tan \left(\frac{\theta -\alpha }{2}\right)\mid -D_{1,2}\ln \mid \tan \left(\frac{\alpha }{2}\right)\mid +\left(D_{1,2}-D_{2,3}\right)\ln \mid \tan \left(\frac{{\theta }_2-\alpha }{2}\right)\mid & \mathrm{for}{\theta }_2\le \theta <{\theta }_3\\ ⋮& \\ D_{i,i+1}\ln \mid \tan \left(\frac{\theta -\alpha }{2}\right)\mid -D_{1,2}\ln \mid \tan \left(\frac{\alpha }{2}\right)\mid +\sum _{m=1}^{i-1}\left(D_{m,m+1}-D_{m+1,m+2}\right)\ln \mid \tan \left(\frac{{\theta }_{m+1}-\alpha }{2}\right)\mid & \mathrm{for}{\theta }_i\le \theta <{\theta }_{i+1}\\ ⋮& \\ D_{n-1,n}\ln \mid \tan \left(\frac{\theta -\alpha }{2}\right)\mid -D_{1,2}\ln \mid \tan \left(\frac{\alpha }{2}\right)\mid +\sum _{m=1}^{n-2}\left(D_{m,m+1}-D_{m+1,m+2}\right)\ln \mid \tan \left(\frac{{\theta }_{m+1}-\alpha }{2}\right)\mid & \mathrm{for}{\theta }_{n-1}\le \theta <{\theta }_n\\ D_{n,1}\ln \mid \tan \left(\frac{\theta -\alpha }{2}\right)\mid -D_{1,2}\ln \mid \tan \left(\frac{\alpha }{2}\right)\mid +\left(D_{n-1,n}-D_{n,1}\right)\ln \mid \tan \left(\frac{{\theta }_n-\alpha }{2}\right)\mid +\sum _{m=1}^{n-2}\left(D_{m,m+1}-D_{m+1,m+2}\right)\ln \mid \tan \left(\frac{{\theta }_{m+1}-\alpha }{2}\right)\mid & \mathrm{for}{\theta }_n\le \theta <2\pi \end{array}$,

$\alpha ={\tan }^{-1}\left(\frac{r_2\sin {\theta }_2-r_1\sin {\theta }_1}{r_2\cos {\theta }_2-r_1\cos {\theta }_1}\right)$,

$D_{p,q}=\frac{r_p{r}_q\sin \left({\theta }_p-{\theta }_q\right)}{{\left[{\left(r_q\cos {\theta }_q-r_p\cos {\theta }_p\right)}^2+{\left(r_q\sin {\theta }_q-r_p\sin {\theta }_p\right)}^2\right]}^{1⁄2}}$,

$I_{\mathit{total}}=\left(D_{n,1}-D_{1,2}\right)\ln \mid \tan \left(\frac{\alpha }{2}\right)\mid +\left(D_{n-1,n}-D_{n,1}\right)\ln \mid \tan \left(\frac{{\theta }_n-\alpha }{2}\right)\mid +\sum _{m=1}^{n-2}\left(D_{m,m+1}-D_{m+1,m+2}\right)\ln \mid \tan \left(\frac{{\theta }_{m+1}-\alpha }{2}\right)\mid$

and 2≤i≤n-1.

Now the only undetermined parameter in Eq. (7) is the constant b. Consider the simplest polygon case, i.e. a triangle with l=10. The amplitude of the modified helical beam is given as ψ(r,θ)=exp(-r ²/$w_0^2$)×exp[iφ(θ)], where w ₀ is the beam waist. It is shown in Fig. 2 that the value of the constant b has a significant effort on the shape of the roundness of the reconstructed perimeter in the far field.

 

Fig. 2. Far-field intensity distributions of modified helical beam with different b

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It is noted that as the value of b decreases the shape of the perimeter tends to be a circle. Obviously an optimized value of b is existed and it can be found out as an empirical formula b=-0.3l valid for l≥10 in the simulation. It is also noted that Eq. (3) is only applied to helical beams with l≫1. So Eq. (7) becomes

According to Eq. (8) and the technique discussed in Ref. 8, we explored and implemented a series of computer-generated holograms for reconstruction of the modified helical beams with arbitrary dark core shapes using an SLM. A TEM₀₀ laser beam shining onto a reflective type SLM (Boulder Nonlinear Systems P512 with 512×512 pixels) was imposed by the designated phase function φ(θ). Then the modified helical beam passed through a focusing lens to generate the far-field intensity distribution as shown in Fig. 3 captured by a CCD beam profiler in the focal plane. A constant phase can be added to cancel out the Gouy phase shift effect in order to keep the patterns the same orientation with that of the designs.

 

Fig. 3. Far-field intensity distribution of the modified helical beams with designated shapes of perimeter such as triangle, Greek cross, star and three letters (N, T, and U). The uppermost row shows the desired shapes of perimeter. The middle low shows the far-field distribution of corresponding modified helical beams through simulation. The bottommost is the experimentally observed far-field patterns. All patterns are designed with overall topological charge l=60.

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The far-field pattern is also employed for optical trapping if the modified helical beam is directed into an optical tweezers system. In practice, the modified helical beam generated by the SLM passes through a telescope system and enters into an inverted microscope. The far-field pattern is formed in the focal plane of a high-NA objective lens mounted in the microscope. As a result of the optical trapping demonstrated in Fig. 4, an optical vortex with a triangle shape of perimeter is surrounded by 3µm diameter polystyrene spheres dispersed in water.

 

Fig. 4. Polystyrene spheres are trapped by modified optical vortex with triangle perimeter. Movie (2.5MB) shows eight polystyrene spheres moving along perimeter of triangle.

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Similar to the conventional optical vortex, the modified optical vortex exerts a torque on the trapped particles. All these particles circulated along the bright rim of the modified optical vortex in the same direction that is determined by the sign of l, which indicates the existence of orbital angular momentum. The movie in Fig. 4 shows that the orbital angular momentum induced rotation of polystyrene spheres trapped in the bright rim of triangle shape optical vortex.

In conclusion, a deterministic method for generation of the modified helical beam with the desired far-field pattern is demonstrated in this paper. Using this technique, we are able to modify the intensity distribution of optical vortex to match the targeted particle in optical tweezers and assemble particle with a single static diffractive optical element. Furthermore, an array of modified optical vortex may have the potential to sort particles by shape.