## Abstract

In an optical vortex (OV) field, the orbital angular momentum (OAM) distribution strongly depends on the intensity, which results in difficulty in OAM independent modulation. To overcome this limitation, we propose a grafted optical vortex (GOV) via spiral phase reconstruction of two or more OVs with different topological charges (TCs). To remain the annular shape of the GOV’s intensity, the Dirac δ-function is employed to restrict the energy in a ring. Theoretical analysis and manipulation experiments of polystyrene microspheres show that the magnitude and direction of the GOV’s local OAM are controllable by modulating the grafted TCs while the intensity remains constant. The results of this work provide an ingenious method to control the local tangential force on the light ring, which will promote potential applications in optical trapping and rotating micro-particles.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In 1992, L. Allen first confirmed that optical vortices (OVs) carry orbital angular momentum (OAM) [1]. The OAM is given by *m*ħ per photon, where *m* is the topological charge (TC) of the OV and ħ is the reduced Planck constant. Subsequently, OVs have been studied in diverse applications, such as micro-particle manipulation [2,3], optical measurements [4–6], image processing [7], OAM-based optical communications [8,9], and others [10].

When manipulating micro-particles via OVs in a transverse plane, the gradient force of the beam is used to trap them, and the OAM is used as an optical wrench to rotate them. After trapping the micro-particles, versatile distributions of the OAM will facilitate more types of manipulations. For a conventional OV (COV), the OAM distribution is a ring, which is the same as its intensity distribution. To control the OAM distribution, a direct method is via the adjustment of the OV’s TC [1]. However, this method only results in uniform OAM distribution on a ring. To obtain versatile OAM distributions, noncanonical optical vortices are proposed, which can generate spiral OAM distributions [11], 3D free style structured OAM distributions [12], and non-uniform hollow circular OAM distributions [13]. Although they can facilitate versatile shapes, it is difficult to independently modulate the local magnitude of the OAM. Thus, an advanced method is employed to control the OAM distribution via mode transformation, such as asymmetric Bessel modes [14], elliptical vortex Hermite–Gaussian beams [15], and elliptic perfect optical vortices [16,17].

However, in the process of summarizing the existing references on OAM modulation, we find that the OAM distribution and magnitude are essentially modulated via a change in the intensity distribution. As a result, the shortcoming of these methods is the strong dependence of the OAM distribution on the intensity. Further, the direction of the local OAM has never been freely modulated to our best knowledge. Therefore, to facilitate more flexible control of the micro-particle rotation movement on the light ring, it is necessary to develop a novel OV with a constant intensity and controllable OAM distribution, including control of the local OAM direction.

In this regard, we hereby propose a novel OV, referred to as a grafted optical vortex (GOV). The GOV has a constant intensity distribution while its local OAM magnitude and direction are controllable.

## 2. Theoretical design of the grafted optical vortex

The generation process of the GOV involves grafting two or more spiral phase of the OVs. As shown in the top row of Fig. 1, the spiral phase generation of the GOV is demonstrated. One spiral phase with a TC of *m*_{1} is cut away from the lower half as the “scion”. The other spiral phase with a TC of *m*_{2} is cut away from the upper half as the “rootstock”. Then, these two halves of the spiral phases are grafted as shown in the Fig. 1(c). However, the radius of the OV becomes larger with an increase of the TC, although the spiral phases are well-grafted. As a result, the intensity of the “scion” and “rootstock” do not graft together. To overcome this drawback, we use a Dirac δ-function to restrict the intensity to a single ring. Finally, the GOV can be expressed by the following formula,

*r*,

*θ*) are the polar coordinates,

*r*

_{0}is the radius of the GOV, rect(.) is the rectangular function,

*N*and

*m*are the amount and TCs of the spiral phases, respectively. For a graft of two vortices i.e.

_{n’}*N*= 2,

*m*

_{1}and

*m*

_{2}correspond to the TCs of the “scion” and “rootstock”, respectively. If

*m*

_{1}=

*m*

_{2}, the GOV will revert back to a COV. For an OV beam, TC is a significant parameter. Therefore, the equivalent TC (ETC) of the GOV is given by

*M*= (

*m*

_{1}+

*m*

_{2})/2 according to the definition of the TC [18,19]. In this analysis, the Dirac δ-function is employed to ensure that OV beams with different TCs have the same radius of the intensity rings [20].

