Simple method for efficient reconfigurable optical vortex beam splitting

Vol. 25, No. 16 | 7 Aug 2017 | OPTICS EXPRESS 18722

Simple method for efficient reconfigurable optical vortex beam splitting ALEXEY P. PORFIREV1,2,* AND SVETLANA N. KHONINA1,2 1

Samara National Research University, 34 Moskovskoe shosse, Samara 443086, Russia Image Processing Systems Institute - Branch of the Federal Scientific Research Centre “Crystallography and Photonics” of Russian Academy of Sciences, 151 Molodogvardejskaya St., Samara 443001, Russia *[email protected] 2

Abstract: In recent years, singular light beams with orbital angular momentum are one of the most striking examples of structured light that have been widely applied in modern science. The transition from the generation of a single vortex beam to the generation of multiple such beams progressed the development of singular optics. This paper presents a new efficient method of vortex laser beam splitting using a two-level pure-phase diffractive optical element. The proposed compact element, which can be easily implemented with a low-cost binary spatial light modulator or fabricated by electron beam lithography or photolithography, is a useful tool for the reconfigurable generation of multiple closed-packed vortex beams. Furthermore, the proposed splitter can efficiently operate in the wavelength range of approximately 8% of the central wavelength, thus providing an efficient method to generate optical vortex arrays with various potential applications in modern optics and photonics. © 2017 Optical Society of America OCIS codes: (050.4865) Optical vortices; (050.1970) Diffractive optics; (050.1380) Binary optics; (140.3300) Laser beam shaping.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

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). N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997). M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). P. Török and P. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express 12(15), 3605–3617 (2004). I. Augustyniak, A. Popiołek-Masajada, J. Masajada, and S. Drobczyński, “New scanning technique for the optical vortex microscope,” Appl. Opt. 51(10), C117–C124 (2012). Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Power-dependent shaping of vortex solitons in optical lattices with spatially modulated nonlinear refractive index,” Opt. Lett. 33(19), 2173–2175 (2008). A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, “Generation of lattice structures of optical vortices,” J. Opt. Soc. Am. B 19(3), 550–556 (2002). A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89(24), 240401 (2002). G. Molina-Terriza, A. Vaziri, J. Rehácek, Z. Hradil, and A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92(16), 167903 (2004). 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). N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabitscale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013). O. J. Allegre, Y. Jin, W. Perrie, J. Ouyang, E. Fearon, S. P. Edwardson, and G. Dearden, “Complete wavefront and polarization control for ultrashort-pulse laser microprocessing,” Opt. Express 21(18), 21198–21207 (2013). S. Hasegawa and Y. Hayasaki, “Holographic femtosecond laser manipulation for advanced material processing,” Adv. Opt. Technol. 5(1), 39–54 (2016). Y. Jin, O. J. Allegre, W. Perrie, K. Abrams, J. Ouyang, E. Fearon, S. P. Edwardson, and G. Dearden, “Dynamic modulation of spatially structured polarization fields for real-time control of ultrafast laser-material interactions,” Opt. Express 21(21), 25333–25343 (2013). X. Li, J. Chu, Q. Smithwick, and D. Chu, “Automultiscopic displays based on orbital angular momentum of light,” J. Opt. 18(8), 085608 (2016).

#295354 Journal © 2017

https://doi.org/10.1364/OE.25.018722 Received 5 May 2017; revised 21 Jul 2017; accepted 24 Jul 2017; published 25 Jul 2017

