1. Introduction

Optical tweezers as novel techniques, which exploit the forces exerted by the strongly focused light beam from precisely created wavefronts of light, can trap and manipulate objects. Optical tweezers have acted the extremely important tool for revealing and manipulating the microscopic (or mesoscopic) world. Progress of optical tweezer technique offers new opportunities for some frontiers of fundamental science and applied researches in the fields of biology, physical chemistry, soft condensed matter physics, medicine and so on [1]. In recent years, there has been considerable interest in the generation of laser beam with helical wavefront. Using an azimuthal phase profile of exp(iℓθ), a plane wavefront can be converted into a helical mode, where ℓ is the topological charge and θ is the azimuthal angle; each photon in a helical mode can carry an orbital angular momentum ℓħ [2–5]. Such helical beams no longer focus to points but produce rings; provided that focused strongly enough, they form toroidal torque optical traps known as optical vortices or optical spanners [6–13]. The salient features of optical vortices open up whole new classes of dynamic optical tweezers and result in potentially widespread technological applications. For instance, the rings of the optical vortices could be used to distributedly drive microfabricated gears for creating micrometer-scale motors [1, 14] and arrays of optical vortices have shown a potential ability to assemble colloidal particles into mesoscopic pumps for microfluidic systems [15,16]. Combination of optical vortices and Bessel beams could form the so-called bottle beams, which are useful for trapping very small dark-seeking objects. Optical vortices have also attracted considerable attention in nonlinear optics, owing to both their appearance formation in laser cavities and to their appearance as solitons in self-defocusing media [17]. The realization of these applications often relies on high efficient and flexible creation of different structures of optical vortices. Helical beams can be readily achieved by conventional modes of light utilizing various mode converters [18]; however, the topological charges are very small [4,5]. In contrast, using the computer-generated holograms displayed on a phase spatial light modulator (PSLM) with high resolution can dynamically generate helical modes up to ℓ = 200 [12,13].

This article addresses the issue of how to efficiently create the desired structures of optical vortices by dynamical computer-generated phase mask. The phase masks utilized have the annulus structures, unlike the simple spoke-like structures in reference [11]. We find the optimal annulus width, at which the subsidiary rings of the optical vortices produced in the focal plane of a lens can be efficiently suppressed. Especially, we also present a valuable method for designing multi-ring structure of optical vortices with the same or opposite angular momentum directions as well as specialized interferometric vortex patterns. We also consider the practical case of Gaussian beam illumination and find that there exists the optimal spot size, which ensures the highest energy/power efficiency for generating the best contrast principal ring.

2. Theory

A helical mode beam can be characterized by the phase distribution of ℓθ-2πint(ℓθ/2π) and generated by a dynamic PSLM, where the function int(x) gives the integer part of x. The field in the focal plane of a lens with a focal length f can be obtained by implementing the Fourier transform to the field on the plane of the PSLM, that is,

where (r,θ) and (ρ,ϕ) are the polar coordinate systems in the front and back focal plane, respectively, J _ℓ (x) is the ℓ -th order Bessel function of the first kind, Γ() is the Gamma function, ₁ F ₂ [ ] is the generalized hypergeometric function, R ₀ is the outer radius of the optical vortex phase mask, λ is the wavelength of the illuminating beam and the dimensionless parameter p = πR ₀ρ/(λf). Fig. 1(a) shows a typical example of a simple spoke-like structure phase mask encoded by ℓ = 40 [6] and Fig. 1(b) is the observed image of the optical vortex in the focal plane. It is found that the image consists of an inner principal ring and some outer subsidiary rings, except for the zero diffraction order at the center. We can assert from Eq. (1) and the properties of the Bessel function that the inner principal ring of the optical vortex is mainly from the contribution of the light coming from the outer section of the phase mask and the inner section of the beam results mainly in the outer subsidiary rings. In many practical applications, however, the subsidiary rings are useless and have to avoid, because they result in the low contrast principal ring and form the origin of disturbance. We believe that removing the central section of the phase mask to form an annulus structure and choosing a suitable annulus width would be beneficial to suppress those subsidiary rings.

