Jump to ContentJump to Main Navigation
Show Summary Details
More options …


Editor-in-Chief: Sorger, Volker

12 Issues per year

CiteScore 2017: 6.57

IMPACT FACTOR 2017: 6.014
5-year IMPACT FACTOR: 7.020

In co-publication with Science Wise Publishing

Open Access
See all formats and pricing
More options …
Volume 7, Issue 3


Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation

Yuanjie Yang
  • School of Astronautics and Aeronautics, University of Electronic Science and Technology of China, Chengdu 611731, China
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Xinlei Zhu / Jun Zeng / Xingyuan Lu / Chengliang ZhaoORCID iD: http://orcid.org/0000-0003-3703-889X / Yangjian Cai
Published Online: 2018-01-06 | DOI: https://doi.org/10.1515/nanoph-2017-0078


Zero-order and higher-order Bessel beams are well-known nondiffracting beams. Namely, they propagate with invariant profile (intensity) and carry a fixed orbital angular momentum. Here, we propose and experimentally study an anomalous Bessel vortex beam. Unlike the traditional Bessel beams, the anomalous Bessel vortex beam carries decreasing orbital angular momentum along the propagation axis in free space. In other words, the local topological charge is inversely proportional to the propagation distance. Both the intensity and phase patterns of the generated beams are measured experimentally, and the experimental results agree well with the simulations. We demonstrate an easy way to modulate the beam’s topological charge to be an arbitrary value, both integer and fractional, within a continuous range. The simplicity of this geometry encourages its applications in optical trapping and quantum information, and the like.

Keywords: orbital angular momentum; fractional topological charge; optical vortex beam; diffraction

1 Introduction

In 1992, Allen et al. recognised that vortex beams with helical phase fronts, described by a transverse phase structure of exp(iℓφ), carry an orbital angular momentum (OAM) of ℓħ per photon [1], where φ is the azimuthal angle, is the topological charge of the field, and ħ is Planck’s constant divided by 2π. The extrinsic nature of the OAM of photons offers an additional degree of freedom, and optical vortex beams have found numerous applications in optical micromanipulation [2], free-space communication [3], and quantum information [4]. It is noted that the phase front of a vortex beam winds by 2π on a closed path around the axis, and the winding number is also possibly a noninteger [5]. When the winding number is a fractional number, the vortex beam has a fractional topological charge. In recent years, much attention has been paid to vortex beams with fractional topological charges [5], [6], [7], [8], [9], [10], which are of importance to multiple applications in quantum optics, quantum information, and microparticle transportation guiding [11], [12], [13].

Bessel beams, one of the most common vortex beams, has attracted increasing interest due to their nondiffracting and self-healing properties. Thus far, generation and propagation properties of a fractional Bessel vortex beam (FBVB) have been studied extensively both theoretically and experimentally [13], [14], [15], [16], [17], [18], [19]. Using a spatial light modulator (SLM), Tao et al. [14] experimentally generated a FBVB in 2003. Soon afterwards, the self-reconstruction and nondiffracting properties of FBVB were examined [15], [16], [17].

On the other hand, it is well known that a vortex beam generally carries a fixed OAM during propagation. However, more recently, increasing interest has been paid to controlling the topological charge of a vortex beam during propagation [20], [21]. Using a class of nondiffracting frozen waves, Dorrah et al. [21] demonstrated that the topological charge of the beam can be controlled along the propagation direction. Recent studies show that the degree of longitudinal control of OAM may find applications in remote sensing, dense data communications, and others [20], [21], [22], [23]. Although it is noted that the method to produce a Bessel vortex beam with decreasing OAM, that is, the topological charge is inversely proportional to the propagation distance, has not been studied yet.

In this study, we combine the two ideas to produce an anomalous Bessel vortex beam. Note that this beam is entirely different from the traditional Bessel beams or FBVBs discussed before. The most appealing property of the anomalous Bessel beam is that the local topological charge varies during propagation. The experimental results demonstrate that such a beam carries a continuously decreasing OAM, both integer and fractional multiple of ħ, along the propagation direction in free space.