Now, let us decompose the OAM states of the GOV to analyze the difference compared to the COV. The bottom row of Fig. 1 shows the OAM state decomposition of two GOV with an integer and half-integer ETCs. The two maximal probabilities of the GOVs are distributed at the states of *m* = *m*_{1} and *m* = *m*_{2}, respectively, which is different to the case of the COVs [21,22]. Furthermore, the other OAM states also have probabilities since only the half spiral phase is retained for both the “scion” and the “rootstock”.

## 3. Experimental setup

While experimental realization of an ideal GOV is impossible, it is possible to approximate a GOV beam experimentally. The idea is to use an approximate function instead of the δ-function. In particular, the numerical axicon method using a spatial light modulator (SLM) is the most flexible technique [23]. Using this method, the phase mask is written as

*ρ*,

*φ*) are the polar coordinates in the SLM plane; and

*α*and

*n*are the cone angle and refraction index, respectively.

*D*is the period of the blazed grating used in the mask. The phase mask generation process is illustrated in the top row of Fig. 2.

The schematic of the experimental setup is also shown in Fig. 2. The light beam comes from a solid-state laser (λ = 532 nm, maximal output power 2W, adjustable, Laserwave Co. Ltd). After collimation through a pinhole filter and a lens, the laser beam is modified using the phase mask displayed in a reflective liquid-crystal SLM (HOLOEYE, PLUTO). The aperture A after the pinhole filter is used to obtain the central part of the output beam. A polarizer P1 before SLM is used to modulate a linearly polarized beam and P2 to eliminate the unregulated light. Then, the modified beam passes through a 4-*f* system which only keeps the + 1st order. The GOVs are generated on the focal plane of L2 (*f* = 100 mm). The following experiment is a typical optical-tweezers setup. The GOV passes through the lens L3 and reflected into the entrance pupil of the micro-objective MO1 (40×, NA = 0.65) by a dichroic mirror (DM). To backlight the trapping area, a LED source (620 ± 10 nm) is used. The image of the manipulation plane is projected onto a CCD camera (Basler acA1600-60gc) through the DM.

## 4. Results and discussions

The intensity, phase and OAM density patterns of the GOVs are demonstrated in the second and fourth columns of Fig. 3. The COVs with the same TCs are also illustrated in the first and third columns of Fig. 3 to distinguish the difference. For the GOV, the “scion” and “rootstock” positions are swapped with each other compared to the light at the object plane due to the focusing of the axicon. Moreover, the experimental intensity patterns well agree with those of the theoretical results, which show that the “scion” and “rootstock” half rings graft well together.

Compared to the COV, the GOV with a half integer ETC also has a gap in the side of the light ring. By calculating the correlation between the GOV’s and COV’s intensity patterns with the same TC, it is possible to determine that the shapes of the GOV and COV are highly similar, given that their correlation coefficients *R* are all higher than 0.95. However, at the joints, the intensity of the GOV has some fluctuation, as shown in Fig. 3(a3)–3(d3), which is particularly obvious for the theoretical intensity. This is because of the interference between the “scion” and “rootstock”, since the intensity has a rotation with respect to the spiral phase due to the combined effect both azimuthal energy flow and radial energy flow [24,25]. To characterize the fluctuation, we calculate the intensity rate between the local extremum at the joints (Q1, Q3, Q4) and the smooth intensity positions (Q2, Q5). The results show that the rates are all greater than 70%, which is larger than 1/(e^2) of the maximum intensity. Consequently, the intensity distribution of the GOV is considered to be smooth and even enough on the light ring.

To verify the quality of the GOV, we calculate the mode purity *ɛ*. Experimentally, the mode purity can be estimated via the correlation coefficients by fitting the experimental intensity pattern and the theoretical intensity pattern [17,26]. The values of the mode purity are shown in the first row of Fig. 3. The results indicate that the mode purity *ɛ* is greater than 0.94 for both the GOV and COV, which indicates that the GOV still conserves a high beam quality during the grafting process.