 

Fig. 1. (a) Spoke-like structure phase mask with ℓ - 40 and (b) corresponding image of optical vortex produced.

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To highlight our scenario, we now consider an annulus structure phase mask with an unchanged outer radius of R ₀ and a changeable inner radius of R ₁ ranging from 0 to R ₀. The field intensity distribution in the focal plane along the radial direction can be calculated from Eq. (1), provided that the lower limit of the integral is replaced by R ₁ instead of zero. For the case of R ₀ = 256 pixels and ℓ = 40 , the calculated results are depicted in Figs. 2(a), (b), (c) and (d) at R ₁ = 0, 125, 200 and 230 pixels, respectively. The parameters used in our theoretical analyses, for example, the wavelength λ = 632.8 nm and the focal length f = 730 mm, are the same as those in the experiments below. It is evident that the too small or too large R ₁ gives rise to the inferior contrast of the inner principal ring, owing to the appearance of the relative strong outer subsidiary rings as background. Consequently, we believe that there should exist an optimal inner radius R ₁ (in other words, an optimal annulus width), at which value the subsidiary rings are significantly suppressed and the inner principal ring exhibits the best contrast.

 

Fig. 2. Intensity distribution of ℓ = 40 optical vortices created by the annulus structure phase mask of R ₀ = 256 pixels, along the radial direction when R ₁ is (a) 0, (b) 125, (c) 200 and (d) 230 pixels, respectively.

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Fig. 3. Dependence of the peak intensity of the principal ring (a) and the first subsidiary ring (b) on the annulus width at different topological charges. The solid, dash, short dash, dash-dot, and dash-dot-dot lines are ℓ = 10, 20, 30, 40, and 50, respectively.

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To find the optimal annulus width of the phase mask and the suitable criterion, we investigate the dependences of the peak intensities of the principal ring and the first subsidiary ring on the annulus widths of ΔR = R ₀ -R ₁ at five different values of ℓ = 10, 20, 30, 40 and 50. Fig. 3 gives the calculated results. As shown in Fig. 3(a), the peak intensity of the principal ring at first increases monotonously as ΔR expanding and then arrives at its maximum saturation value once ΔR is over a certain critical value. In contrast, with increasing ΔR, the peak intensity of the first subsidiary ring at first rapidly grows up to its sub-maximum, then drops down to a minimum (near zero) at a characteristic annulus width (which is marked as ΔR_opt) and finally arrives at its maximum again, as shown in Fig. 3(b). ΔR_opt is called the optimal annulus width, originating from the two dominant factors: (i) the first subsidiary ring is completely subdued and the other subsidiary rings have the extremely low intensity simultaneously, making the principal ring exhibits the best contrast and (ii) the principal ring also has the very high intensity being at least about 90% of the maximum value. The criterion for finding this optimal annulus width is that the intensity of the first subsidiary ring has the lowest level. Based on this criterion and further numerical fitting, a simple scaling relation of ΔR_opt. with R ₀ and ℓ is found that

It allows us to design the annulus phase mask with the helical wavefront and the optimal annulus width. The optimal annulus width ΔR_opt increases linearly with the increase of the outer radius R ₀ of the annulus phase mask while decreases nonlinearly with the topological charge ℓ increasing; that is to say, the large R ₀ requires the large ΔR_opt, while the large ℓ gives rise to the small ΔR_opt. In addition, we also reveal the dependence of the radius ρ _P of the principal ring of optical vortex on the annulus width ΔR, which exhibits the analogous behavior as the peak intensity of the principal ring. When ΔR > ΔR_opt, ρ _P becomes to be independent of ΔR. In fact, this is a very important nature and the third reason choosing the above criterion of the optimal annulus width, and allows to create a simple relation of ρ _P with R ₀ and ℓ, that is,

Obviously, the adoption of the optimal annulus structure still keeps the feature of linear scaling of ρ _P versus ℓ, like in the traditional spoke-like structure [6]. ρ _P can be controlled by choosing R ₀ and ℓ. The direction (clockwise or anti-clockwise) of the orbital angular momentum carried on the optical vortex can be also arbitrarily selected, by changing the sign of ℓ.