Vol. 25, No. 16 | 7 Aug 2017 | OPTICS EXPRESS 18723

16. T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015). 17. K. Ladavac and D. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12(6), 1144–1149 (2004). 18. C.-S. Guo, Y.-N. Yu, and Z. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283(9), 1889–1893 (2010). 19. C.-F. Kuo and S.-C. Chu, “Numerical study of the properties of optical vortex array laser tweezers,” Opt. Express 21(22), 26418–26431 (2013). 20. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1–6), 169–175 (2002). 21. L.-G. Wang, L.-Q. Wang, and S.-Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282(6), 1088–1094 (2009). 22. S. Vyas and P. Senthilkumaran, “Vortex array generation by interference of spherical waves,” Appl. Opt. 46(32), 7862–7867 (2007). 23. J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1– 3), 21–27 (2001). 24. S.-C. Chu, Y.-T. Chen, K.-F. Tsai, and K. Otsuka, “Generation of high-order Hermite-Gaussian modes in endpumped solid-state lasers for square vortex array laser beam generation,” Opt. Express 20(7), 7128–7141 (2012). 25. S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46(15), 2893–2898 (2007). 26. J. Masajada, A. Popiolek-Masajada, and M. Leniec, “Creation of vortex lattices by a wavefront division,” Opt. Express 15(8), 5196–5207 (2007). 27. S.-C. Chu, C.-S. Yang, and K. Otsuka, “Vortex array laser beam generation from a Dove prism-embedded unbalanced Mach-Zehnder interferometer,” Opt. Express 16(24), 19934–19949 (2008). 28. G. Ruffato, M. Massari, and F. Romanato, “Diffractive optics for combined spatial- and mode- division demultiplexing of optical vortices: design, fabrication and optical characterization,” Sci. Rep. 6(1), 24760 (2016). 29. G.-X. Wei, L.-L. Lu, and C.-S. Guo, “Generation of optical vortex array based on the fractional Talbot effect,” Opt. Commun. 282(14), 2665–2669 (2009). 30. S. Fu, T. Wang, and C. Gao, “Perfect optical vortex array with controllable diffraction order and topological charge,” J. Opt. Soc. Am. A 33(9), 1836–1842 (2016). 31. Y. Lu, B. Jiang, S. Lu, Y. Liu, S. Li, Z. Cao, and X. Qi, “Arrays of Gaussian vortex, Bessel and Airy beams by computer-generated hologram,” Opt. Commun. 363, 85–90 (2016). 32. A. Kapoor, M. Kumar, P. Senthilkumaran, and J. Joseph, “Optical vortex array in spatially varying lattice,” Opt. Commun. 365, 99–102 (2016). 33. P. García-Martínez, M. M. Sánchez-López, J. A. Davis, D. M. Cottrell, D. Sand, and I. Moreno, “Generation of Bessel beam arrays through Dammann gratings,” Appl. Opt. 51(9), 1375–1381 (2012). 34. Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A 83(5), 053813 (2011). 35. G. Ruben and D. M. Paganin, “Phase vortices from a Young’s three-pinhole interferometer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(6), 066613 (2007). 36. A. Sabatyan, S. M. Taheri Balanoji, and S. M. Taheri Balanoji, “Square array of optical vortices generated by multiregion spiral square zone plate,” J. Opt. Soc. Am. A 33(9), 1793–1797 (2016). 37. A. P. Porfirev, A. V. Ustinov, and S. N. Khonina, “Polarization conversion when focusing cylindrically polarized vortex beams,” Sci. Rep. 6(1), 6 (2016). 38. R. Barboza, U. Bortolozzo, G. Assanto, E. Vidal-Henriquez, M. G. Clerc, and S. Residori, “Harnessing optical vortex lattices in nematic liquid crystals,” Phys. Rev. Lett. 111(9), 093902 (2013). 39. E. Brasselet, “Tunable optical vortex arrays from a single nematic topological defect,” Phys. Rev. Lett. 108(8), 087801 (2012). 40. B. Yang and E. Brasselet, “Arbitrary vortex arrays realized from optical winding of frustrated chiral liquid crystals,” J. Opt. 15(4), 044021 (2013). 41. J. Jin, M. Pu, Y. Wang, X. Li, X. Ma, J. Luo, Z. Zhao, P. Gao, and X. Luo, “Multi-channel vortex beam generation by simultaneous amplitude and phase modulation with two-dimensional metamaterial,” Adv. Mater. Technol. 2(2), 1600201 (2017). 42. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006). 43. M. Beresna, M. Gecevicius, and P. G. Kazansky, “Polarization sensitive elements fabricated by femtosecond laser nanostructuring of glass,” Opt. Mater. Express 1(4), 783–795 (2011). 44. S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39(5), 1147–1154 (1992). 45. D. Garoli, P. Zilio, Y. Gorodetski, F. Tantussi, and F. De Angelis, “Optical vortex beam generator at nanoscale level,” Sci. Rep. 6(1), 29547 (2016). 46. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. P. Turunen, “Generation of GaussHermite modes using binary DOEs,” Proc. SPIE 4016, 234–239 (1999).