The phase patterns of the GOV and COV are shown in the fourth row of Fig. 3, in which the positive TCs are encircled by red circles. Moreover, the initial spiral phases in the object plane are shown in the insets of Fig. 3(a1)–3(d1). For integer ETC, the TC magnitude of the GOV is 4, which is in agreement with the result of *M *= (*m*_{1}+*m*_{2})/2 = (2 + 6)/2 = 4. The TC of the GOV splits into four unit TCs which is different to that of the COV shown in Fig. 3(a4). When the ETC is a half integer, one of the singularities of the GOV is located at the annular gap as shown in Fig. 3(d4). This phenomenon is the same as the fractional OV. The reason is that a fractional phase jump occurs at the object plane when the ETC is equal to a half-integer, as shown in the inset of Fig. 3(d1). Moreover, the TC distribution of the GOV shifts to the side where the “rootstock” with the larger TC is located as shown in Fig. 3(b4) and 3(d4).

The bottom row of Fig. 3 illustrates the corresponding OAM distribution of the GOV and COV which is numerically calculated by employing the method of Refs. [1,27]. The OAM distribution of the COV is uniform along the azimuthal direction for integer TC. This is different from the OAM ring of the GOV which is distinctly divided into two half circles, where the larger TC of the OV results in a larger OAM. At the joints, the larger OAM rapidly decreases and transfers to the lower OAM, which is illustrated as the ×2 magnification insets in Fig. 3(b5). For half integer TC, this rule still applies as shown in Fig. 3(c5) and 3(d5).

Now, we can summarize the major properties of the GOV: the intensity ring remains constant while the OAM is uneven, and can be controlled by the TC of the GOVs. This property can facilitate more degrees of freedom for manipulating micro-particles. From the viewpoint of the force field, the GOV can simultaneously provide a constant gradient force and a controllable tangential force for the trapped micro-particles. The focus of the next few sections is to determine the modulating characteristics of the GOV’s local OAM.

Let us consider the modulation of the GOV’s local OAM with a constant ETC (*M* = 3) but different *m*_{1} and *m*_{2}, which are shown in Fig. 4. The GOV’s intensity is remained at a constant for different grafted TCs which have higher correlation *R *> 0.92 through correlating each sub-image with Fig. 4(a1). However, the OAM distributions on the rings are distinct from each other as shown in Fig. 4(a2) and 4(b2). Furthermore, it is well-known that the OAM of light is a vector. As are shown in Fig. 4(c2) and 4(d2), if the *m*_{1} and *m*_{2} own different signs, the direction of the local OAM can be modulated. However, the OAM first decreases to zero and then increases in the opposite direction, which is different for the case where *m*_{1} and *m*_{2} have the same sign at the joints. The dependence of the OAM value on the grafted TCs is linear through the fitting of the “*”-marked data, which is shown in the inset of Fig. 4(d2).

To quantitatively analyze the properties of the GOV, the center profiles of their intensity and the OAM rings of Fig. 4 are plotted in Fig. 5. For theoretical simulation results, there is a valley and a peak at the two joints on each intensity profile curve. The reason for this is that destructive and constructive interference occurs. Moreover, the magnitudes of the valley and peak are proportional to the values of |*m*_{1}-*m*_{2}|. In addition, the experimental intensity results have similar trend compared with the theoretical results except for some fluctuations. For all OAM profiles, the local OAM smoothly transits from one stage to the other, which is an optimal candidate OV to freely control the local OAM distribution on the ring.

To experimentally verify that the OAM distributions of the proposed GOVs are controllable, 3µm-diameter polystyrene microspheres suspended in distilled water are trapped and manipulated by the GOVs with different grafted TCs. Experimental setup is depicted in Fig. 2 and the laser power is set to 30.58 mW. Owing to the liquid environmental impact and slight out-of-focus in the trapping plane, the beams look a little different between in air and in the aqueous solution [28]. As are shown in Fig. 6(b1)–6(f1), a microsphere is trapped on the intensity ring of the GOV (*m*_{1} = 20, *m*_{2} = 14). The average velocity of the trapped microsphere move in the lower part is 5.131 µm/s which is higher than 2.719 µm/s in the upper part (details, see Visualization 1). The bigger average velocity owing to the bigger local OAM density and the velocity of the trapped microsphere is not uniform due to the Brownian movement and environmental disturbance [29]. A brighter intensity appears in the left part due to the two different magnitude energy flows which result in a higher velocity.