2 Principle and methods

We know that a zeroth-order Bessel beam can be thought of as the Fourier transform of an annular slit. Therefore, for the first time, Durnin et al. [24] observed a Bessel beam by placing an annular slit in the back focal plane of a converging lens. Due to the circular symmetry of the slit, we obtain a bright spot in the centre of the diffraction pattern, that is, zero-order Bessel beam. However, if the annular slit is changed into a spiral slit, then the transmitting wavelets from different parts of the spiral slit undergo different optical paths and reach the centre of the observation plane. In other words, we can introduce a continuous phase shift using a spiral slit, and then a phase singularity can be seen on the beam axis [25]. Recently, spiral slits [26], [27], [28], [29] and plasmonic nanosieves [30], [31] have been adopted to produce plasmonic vortices and manipulate OAM. Cho et al. [27] showed that we can sculpture the fractional plasmonic vortex by modifying the structure of the spiral slit or tuning the operating wavelength. Wang et al. [28] theoretically analysed a method for fractional plasmonic vortex sculpturing, utilising the radial phase gradient induced by incident Laguerre–Gaussian beam. Here, we show that a Fermat’s spiral slit can generate a novel anomalous vortex beam, and the OAM can be manipulated with propagation easily.

We start by studying the proposed method theoretically. A spiral slit located in z=0 plane is plotted in Figure 1, where r0 is the initial radius of the spiral slit, rα and α are the radial and the angular coordinates, respectively. Let us suppose that ρα is the distance from point (rα, α, 0) to the centre of observation plane (0, 0, z). Then, if a plane wave is incident on the screen, the transmitted wavelets from two close points, (rα, α) and (rα+Δα, αα), along the slit with a little increment of angle Δα undergo different optical paths and arrive at the centre of the observation plane (0, 0, z). The phase difference at the point (0, 0, z) can be written as Δθ=2πΔρ/λ, where Δρ=ρα+Δαρα, and λ is the wavelength of the plane wave. It is noted that if Δρ=ℓ λΔα/2π, then, the total phase shift caused by the spiral slit can be calculated as 02πdθ=2πλ02πλ2πdα=2π, where ℓ is a constant. This means that the phase front of the beam in the centre of the observation plane winds by 2π on a closed path around the axis, namely, there will be an optical vortex in the centre of the observation plane [32], and the topological charge of which is . It is noted that the topological charge is dependent on distance z and describes the vortex located at the beam centre; therefore, it is a local topological charge. To meet this requirement for forming a helical phase front, the structure of a spiral slit can be written as

Geometry and notation of the spiral slit for achieving an anomalous Bessel vortex beam.
Figure 1:

Geometry and notation of the spiral slit for achieving an anomalous Bessel vortex beam.


The aforementioned analysis shows that if there is a spiral slit as described in Eq. 1, namely, a Fermat’s spiral, then we can get a vortex beam with topological charge at observation plane z. Interestingly, Eq. 1 indicates that for a given spiral slit, the value of ℓ can be regarded as a constant, which indicates that we can get a decreasing topological charge with the increasing propagation distance z. In other words, for a given optical beam (the wavelength is fixed), the topological charge of the generated vortex beam is inversely proportional to the propagation distance z. In the Fresnel approximation, the complex amplitude of the diffracted beam at a distance z can be obtained from the diffraction integral [33]


where T(x,y) denotes the aperture function of the spiral slit.

3 Results and discussion

According to the analysis in Section 2, we know that one can obtain a vortex beam using a Fermat’s spiral. If we want to obtain an anomalous Bessel vortex beam, the spiral slit must meet another requirement: the gap of the spiral slit is much smaller than its radius, that is, ρρ0=ρ0. If so, the spiral slit can be regarded as an annular slit approximately; accordingly, the field distribution of the generated vortex beam can be described by a Bessel function at some specific propagation distances. The simulations of the intensity and the phase patterns for different propagation distances are shown in Figure 2A. The spiral slit parameters are set to λ=532 nm, =2, z=1 m, r0=3 mm, and d=0.07 mm, where d is the width of the slit. Although the slit width d does not appear in Eq. 1, it plays an important role in generating anomalous Bessel vortex beams. Similar to the annular aperture for producing Bessel beams [24], the width of the slit should be much smaller than the radius, that is, d=ρ0. Figure 2A shows that we can obtain a Bessel beam with topological charge =2 in the plane z=1 m. Interestingly, we can see that the topological charge of the vortex beam decreases continually with increasing propagation distance, both integer and fractional. Furthermore, when the topological charges are not an integer, the intensity patterns are no longer rotationally symmetrical rings. Figure 2B shows the corresponding experimental results for different propagation distances, which agree well with the simulations in Figure 2A. Therefore, we can see clearly that the topological charges of the vortex beam are inversely proportional to the propagation distance.