Vol. 25, No. 16 | 7 Aug 2017 | OPTICS EXPRESS 18724

47. E. G. Abramochkin and V. G. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83(1–2), 123–125 (1991). 48. M. W. Beijersbergen, L. Allen, H. E. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1–3), 123–132 (1993). 49. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Vortex Hermite-Gaussian laser beams,” Opt. Lett. 40(5), 701–704 (2015). 50. A. Kuchmizhak, E. Pustovalov, S. Syubaev, O. Vitrik, Y. Kulchin, A. Porfirev, S. Khonina, S. Kudryashov, P. Danilov, and A. Ionin, “On-fly femtosecond-laser fabrication of self-organized plasmonic nanotextures for chemo- and biosensing applications,” ACS Appl. Mater. Interfaces 8(37), 24946–24955 (2016). 51. A. Kuchmizhak, S. Gurbatov, O. Vitrik, Y. Kulchin, V. Milichko, S. Makarov, and S. Kudryashov, “Ion-beam assisted laser fabrication of sensing plasmonic nanostructures,” Sci. Rep. 6(1), 19410 (2016). 52. S. Syubaev, A. Zhizhchenko, A. Kuchmizhak, A. Porfirev, E. Pustovalov, O. Vitrik, Y. Kulchin, S. Khonina, and S. Kudryashov, “Direct laser printing of chiral plasmonic nanojets by vortex beams,” Opt. Express 25(9), 10214–10223 (2017). 53. T. Alieva, M. J. Bastiaans, and M. L. Calvo, “Fractional transforms in optical information processing,” EURASIP J. Adv. Signal Process. 10, 1–22 (2005). 54. S. N. Khonina, S. A. Balalayev, R. V. Skidanov, V. V. Kotlyar, B. Paivanranta, and J. Turunen, “Encoded binary diffractive element to form hyper-geometric laser beams,” J. Opt. A, Pure Appl. Opt. 11(6), 065702 (2009). 55. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001). 56. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966). 57. G. Molina-Terriza, J. Recolons, and L. Torner, “The curious arithmetic of optical vortices,” Opt. Lett. 25(16), 1135–1137 (2000). 58. J. N. Mait, “Design of Dammann gratings for two-dimensional, nonseparable, noncentrosymmetric responses,” Opt. Lett. 14(4), 196–198 (1989). 59. D. Huang, H. Timmers, A. Roberts, N. Shivaram, and A. S. Sandhu, “A low-cost spatial light modulator for use in undergraduate and graduate optics labs,” Am. J. Phys. 80(3), 211–215 (2012). 60. S. P. Kotova, V. V. Patlan, and S. A. Samagin, “Focusing light into a line segment of arbitrary orientation using a four-channel liquid crystal light modulator,” J. Opt. 15(3), 035706 (2013). 61. A. Cofré, P. Garcia-Martinez, A. Vargas, and I. Moreno, “Vortex beam generation and other advanced optics experiments reproduced with a twisted-nematic liquid-crystal display with limited phase modulation,” Eur. J. Phys. 38(1), 014005 (2017). 62. V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52(8), 429–431 (1990). 63. E. Brasselet, M. Malinauskas, A. Zukauskas, and S. Juodkazis, “Photopolymerized microscopic vortex beam generators: Precise delivery of optical orbital angular momentum,” Appl. Phys. Lett. 97(21), 211108 (2010). 64. D. Garoli, P. Zilio, Y. Gorodetski, F. Tantussi, and F. De Angelis, “Beaming of helical light from plasmonic vortices via adiabatically tapered nanotip,” Nano Lett. 16(10), 6636–6643 (2016). 65. X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338(6105), 363–366 (2012). 66. S. Paul, V. S. Lyubopytov, M. F. Schumann, J. Cesar, A. Chipouline, M. Wegener, and F. Küppers, “Wavelength-selective orbital-angular-momentum beam generation using MEMS tunable Fabry-Perot filter,” Opt. Lett. 41(14), 3249–3252 (2016). 67. H. Li, D. B. Phillips, X. Wang, Y.-L. D. Ho, L. Chen, X. Zhou, J. Zhu, S. Yu, and X. Cai, “Orbital angular momentum vertical-cavity surface-emitting lasers,” Optica 2(6), 547–552 (2015).