In Fig. 6(b2)–6(f2), two microspheres are trapped and rotate simultaneously along the intensity trajectory until they meet together to stop movement, (details, see Visualization 2). Their movements are not completely synchronized on account of their different initial positions and two interference areas in left and right of the GOV due to the two inverse azimuthal energy flow [24] which cause the microspheres to stop. In short, particle manipulating experiments demonstrate that the magnitude and directions of the GOVs’ local OAM on the ring are controllable while the intensity remains constant to provide enough gradient force, simultaneously.

Note that the aberration is existed in our manipulation experiments due to the imperfections of the optical tweezer system [30], which reduces the beam quality of the GOV. Consequently, this issue should be concerned as the GOV is used to conduct more accurate manipulation. In this case, a flexible method of wavefront correction based on SLM is a good candidate to decrease the aberration and increase the beam quality [31,32].

Regarding the particle manipulation of the GOV, versatile combination of the grafted TCs can provide more alternative options. However, the radius of the OV slowly increases with increasing TC by the axicon method [33], which would lead to a low-quality GOVs when the grafted TCs *m*_{1} and *m*_{2} are greatly different. An effective method to eliminate this influence is to carefully adjust the cone angle of the axicon before grafting. If the difference is greater than 20, other techniques should be developed to ensure perfect grafting.

If necessary, more versatile GOVs can be obtained as required. Compared to the GOVs grafted via two OVs, the OAM distributions of GOVs grafted using three or more OVs [i.e. *N *= 3… in Eq. (1)] have more versatile combinations.

## 5. Conclusions

In conclusion, we have proposed a grafted optical vortex (GOV) with an OAM distribution that is easily modulated while the intensity remains constant. The magnitude and direction of the local OAM on the ring are freely controlled by changing the TCs of two or more grafted OVs. Moreover, the particular properties of the proposed GOVs are verified again by the experiments of the micro-particle manipulation. This grafted method of the spiral phases provided a controllable local tangential force on the light ring which can realize an accelerate, decelerate and opposite movement about the trapped micro-particles in optical tweezers.

## Funding

National Natural Science Foundation of China (NSFC) (61775052, 11704098, 11525418, 91750201).

## Acknowledgments

The authors thank Dr. Jeffrey Melzer at the University of Arizona for constructive suggestions in particle manipulation.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

## References

**1. **L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**(11), 8185–8189 (1992). [CrossRef]

**2. **D. G. Grier, “A revolution in optical manipulation,” Nature **424**(6950), 810–816 (2003). [CrossRef]

**3. **M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics **5**(6), 343–348 (2011). [CrossRef]

**4. **M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science **341**(6145), 537–540 (2013). [CrossRef]

**5. **A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics **8**(2), 200–227 (2016). [CrossRef]

**6. **J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, “Quantitative orbital angular momentum measurement of perfect vortex beams,” Opt. Lett. **44**(11), 2736–2739 (2019). [CrossRef]

**7. **A. Aleksanyan, N. Kravets, and E. Brasselet, “Multiple-star system adaptive vortex coronagraphy using a liquid crystal light valve,” Phys. Rev. Lett. **118**(20), 203902 (2017). [CrossRef]

**8. **J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics **6**(7), 488–496 (2012). [CrossRef]

**9. **M. McLaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express **20**(21), 23589–23597 (2012). [CrossRef]

**10. **H. Li, X. Huang, Q. Cao, Y. Zhao, P. Li, C. Wan, and A. Chong, “Generation of three-dimensional versatile vortex linear light bullets (Invited Paper),” Chin. Opt. Lett. **15**(3), 030009 (2017). [CrossRef]