Intensity and the phase patterns of vortex beams with different topological charge at different distance z. (A) Simulations based on the spiral slit; (B) experimental results; (C) theoretical results based on Eq. 9 from the article by Gutiérrez-Vega and López-Mariscal [17].
Figure 2:

Intensity and the phase patterns of vortex beams with different topological charge at different distance z.

(A) Simulations based on the spiral slit; (B) experimental results; (C) theoretical results based on Eq. 9 from the article by Gutiérrez-Vega and López-Mariscal [17].

To verify the integer and FBVBs generated by a Fermat’s spiral slit, the intensity and phase patterns of the exact FBVB with the same topological charges, using the analytical expression Eq. 9 in the article by Gutiérrez-Vega and López-Mariscal [17], are shown in Figure 2C. The comparison shows excellent agreement between the results obtained by the proposed method and the analytical results based on the work of Gutiérrez-Vega and López-Mariscal [17]. Figure 2 indicates that using a Fermat’s spiral slit, we can obtain an FBVB with decreasing topological charge during propagation, namely, an anomalous Bessel vortex beam. Furthermore, it is noted that the topological charge decreases from 2 to 1 within a 1-m distance. Although if the spiral slit is designed for producing a vortex beam with topological charge =2 at z=0.5 m, then we will get a vortex beam with charge =1 at z=1 m. That is, the topological charge decreases from 2 to 1 within a 0.5-m distance. Therefore, we can engineer the OAM decreasing rate by tuning the structure of the spiral slit.

Figure 3 shows the experimental setup for generating an anomalous Bessel vortex beam and measuring its intensity and phase patterns. The laser beam, which is expanded by a beam expander, illuminates a Fermat’s spiral slit and is displayed by SLM1. The output beam can be split into the transmitted and reflected beams by the first beam splitter (BS1). The distance between BS1 and SLM2 is equal to that between BS1 and the beam profile analyser. The intensity profile generated by the anomalous Bessel vortex beam is captured by the beam profile analyser. The phase pattern of the generated anomalous Bessel vortex beam can be measured by BS2, SLM2, and CCD based on a new method that we proposed recently [34].

Experimental setup for generating an anomalous Bessel vortex beam and measuring its phase patterns. BE, beam expander; SLM1 and SLM2, spatial light modulators; BS1 and BS2, beam splitters; BPA, beam profile analyser; CCD, charge-coupled device.
Figure 3:

Experimental setup for generating an anomalous Bessel vortex beam and measuring its phase patterns.

BE, beam expander; SLM1 and SLM2, spatial light modulators; BS1 and BS2, beam splitters; BPA, beam profile analyser; CCD, charge-coupled device.

Furthermore, from Eq. 1, we can see that the geometric structure of the spiral slit is dependent on the wavelength as well. Therefore, for a given Fermat’s spiral slit, if we change the wavelength of incident beams, we can obtain different topological charges at the same propagation distance. Simulations of the intensity and phase patterns with different wavelengths and the same propagation distance are shown in Figure 4A, in which the parameters are set to z=1 m, r0=3.17 mm, and d=0.028 mm. The corresponding experimental results are shown in Figure 4B. From Figure 4, we can see that the topological charge of the vortex beam decreases with increasing wavelength.

Simulations and experimental results of anomalous Bessel vortex beams with different wavelengths at same propagation distance. (A) Simulations; (B) experimental results.
Figure 4:

Simulations and experimental results of anomalous Bessel vortex beams with different wavelengths at same propagation distance.