1. Introduction Since the Laguerre-Gaussian laser modes were shown to have a non-zero orbital angular momentum (OAM) [1], the so-called vortex laser beams with OAM have become one of the most rapidly growing and key areas of research in optics and photonics. Due to their unique properties, such beams have found applications in optical manipulation [2, 3], superresolution microscopy [4, 5], nonlinear optics [6, 7], quantum cryptography [8,9] and optical data transmission [10, 11]. The transition from a single vortex beam to multiple vortex beams made it possible to develop new high-performance methods of laser nanofabrication [12–14] and technologies for creation of an automultiscopic 3D display [15], to significantly increase the speed of the modern optical communication systems [16], to create micromechanical pumps [17] and systems of passive optical sorting of microscale objects [18] as well as other interesting technologies in the field of optical tweezers [19, 20]. Various techniques have been used to generate multiple vortex beams such as coherent laser beam arrays [21], various interferometric methods that require a sufficiently accurate adjustment of the optical setup [22–27], multi-value holograms implemented both with the

Vol. 25, No. 16 | 7 Aug 2017 | OPTICS EXPRESS 18725

help of static non-tunable diffractive optical elements (DOE) [28] and the use of commercial spatial light modulators (SLMs) [29–33], which allow for dynamic adjustment, but with significantly less efficiency compared to DOE. The application of various static phase or amplitude masks is also commonly used to create arrays of optical vortices [34–37]. Barboza et al. [38] utilized self-induced vortex like defects in the nematic liquid crystal layer of a light valve to demonstrate the realization of programmable lattices of optical vortices with arbitrary distribution in space. The possibility to generate arrays of optical vortices in a controllable manner and at the microscale was also demonstrated by using umbilical defects in nematic liquid crystals [39] and defect structures in frustrated cholesteric films [40]. Most recently, the possibility of multi-channel vortex beam generation with two-dimensional metamaterials was demonstrated [41]. Most of the above mentioned methods do not allow the reconfigurable formation of optical vortex arrays. For those that do, this is achieved by commercial SLM, which has a number of disadvantages such as high cost, low efficiency, polarization dependence and not suitable for use in very compact devices because of their size. In addition, most of these approaches do not allow the generation of arrays of closely-packed vortex beams. We propose an alternative method combining a single vortex beam generator (for example, a q-plate [42], an S-waveplate [43], a spiral phase plate [44], or even plasmonic optical vortex emitter [45]) and a two-level pure-phase splitter (Hermite-Gaussian (HG) mode generators [46]), allowing the generation of embedded vortex Hermite-Gaussian (EVHG) beams. Beams defined in the Cartesian coordinate system, such as HG beams or Airy beams, are not vortex, although their superpositions are used to generate vortex beams [47, 48]. Kotlyar et al. [49] considered the so-called elliptical vortex Hermite–Gaussian (vHG) beams, depending on the complex argument. They significantly differ from the EVHG beams considered in this paper. This paper reports the design and experimental investigation of a two-level pure-phase optical vortex beam splitter for reconfigurable generation of multiple closed-packed optical vortex beams. The multi-level reflective SLM was utilized to perform the proof-concept experiments. However, such a splitter can be implemented as a dielectric element (for example, on fused silica substrate) to increase its efficiency and the damage threshold. The results show the potential of the proposed element to generate variable closed-packed optical vortex beams with topological charge l = ± 1. Such generation of closed-packed optical vortices is particularly useful for high-performance parallel laser fabrication of bio- and chemo-sensors, since it allows the increased density of manufactured resonant plasmonic nanostructures for surface-enhanced spectroscopy [50–52]. In addition, the possibility to use such splitters to dynamically change the spatial structure of intensity distribution of the initial single the donut-shape vortex beam is also demonstrated. 2. Theoretical background An EVHG beam in the initial plane (z = 0) can be described by  x2 + y 2   x  s  y Ψ nms ( x, y ) = exp  −  H n   H m   ( x + iy ) , 2 2 σ σ σ      

(1)

where H n ( x ) and H m ( y ) are the Hermite polynomials of the n-th and m-th orders, σ is the waist radius, s is the topological charge. For the EVHG beams with an embedded first-order optical vortex (s = 1), instead of singular lines (linear phase jumps of π), sets of isolated first-order optical vortices are formed in the focal plane of a converging lens (see Fig. 1, right column).

Vol. 25, No. 16 | 7 Aug 2017 | OPTICS EXPRESS 18726

Fig. 1. Combination of a SPP and the HG-mode generator to generate EVHG beams. Left column represents the light field distributions in the initial plane (z = 0). Right column represents the generated light field distributions in the back focal plane of a converging lens.