In the above theoretical analyses, our consideration was confined to the case of the uniform illuminating beam (a plane wave) for simplicity. As viewed from practical applications, however, the obtaining of ideal uniform plane wave is very difficult and even impossible. Despite we can generate the near uniform plane wave, for instance, through expanding beam and picking up just the center of the beam, the massive power of beam used is cut off. Therefore, the investigation is of great importance for the Gaussian beam illumination, because the output beams of many lasers have the near Gaussian profiles, such as He-Ne lasers. For the theoretical treatment of Gaussian beam illumination, we can divide a helical phase mask into N narrow annuluses with the same width of Δr; Eq. (1) can be rewritten as

where u ₀(r,ω)) is the amplitude of the illuminating Gaussian beam

Here A ₀ is a constant and ω is the spot size (half beam size on the level of e ^-2) of the Gaussian beam at the input plane. In our consideration the illuminating beam is always collimated; in this condition, ω is approximately equal to the beam waist. On the basis of the properties of the ℓ th order Bessel function, we can still infer that the inner principal ring in the focal plane is mainly produced by the light coming from the outer section of the phase mask, despite the illuminating beam is a Gaussian beam instead of a uniform plane wave. The inner section of the phase mask has the main contribution to the outer subsidiary rings, which could be depressed by adopting the annulus phase mask in the same way described above. Our numerical analyses and related experiments demonstrated that the expression Eq. (2) for determining the optimal annulus width is still available for the Gaussian beam illumination, especially when the spot size is not less than the radius of the phase mask. The interesting and important problem is the energy/power efficiency for producing the optical vortices. We discuss first two extreme cases: (i) the spot size ω →∞ corresponds to the fact that the Gaussian beam must be infinitely expanded (i.e., it becomes a uniform plane wave), it is clear that the available energy/power efficiency must be extremely low; (ii) the spot size ω <R ₀ gives rise to that most of the input energy passes through the central part and forms the useless central spot. Hence we can assert the fact that there should exist an optimal spot size for the maximum energy/power efficiency when a Gaussian beam illuminates the annulus phase mask. Through numerically finding the maximum intensity of the principal ring by Eq. (4), we obtained the optimal spot size ω_opt for the collimated Gaussian beam illumination to be

where R_m is the average radius of the annulus phase mask with the optimal annulus width ΔR_opt, i.e., R_m =R ₀-ΔR_opt/2 or = R ₁+ΔR_opt/2. We give a simulated test regarding the dependence of the intensity of the principal ring on the spot size of Gaussian beam, where an optimal annulus phase mask (R ₀ = 256 pixels, ℓ = 40 and ΔR_opt = 50 pixels) is designed by Eq. (2). The solid line in Fig. 4 shows our simulated result. We can see that when the spot size of Gaussian beam is near 330 pixels, the intensity of the principal ring in the focal plane arrives at its maximum, which is in good agreement with the value predicted by Eq. (6).

 

Fig. 4. Variation of the peak intensity of the principal ring with the spot size of Gaussian beam when R ₀ =256 pixels, ℓ = 40 and ΔR=ΔR_opt = 50 pixels. The solid line is the theoretical values, while the circles are the experimental results.