(A) Simulations; (B) experimental results.

In the previous scenario, we have demonstrated the possibility of modulating the topological charge as a vortex beam propagates. It should be emphasised that the topological charge modulation discussed here does not violate the conservation of angular momentum. To fully understand this phenomenon, let us briefly review one of the most popular methods for the generation of vortex beams from a plane wave, that is, using spiral zone plates [35]. It is known that if a spiral zone plate is designed for producing a vortex with topological charge , then, such a plate can produce a series of vortex beams, with topological charge ±, ±2, ±3…, at different propagation planes. It is noted that, here, the topological charges refer to the local one in the beam centre rather than the whole beam. Because the vortex beam generated by this method can be regarded as two parts, the dominant structure located in the central region and the background has a lower intensity spread over a larger area in the outer region. In other words, the generated beam is a mixture of vortex beams with different OAMs, only one of which is focused in the centre in a specific propagation plane, and others are out of focus and are in the background [35]. Therefore, the total angular momentum in any plane remains constant. It is worth noting that what we discussed in the previous scenario is the local topological charge near the propagation axis. Similar to a spiral zone plate, the spiral slit proposed in our study just changes the distribution of OAMs in different propagation planes, but the total OAMs of the entire optical field in each propagation plane remains unvaried. That is, whereas the local topological charge of the beam in the central part changes, charges in the outer part change accordingly and the total momentum is always conserved [20], [21]. Therefore, for the most part, what we can modulate is the local OAM rather than total OAM. For some special cases, the local OAM may even change sign whereas the total OAM of the beam remains constant [36]. Moreover, we know that the rotation rate of particles, trapped within a vortex beams, is dependent on the transfer of local angular momentum rather than total angular momentum [37].

4 Conclusion

In summary, we have studied a simple method for the generation of anomalous Bessel vortex beams both numerically and experimentally. An appealing propagation property of such a beam is that the local topological charge of the central beam is inversely proportional to the propagation distance, and the OAM deceasing rate can be engineered by tuning the structure of the spiral slit. We have demonstrated that one can control the OAM longitudinally by a simple spiral slit, which may find applications in optical trapping, quantum communications, dense data communications, and so on. Moreover, in this study, we retrieved the phase pattern experimentally; therefore, we can obtain the topological charge intuitively, rather than by analyzing the intensity patterns of the vortex beam indirectly [38], [39].


This work was supported by the National Natural Science Fund for Distinguished Young Scholars (11525418), the National Natural Science Foundation of China (11474048, 11374222, 91750201 and 11774250), and the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.


  • [1]

    Allen L, Beijersbergen MW, Spreeuw R, Woerdman J. Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes. Phys Rev A 1992;45:8185–89. CrossrefPubMedGoogle Scholar

  • [2]

    Dholakia K, Čižmár T. Shaping the future of manipulation. Nat Photon 2011;5:335–42. CrossrefGoogle Scholar

  • [3]

    Wang J, Yang J, Fazal IM, et al. Terabit free-space data transmission employing orbital angular momentum multiplexing. Nat Photon 2012;6:488–96. CrossrefGoogle Scholar

  • [4]

    Molina-Terriza G, Torres JP, Torner L. Twisted photons. Nat Phys 2007;3:305–10. CrossrefWeb of ScienceGoogle Scholar

  • [5]

    Berry M. Optical vortices evolving from helicoidal integer and fractional phase steps. J Opt A 2004;6:259–68. CrossrefGoogle Scholar

  • [6]

    Li X, Tai Y, Lv F, Nie Z. Measuring the fractional topological charge of LG beams by using interference intensity analysis. Opt Commun 2015;334:235–39. CrossrefWeb of ScienceGoogle Scholar

  • [7]

    O’Dwyer D, Phelan C, Rakovich Y, Eastham P, Lunney J, Donegan J. Generation of continuously tunable fractional optical orbital angular momentum using internal conical diffraction. Opt Express 2010;18:16480–85. Web of ScienceCrossrefPubMedGoogle Scholar

  • [8]

    Leach J, Yao E, Padgett MJ. Observation of the vortex structure of a non-integer vortex beam. New J Phys 2004;6:71. CrossrefGoogle Scholar