It is well known that vortex beams, when propagating in a homogeneous medium, and also passing through lenses, retain a vortex phase and a zero-intensity value on the optical axis [1–3, 44]. However, in anisotropic media [42] or with astigmatic transformations [47, 48], such beams lose their axial symmetry in the intensity distribution. Since EVHG beams do not have axial symmetry, they can undergo changes analogous to anisotropic or astigmatic transformation when propagating in free space or passage through the lens. The propagation of an EVHG beam through a paraxial lens system can be described using fractional Fourier transform [53] in polar coordinates: E ( ρ ,θ , z ) = −

  ik ik ρ 2 exp ( ikz ) exp   × 2π f sin (α z )  2 f tan (α z ) 

    ikr 2 ik ρ r ×   E0 ( r , ϕ ) exp  cos (θ − ϕ )  r d r d ϕ ,  exp  −  f sin (α z )  0 0  2 f tan (α z )  2π R

(2)

where α = π/(2f), f is the focal length, k = 2π/λ is the wavenumber, λ is the wavelength, R is the input beam radius. Figure 2 shows a propagation of the EVHG beams from the input plane (z = 0) through the focal plane (z = f = 300 mm) to the output plane (z = 2f = 600 mm).

Vol. 25, No. 16 | 7 Aug 2017 | OPTICS EXPRESS 18727

Fig. 2. Transformations of various EVHG beams as they pass through the lens system. A converging lens with a focal length of f = 300 mm is located in the initial plane (z = 0). The laser beam propagates from left to right.

From Fig. 2 it is evident that the EVHG beams without vortex component (s = 0) are standard HG modes and propagate through the lens system retaining its structure. When the vortex phase is embedded into the HG modes, they change their structure during propagation. Note that the presence of zero intensity on the optical axis, which is characteristic for beams with a vortex phase singularity, is not guaranteed in this case and depends on the ratio of the order of the optical vortex s and the HG mode indexes n and m. Regarding the individual terms in Eq. (1), although all terms for Hermite polynomials have the same degree parity, for the sake of generality, the case of an arbitrary polynomial in two variables was considered: Pnms ( x, y ) = A(r )Cnms x n y m ( x + iy ) , s

(3)

where A(r) is an arbitrary axisymmetric function (depending only on the radius), Cnms is a normalizing factor. For the convenience of further analysis, Eq. (3) is written in polar coordinates:

Pnms (r , ϕ ) = A(r )Cnms ( r cos ϕ ) ( r sin ϕ ) r s exp ( isϕ ) . n

m

(4)

The Fourier transform of the function in Eq. (4) is as follows: ∞ 2π  i 2π  2π ρ r cos(ϕ − θ )  r d r d ϕ . Pnms (r , ϕ ) exp  −   λf 0 0  λf  Equation (5) is rewritten in the form:

Fnms ( ρ , θ ) =

Fnms ( ρ , θ ) =

2π C × λ f nms

∞ 2π  i 2π  n m ρ r cos(ϕ − θ )  d ϕ d r. × A(r ) ⋅ r n + m + s +1  ( cos ϕ ) ( sin ϕ ) exp ( isϕ ) exp  − λ f   0 0 In the case when m = 0, the integral by the angle in Eq. (6) takes the simpler form:

Gns ( ρ , θ ) =

2π

 ( cos ϕ ) 0

n

 ik  exp ( isϕ ) exp  − ρ r cos(ϕ − θ )  d ϕ = f  

(5)

(6)

(7)  ik  = exp(isθ )  ( cos(t + θ ) ) exp ( ist ) exp  − ρ r cos ( t )  d t.  f  0 Using the trigonometric formulas, Eq. (7) can be reduced to a combination of table integrals of the form: 2π

n

Vol. 25, No. 16 | 7 Aug 2017 | OPTICS EXPRESS 18728

2π

 ik

k  ρr . f    0 Note that the integrals with sin(nx) analogous to Eq. (8) are equal to zero. The simplest result is obtained for θ = 0:

 exp − f



ρ r cos x  cos(nx)dx = 2π e −inπ / 2 J n 

r2

∞

− 2 2π  n  Fn ,0, s ( ρ , 0) = Cn ,0, s  e 2σ r n + s +1   al J s − n + 2l ( k ρ r / f )  d r. λf  l =0  0 Then, using the following equation: ∞

x

(ν + 2 + 2 t ) −1 − nx 2

e

(8)