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3. Experiment

In all our experiments, a He-Ne laser at λ = 632.8 nm was used as light source and the designed phase mask was displayed on a twisted nematic liquid crystal display (LCD) (with 1024×768 pixels, each pixel with 18μm×18μm) and a lens of focal length f = 730 mm was used. Phase-only operation for the LCD can be achieved by use of a combination of two polarizers and two quarter-wavelength plates [19]. The output beam of the He-Ne laser has a good Gaussian profile, which has been characterized by the beam profile analyzer. First, we investigate experimentally the dependence of the energy/power efficiency of the principal ring on the spot size of Gaussian beam, for the optimal annulus phase mask with R ₀ = 256 pixels, ℓ = 40 and ΔR = ΔR_opt = 50 pixels. The circles in Fig. 4 show the measured results, which agree with the theory. In all the experiments below we adjust the spot size of the input Gaussian beam to satisfy the optimal condition of Eq. (6). The optimal annuls phase mask (as shown in Fig. 5(a)) was displayed on the LCD. Fig. 5(b) and (c) are the simulated intensity and phase distribution of the optical vortex ring in the focal plane, respectively. The center spot in Fig.5 (b) is formed by the light coming from the center section of the annular phase mask, because we did not block it in this simulation. The simulated result indicates that the radius of the principal ring produced in the focal plane is well in agreement with the value predicted by Eq. (3). The thick dashed circle in Fig. 5(c) gives the corresponding position of the principal ring in the focal plane, which indicates the helical wavefront of the vortex ring. Fig. 5(d) shows the observed optical vortex ring in real experiments, which is consistent with the simulated result shown in Fig. 5(b). Compared with Fig. 1(b) of the conventional spokelike structure phase mask with the same R ₀ and ℓ, it can be seen that the subsidiary rings was successfully suppressed and the principal ring becomes more clear.

 

Fig. 5. (a) An example of the helical mode annulus phase mask, (b) simulated intensity pattern of optical vortex, (c) simulated phase pattern of optical vortex, and (d) experimentally observed intensity pattern of optical vortex.

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The idea mentioned above provides a valid avenue to design complicated vortex structures such as multi-ring structures and interferometric structures. As an example here, we introduce the method for designing a bi-annulus phase mask to construct a bi-optical vortex rings. The procedure is as follows: (i) Choosing the design parameters: the maximum outer radius R ₀ of phase mask, the wavelength λ, the focal length f, the desired radii ρ _P(ℓ₁) and ρ _P(ℓ₂) of two principal rings of the bi-optical vortices. R ₀ is usually determined by the size of PSLM used in experiments. ρ _P(ℓ₁) and ρ _P(ℓ₂) depend upon the experimental aims. (ii) Calculating the topological charge ℓ₁ of the outer annulus of the phase mask according to Eq. (3) and its inner radius R ₁ =R ₀ -ΔR (ℓ₁) by Eq. (2). (iii) Calculating the topological charge ℓ₂ of the inner annulus using Eq. (3) and its inner radius R ₂ = R ₁-ΔR(ℓ₂) by Eq. (2), but R ₀ and ρ _P(ℓ₁) should be replaced by R ₁ and ρ _P(ℓ₂), respectively.

Figure 6(a) shows a bi-annulus structure phase mask designed by the procedure mentioned above, where R ₀ =256 pixels, ρ _p(ℓ₁) = 750 μ m and ρ _P(ℓ₂) = 550 μ m. We estimate ℓ₁ ≈41, R ₁ -207 pixels, ℓ₂ ≈ 20 and R ₂ -149 pixels, respectively. Figure 6(b) indicates the observed pattern of bi-ring optical vortex produced in the focal plane. In this bi- ring structure situation, although the spot size of the incident Gaussion beam cannot be directly determined by Eq. (6), the calculated values offer a useful instruction for experiments. In our experiment the spot size is approximately equal to the optimal spot size calculated only according to the parameters of the outer annulus, which makes a smaller difference between the intensities of the two vortex rings. Because the two principal rings are from two different annulus structures of the phase mask, their orbital angular momentums carried could be set to the same or opposite directions, depending on the relative signs of the topological charges ℓ₁ and ℓ₂ of the two annulus. Figure 6(a) and (b) are the case in which the orbital angular momentums carried by the two principal rings have the opposite directions. Figure 6(c) and (d) are the other designed example, the difference from the former is that the latter has ρ _P(ℓ₁) - ρ _P(ℓ₂) = 750 μ m and the same sign of ℓ ₁ = 41 and ℓ ₂ = 33. In this situation, the two vortex rings produced by this bi-annulus structure are overlapped each other in the focal plane. The interference between these overlapped principal rings produces a new optical vortex distribution. It is clear that more complicated structure phase mask could be created with the present method.