  • [9]

    Götte JB, O’Holleran K, Preece D, et al. Light beams with fractional orbital angular momentum and their vortex structure. Opt Express 2008;16:993–1006. PubMedCrossrefWeb of ScienceGoogle Scholar

  • [10]

    Fang Y, Lu Q, Wang X, Zhang W, Chen L. Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale. Phys Rev A 2017;95:023821. Web of ScienceCrossrefGoogle Scholar

  • [11]

    Chen L, Lei J, Romero J. Quantum digital spiral imaging, Light: Sci Appl 2014;3:e153. Google Scholar

  • [12]

    Oemrawsingh S, Ma X, Voigt D, Aiello A, Eliel E, Woerdman J. Experimental demonstration of fractional orbital angular momentum entanglement of two photons. Phys Rev Lett 2005;95:240501. CrossrefPubMedGoogle Scholar

  • [13]

    Tao S, Yuan X, Lin J, Peng X, Niu H. Fractional optical vortex beam induced rotation of particles. Opt Express 2005;13:7726–31. PubMedCrossrefGoogle Scholar

  • [14]

    Tao S, Lee W, Yuan X. Dynamic optical manipulation with a higher-order fractional Bessel beam generated from a spatial light modulator. Opt Lett 2003;28:1867–69. CrossrefPubMedGoogle Scholar

  • [15]

    Tao SH, Lee WM, Yuan X. Experimental study of holographic generation of fractional Bessel beams. Appl Opt 2004;43: 122–26. CrossrefPubMedGoogle Scholar

  • [16]

    Tao SH, Yuan X. Self-reconstruction property of fractional Bessel beams. J Opt Soc Am A 2004;21:1192–97. CrossrefGoogle Scholar

  • [17]

    Gutiérrez-Vega JC, López-Mariscal C. Nondiffracting vortex beams with continuous orbital angular momentum order dependence. J Opt A 2008;10:015009. CrossrefWeb of ScienceGoogle Scholar

  • [18]

    Mitri F. High-order Bessel nonvortex beam of fractional type α. Phys Rev A 2012;85:025801. CrossrefWeb of ScienceGoogle Scholar

  • [19]

    Gao J, Zhang Y, Dan W, Hu Z. Turbulent effects of strong irradiance fluctuations on the orbital angular momentum mode of fractional Bessel Gauss beams. Opt Express 2015;23:17024–34. PubMedWeb of ScienceCrossrefGoogle Scholar

  • [20]

    Davis JA, Moreno I, Badham K, Sánchez-López MM, Cottrell DM. Nondiffracting vector beams where the charge and the polarization state vary with propagation distance. Opt Lett 2016;41:2270–73. Web of ScienceCrossrefPubMedGoogle Scholar

  • [21]

    Dorrah AH, Zamboni-Rached M, Mojahedi M. Controlling the topological charge of twisted light beams with propagation. Phys Rev A 2016;93:063864. CrossrefWeb of ScienceGoogle Scholar

  • [22]

    Rui G, Zhan Q. Tailoring optical complex fields with nano-metallic surfaces, Nanophotonics 2015;4:2. Web of ScienceGoogle Scholar

  • [23]

    Han Y, Liu Y, Wang Z, et al. Controllable all-fiber generation/conversion of circularly polarized orbital angular momentum beams using long period fiber gratings. Nanophotonics 2018;7:287–93. Web of ScienceGoogle Scholar

  • [24]

    Durnin J, Miceli Jr J, Eberly J. Diffraction-free beams. Phys Rev Lett 1987;58:1499. CrossrefPubMedGoogle Scholar

  • [25]

    Li Z, Zhang M, Li X, Liu C, Cheng C. Topological charge conversion with spiral-slit screens. Chin Phys Lett 2013;30:104206. Web of ScienceCrossrefGoogle Scholar

  • [26]

    Tsai W-Y, Huang J-S, Huang C-B. Selective trapping or rotation of isotropic dielectric microparticles by optical near field in a plasmonic archimedes spiral. Nano Lett 2014;14:547–52. Web of ScienceCrossrefGoogle Scholar