Jν (cx)dx =

0

 c2  t !cν exp  −  Lνt t +ν +1 2 n  4n  ν +1

 c2   ,  4n 

(9)

(10)

where Lνt ( x ) is generalized Laguerre polynomial, thereby, obtaining Eq. (11) instead of Eq. (9): Fn ,0, s ( ρ , 0) =

 1  k ρσ  2  2 s + 2 2π σ × Cn ,0, s exp  −   2  f   λf  

(11)  1  k ρσ  2    . × al 2 L  2  f   l =0   Equation (11) is correct for s ≥ n, otherwise, the Bessel functions with negative index should be replaced by positive, so the sum will start from l = (n - s)/2 and some coefficients al will change. For some of the simplest cases, the calculation of the field over the entire focal plane using Eq. (6) for an arbitrary angle θ were considered. n

n −l

 kρ  (n − l )!σ    f  2l

s − n + 2l

s − n + 2l n −l

1) n = 1, m = 0. F1,0, s ( ρ , θ ) = C1,0, s

 1  k ρσ  2  i 4π 2 (k ρ ) s −1σ 2 s + 2 −  × exp  2  f   λf f s −1  

2   k ρσ    π   × exp is  θ −    s exp ( −iθ ) −  cos θ  .   2       f  

(12)

It is assumed that s ≥ 1 in Eq. (12), in particular, when s = 1: F1,0,1 ( ρ ,θ ) = C1,0,1

2  1  k ρσ  2     k ρσ  4π 2σ 4   θ θ ⋅ − − exp  −  exp exp cos θ  . (13) i i ( ) ( )     2 f     λf  f     

As seen from Eq. (13), there is no zero at the origin. The factor in parentheses was rewritten by dividing the real and imaginary parts: [1 - (kρσ/f)2]cosθ-isinθ. Thus, there are two zeros at the points where two conditions are satisfied: θ = 0° (horizontal axis) and (kρσ/f)2 = 1, i.e. ρ1,2 = ± f/(kσ) (the “+” sign corresponds to the angle θ = 0°, and the “-” sign corresponds to the angle θ = 180°). Thus, the distance between these zeros of intensity (2f/(kσ) = λf/(πσ)) equals the waist diameter of a focused Gaussian beam. 2) n = 2, m = 0.

Vol. 25, No. 16 | 7 Aug 2017 | OPTICS EXPRESS 18729

F2,0, s ( ρ ,θ ) = −C2,0, s

 1  k ρσ  2  4π 2 (k ρ ) s − 2 σ 2 s + 2   π  −   exp is  θ −   × exp  2  f   2  λf f s−2    

2 4   k ρσ  1 + cos 2θ  k ρσ   ×  s ( s − 1) exp ( −i 2θ ) − s 1 + exp ( −i 2θ )  + 1   +   . 2   f   f  

(

(14)

)

Equation (14) is correct for s ≥ 2. For s = 1 it is necessary to use: F2,0,1 ( ρ ,θ ) = −C2,0,1

 1  k ρσ  2  i 2π 2 k ρσ 6 × exp  −   2  f   λf f  

2    k ρσ  × exp ( iθ )  4 + 2 exp ( −i 2θ ) −   (1 + cos 2θ )  .    f 

(15)

From Eq. (15) one can see the presence of zero at the origin. Rewriting the expression in parentheses with a division into the real and imaginary parts obtains: [4 + 2cos2θ - (kρσ/f)2(1 + cos2θ)] - i2sinθ. Thus, there are two additional zeroes at the points where two conditions are satisfied: θ = 0° and (kρσ/f)2 = 3, i.e. ρ1,2 = ± 3 f (kσ) , similarly to the previous case. In this case, the distance between zeros of intensity 3λ f (2π σ) (considering the zero at the origin) is less than the waist diameter of a focused Gaussian beam. 3) n = 1, m = 1. F1,1, s ( ρ ,θ ) = −C1,1, s

 1  k ρσ  2  i 2π 2 (k ρ ) s − 2 σ 2 s + 2   π  exp  −    exp is  θ −   × s−2  λf 2  f    2 f  

2 4   k ρσ   k ρσ   ×  2s ( s − 1) exp ( −i 2θ ) − 2 s exp ( −i 2θ )  − ⋅ θ i sin 2    .  f   f   

(16)