**11. **P. Li, S. Liu, T. Peng, G. Xie, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express **22**(7), 7598–7606 (2014). [CrossRef]

**12. **J. A. Rodrigo and T. Alieva, “Freestyle 3D laser traps: tools for studying light-driven particle dynamics and beyond,” Optica **2**(9), 812–815 (2015). [CrossRef]

**13. **Y. Zhang, Y. Xue, Z. Zhu, G. Rui, Y. Cui, and B. Gu, “Theoretical investigation on asymmetrical spinning and orbiting motions of particles in a tightly focused power-exponent azimuthal-variant vector field,” Opt. Express **26**(4), 4318–4329 (2018). [CrossRef]

**14. **V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Asymmetric Bessel modes,” Opt. Lett. **39**(8), 2395–2398 (2014). [CrossRef]

**15. **V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Vortex Hermite-Gaussian laser beams,” Opt. Lett. **40**(5), 701–704 (2015). [CrossRef]

**16. **A. A. Kovalev, V. V. Kotlyar, and A. P. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. **110**(26), 261102 (2017). [CrossRef]

**17. **X. Li, H. Ma, C. Yin, J. Tang, H. Li, M. Tang, J. Wang, Y. Tai, X. Li, and Y. Wang, “Controllable mode transformation in perfect optical vortices,” Opt. Express **26**(2), 651–662 (2018). [CrossRef]

**18. **J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A **336**(1605), 165–190 (1974). [CrossRef]

**19. **G. Gbur, “Fractional vortex Hilbert's hotel,” Optica **3**(3), 222–225 (2016). [CrossRef]

**20. **A. S. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizón, “Generation of the “perfect” optical vortex using a liquid-crystal spatial light modulator,” Opt. Lett. **38**(4), 534–536 (2013). [CrossRef]

**21. **S. Frankearnold, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. **54**(12), 1723–1738 (2007). [CrossRef]

**22. **G. Tkachenko, M. Chen, K. Dholakia, and M. Mazilu, “Is it possible to create a perfect fractional vortex beam?” Optica **4**(3), 330–333 (2017). [CrossRef]

**23. **P. Vaity and L. Rusch, “Perfect vortex beam: Fourier transformation of a Bessel beam,” Opt. Lett. **40**(4), 597–600 (2015). [CrossRef]

**24. **J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. **50**(10), 1573–1580 (2003). [CrossRef]

**25. **X. Li, H. Ma, H. Zhang, M. Tang, H. Li, J. Tang, and Y. Wang, “Is it possible to enlarge the trapping range of optical tweezers via a single beam?” Appl. Phys. Lett. **114**(8), 081903 (2019). [CrossRef]

**26. **Y. Ohtake, T. Ando, N. Fukuchi, N. Matsumoto, H. Ito, and T. Hara, “Universal generation of higher-order multiringed Laguerre-Gaussian beams by using a spatial light modulator,” Opt. Lett. **32**(11), 1411–1413 (2007). [CrossRef]

**27. **A. O’Neil, I. MacVicar, L. Allen, and M. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. **88**(5), 053601 (2002). [CrossRef]

**28. **M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. **38**(22), 4919–4922 (2013). [CrossRef]

**29. **J. Melzer and E. McLeod, “Fundamental limits of optical tweezer nanoparticle manipulation speeds,” ACS Nano **12**(3), 2440–2447 (2018). [CrossRef]

**30. **A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express **15**(9), 5801–5808 (2007). [CrossRef]

**31. **T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics **4**(6), 388–394 (2010). [CrossRef]

**32. **Y. Liang, Y. Cai, Z. Wang, M. Lei, Z. Cao, Y. Wang, M. Li, S. Yan, P. R. Bianco, and B. Yao, “Aberration correction in holographic optical tweezers using a high-order optical vortex,” Appl. Opt. **57**(13), 3618–3623 (2018). [CrossRef]

**33. **Y. Liang, M. Lei, S. Yan, M. Li, Y. Cai, Z. Wang, X. Yu, and B. Yao, “Rotating of low-refractive-index microparticles with a quasi-perfect optical vortex,” Appl. Opt. **57**(1), 79–84 (2018). [CrossRef]