 

Fig. 6. (a) and (c) are two examples of bi-annulus phase masks, and (b) and (d) are the respective optical vortices produced. Bi-optical vortices have the opposite angular momentum directions in Fig. 6(b) and have the same radii and directions in Fig. 6(d).

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4. Discussion

We find from Figs. 1, 5 and 6 that there occurs a phenomenon of extremely weak azimuthal periodic modulation in the experimentally observed intensity distribution of optical vortices. We think that this kind of periodic modulation should be mainly imputed to the phase deviation from the perfect phase of 2π at the phase step points of phase mask. For the convenience of description, we define the phase deviation Δϕ= ϕ - 2π, where ϕ is the maximum phase at the phase step points. For an ideal programmable PSLM, ϕ is exactly equal to 2π, that is, Δϕ = 0 . In most practical situations, however, the phase deviation Δϕ is often either smaller or larger than 0, which results in the periodic modulation of the reconstructed optical vortices.

 

Fig. 7. Observed patterns of optical vortices in experiments for the non-perfect phase masks with the phase variation ranging from zero to 1.5π. (a), (b), (c) and (d) correspond to Fig. 1, Fig. 5, Fig. 6(b) and Fig. 6(d), respectively.

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Fig. 8. Dependence of the depth of the azimuthal modulation on the phase deviation for the optimal annulus phase mask with R ₀ = 256 pixels, ℓ = 40 and ΔR_opt((40) = 50 pixels.

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To confirm this mechanism, we choose on purpose the gray levels of the phase masks involved in Fig. 1, Fig. 5 and Fig. 6, making Δϕ = -0.5π when these phase masks are displayed onto the LCD described above. Figures 7(a), (b), (c) and (d) show the observed images of the optical vortex rings produced in the focal plane. Compared with Fig. 1(b), Fig. 5(d), Fig. 6(b) and Fig. 6(d), the corresponding optical vortices shown in Fig.7 exhibit very serious intensity modulations, which are obvious because of the introduction of the large phase deviation. To further verify this idea, we study the effect of the phase deviation Δϕ on the depth of the azimuthal modulation in detail; as an example, the theoretical result is plotted in Fig. 8 for the optimal annalus phase mask with R ₀ = 256 pixels, ℓ = 40 and ΔR_opt (40) = 50 pixels. We find that the depth of the azimuthal modulation increases nearly with exacerbating the phase deviation δϕ. This fact could be useful in adjusting the modulation depth for studying Brownnian transport in modulated potentials.

5. Conclusion

We have presented and demonstrated the idea of the optimal annulus structure phase mask, which enables us to construct optical vortex rings with the best contrast principal ring and the largely suppressed subsidiary rings. The idea can be extended to complicated multi-annulus structures, which provide a novel realization of the impossible multi-optical vortices in the simple spoke structure. In such multi-optical vortex rings, they can be designed to carry the same or different orbital angular momentums in the magnitude and direction. The novel optical ratchet potentials in apodized single or multi-optical vortex rings can be controlled through choosing the phase deviations at the phase step points in the phase mask. We believe that all the novel properties will make the multi-optical vortices as the dynamic multiple optical tweezers into a useful tool for scientific research and technological applications. We also found the optimal spot size for obtaining the highest energy/power efficiency when the Gaussian beam illumination is used.