Equation (16) is correct for s ≥ 2. For s = 1 it is necessary to use:  1  k ρσ  2  2π 2 k ρσ 6 × exp  −  F1,1,1 ( ρ ,θ ) = C1,1,1  2  f   λf f   2    k ρσ  × exp ( iθ )  2 exp ( −i 2θ ) + i   sin 2θ  .    f 

(17)

As can be seen from Eq. (17), there is a zero of the first order at the origin. The square bracket is equal to 2cos2θ + i[(kρσ/f)2 - 2]sin 2θ. Hence, there is a root at the points when θ = 45°(135°) and (kρσ/f)2 = 2 . Thus, there are four zeros ρ1,2 = ± 2 f (kσ) (the “+” sign corresponds to the angles θ = 45° and 135°, the “-” sign corresponds to the angles θ = 225° and 315°). Zeros lie in the vertices of a square (on diagonals). 3. Experimental details

A so-called S-waveplate (Altechna), that allows the conversion of an incident circularly polarized Gaussian beam into a first-order vortex beam, was used to generate an initial single first-order optical vortex beam. The combination of a linear polarizer and a quarter-wave

Vol. 25, No. 16 | 7 Aug 2017 | OPTICS EXPRESS 18730

retarder was utilized to generate a circularly polarized laser beam and a pinhole was used for spatial filtering of the output laser beam (see Fig. 3). A second quarter-wave retarder was utilized to convert the generated circularly polarized vortex beam to linearly polarized one. Then, the generated first-order optical vortex beam was collimated with a first lens (Lens1) and directed at the SLM. A phase-only PLUTO VIS SLM based on a reflective LCOS microdisplay with a spatial resolution of 1920 × 1080 pixels and a pixel size of 8 mm was used. The illumination angle of the SLMs was approximately 8.5 degrees. Splitter two-level phase patterns were applied to the liquid crystal display of the SLM to perform the proofconcept experiments. The resolution of the phase patterns prepared and displayed on the SLMs was

Fig. 3. (A) Schematic of the experimental setup for investigation of two-level pure-phase optical vortex beam splitter. (B) Beam intensity distributions and interferograms measured in different planes of the experimental setup. In the transition from the plane 1 to the plane 2, a phase singularity is embedded into the laser beam and it acquires a donut-shape corresponding to a first-order vortex beam. The HG33 mode generator (the phase pattern is shown in the inset) splits the first-order optical vortex beam in the focal plane of the fourth lens (plane 3).

1024 × 1024 pixels. The 4f optical system consists of the second and third lenses (Lens2 and Lens3) and a diaphragm for spatial filtering of the laser beam reflected from the SLM. Using a beam splitter BS1, the output laser beam was split into two arms to create an interferometer. A forth lens (Lens4) collimated a reference beam in the second arm. Then, the object beam and the reference beam were recombined by a second beam splitter BS2. Then, a fifth lens (Lens5) focused the shaped light field on the display of the CCD-camera LOMO TC-1000 (3664⨯2740 pixel resolution with pixel size of 1.67 × 1.67 mm). For the design of the splitter phase patterns, the following formula was used:  2y   x2 + y 2   2 x  (18) Tnm ( x, y ) = exp  −  H m   .  H n  2 σ    σ   σ  In the case of n < 2 and m < 2, the amplitude of light field distribution described by Eq. (18) was ignored to design a pure-phase splitter, whereas when n ≥ 2 and m ≥ 2, amplitude encoding using a partial encoding technique [54] was performed. This technique allowed us to take into account the amplitude distribution from Eq. (18), while the efficiency of the generating light fields did not decrease significantly.

Vol. 25, No. 16 | 7 Aug 2017 | OPTICS EXPRESS 18731

4. Results and discussion

Figure 4 shows various phase patterns of the two-level splitters utilized to generate multiple optical vortex beams. Figure 5 shows images of multiple optical vortex beams generated using a different combination of S-waveplate and proposed phase patterns. Experimentally measured interferograms obtained as a result of interference of the generated vortex beam

Fig. 4. Phase pattern of the two-level splitters utilized to generate multiple optical vortex beams (black, 0; white, π).