  • [27]

    Cho S, Park J, Lee S, Kim H, Lee B. Coupling of spin and angular momentum of light in plasmonic vortex. Opt Express 2012;20:10083–94. CrossrefWeb of SciencePubMedGoogle Scholar

  • [28]

    Wang Y, Zhao P, Feng X, et al. Dynamically sculpturing plasmonic vortices: from integer to fractional orbital angular momentum. Sci Rep 2016;6:36269. PubMedCrossrefWeb of ScienceGoogle Scholar

  • [29]

    Mehmood MQ, Liu H, Huang K, et al. Broadband spin-controlled focusing via logarithmic-spiral nanoslits of varying width. Laser Photon Rev 2015;9:674–81. Web of ScienceCrossrefGoogle Scholar

  • [30]

    Mei S, Mehmood MQ, Hussain S, et al. Flat helical nanosieves. Adv Funct Mater 2016;26:5255–62. CrossrefWeb of ScienceGoogle Scholar

  • [31]

    Mehmood M, Mei S, Hussain S, et al. Visible-frequency metasurface for structuring and spatially multiplexing optical vortices. Adv Mat 2016;28:2533–39. CrossrefGoogle Scholar

  • [32]

    Li Z, Zhang M, Liang G, Li X, Chen X, Cheng C. Generation of high-order optical vortices with asymmetrical pinhole plates under plane wave illumination. Opt Express 2013;21:15755–64. CrossrefPubMedWeb of ScienceGoogle Scholar

  • [33]

    Goodman JW. Introduction to fourier optics. Greenwood Village, Roberts and Company Publishers, 2005. Google Scholar

  • [34]

    Shao Y, Lu X, Konijnenberg S, Zhao C, Cai Y, Urbach HP. Spatial coherence measurement and partially coherent diffractive imaging. arXiv preprint arXiv:1706.07203 2017. Google Scholar

  • [35]

    Saitoh K, Hasegawa Y, Tanaka N, Uchida M. Production of electron vortex beams carrying large orbital angular momentum using spiral zone plates, J Electron Microsc 2012;61:171–77. Web of ScienceGoogle Scholar

  • [36]

    Padgett M, Allen L. Orbital angular momentum exchange in cylindrical-lens mode converters. J Opt B Quantum Semiclassical Opt 2002;4:S17. CrossrefGoogle Scholar

  • [37]

    Garcés-Chávez V, McGloin D, Padgett M, Dultz W, Schmitzer H, Dholakia K. Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle. Phys Rev Lett 2003;91:093602. CrossrefGoogle Scholar

  • [38]

    Prabhakar S, Kumar A, Banerji J, Singh R. Revealing the order of a vortex through its intensity record. Opt Lett 2011;36: 4398–400. PubMedWeb of ScienceCrossrefGoogle Scholar

  • [39]

    Guo C, Yue S, Wei G. Measuring the orbital angular momentum of optical vortices using a multipinhole plate. Appl Phys Lett 2009;94:231104. Web of ScienceCrossrefGoogle Scholar

About the article

Received: 2017-08-04

Revised: 2017-11-16

Accepted: 2017-11-28

Published Online: 2018-01-06

Published in Print: 2018-02-23

Citation Information: Nanophotonics, Volume 7, Issue 3, Pages 677–682, ISSN (Online) 2192-8614, DOI: https://doi.org/10.1515/nanoph-2017-0078.

Export Citation

©2018 Chengliang Zhao and Yangjian Cai et al., published by De Gruyter, Berlin/Boston. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Xiansheng Hu, Zhaxibamao Gezhi, Osami Sasaki, Ziyang Chen, and Jixiong Pu
Applied Optics, 2018, Volume 57, Number 35, Page 10300
Ahmed H. Dorrah, Carmelo Rosales-Guzmán, Andrew Forbes, and Mo Mojahedi
Physical Review A, 2018, Volume 98, Number 4
Jun Zeng, Xianlong Liu, Fei Wang, Chengliang Zhao, and Yangjian Cai
Optics Express, 2018, Volume 26, Number 21, Page 26830

Comments (0)

Please log in or register to comment.
Log in