arrays and a tilted plane wave indicate the presence of first-order phase singularities (forkshaped interference fringes), both in the case of the incident single vortex beam (see Fig. 3(B)) and a vortex beam array generated by a splitter. From the analysis of these interferograms, one can conclude that the number of phase singularities that indicate the presence of multiple optical vortices formed by the splitter is equal to: N = (n + 1)(m + 1) + nm. (19) These vortices have a topological charge equal to 1, but different signs, the number of positively-charged optical vortices is equal to (n + 1)(m + 1), and the number of negativelycharged optical vortices is nm. The total normalized OAM of such optical field can be calculated from the following equation [55]: ∞

μ=

 l⋅I

l =−∞ ∞

I

l =−∞

l

(20)

,

l

where Il is the partial energy of the l-th angular component of the light field. In this case, there are several vortex components of the first order with different energies: ( n +1)( m +1)

μ=

 t1 =1

( n +1)( m +1)

 t1 =1

nm

I t1 − I t2 t2 =1 nm

,

(21)

I t1 +  I t2 t2 =1

where I t1 is the energy of the vortex components of the positive first order, I t2 is the energy of the vortex components of the negative first. In this case, I t1 and I t2 do not have the same values.

Vol. 25, No. 16 | 7 Aug 2017 | OPTICS EXPRESS 18732

As is known, the HGnm mode is a set of (n + 1)(m + 1) intensity maxima [56]. The purephase HGnm mode former can be represented as a two-dimensional low-frequency or degenerate diffraction grating. For example, in the case where n = 0 and m = 1, the phase transmission function of the HGnm mode former and the phase transmission function of the binary diffraction grating generating two separate intensity maxima located at the minimum possible distance between them have identical profiles (see Fig. 4). Therefore, using the

Fig. 5. Experimentally obtained intensity distributions and interferograms of multiple optical vortex beams generated using a combination of a first-order optical vortex generator and an HGnm mode former. The arrows indicate the positions of the phase singularities in the images of the interferograms. The scale bar is 1 mm.

Vol. 25, No. 16 | 7 Aug 2017 | OPTICS EXPRESS 18733

proposed splitting method, one can expect the appearance of (n + 1)(m + 1) optical vortices. Where then do additional nm phase singularities come from on the simulated and experimentally measured interferograms? It has been shown previously that the superposition

Fig. 6. (A) Schematic representation of the controlled change in the ratio of the radius of the incident first-order vortex beam Rbeam and the radius of the utilized splitter R0. (B)-(С) Simulated and experimentally obtained intensity distributions and interferograms of multiple optical vortex beams generated at different values Rbeam/R0 for splitters with parameters n = 2, m = 2 and n = 3, m = 3. The arrows indicate the positions of the phase singularities in the images of the interferograms. The change in the number of phase singularities with decrease of Rbeam/R0 can be clearly seen. The scale bar is 1 mm.

of two noncoaxial closely spaced positively- or negatively-charged first-order optical vortices can create an additional negatively- or positively-charged first-order optical vortex located between them [57]. This situation is clearly visible, for example, in the case of the splitter with n = 1 and m = 1. In this case, and in cases of large values of the parameters n and m, this phenomenon leads to the appearance of additional nm optical vortices. As was noted in the previous section, the distance between these vortices is approximately the diameter of the waist of the focused Gaussian beam, which cannot be achieved with diffraction gratings, e.g., Dammann gratings [58]. The use of such an approach to the generation of multiple optical vortex beams, when the element forming the initial first-order vortex beam and the element performing the splitting are separated, makes it possible to transform the generated light field distribution only by changing the ratio of the radius of the incident first-order vortex beam Rbeam and the radius of the splitter (see Fig. 6). In fact, in this case, the splitter phase pattern passively changes, since the laser beam is no longer incident on all sections of the element. Consequently, the same splitter can be used to form optical vortex arrays with a controllable number of vortex beams. So, when the ratio of the radius of the incident first-order vortex beam Rbeam and the radius of the utilized splitter (n = 2, m = 2) R0 decreases, the number of generated optical vortex N also

Vol. 25, No. 16 | 7 Aug 2017 | OPTICS EXPRESS 18734

decreases from N = 13 for Rbeam/R0 = 1.0 to N = 4 for Rbeam/R0 = 0.5 and N = 1 for Rbeam/R0 < = 0.3 as shown in Fig. 6(B). Figure 6(C) shows the results of tunable generation of optical vortex arrays with the help of another splitter (n = 3, m = 3), also with a decrease of Rbeam/R0, the number of formed optical vortices changes sequentially from N = 25

Fig. 7. The simulation results for an effect of the wavelength of the initial single first-order vortex beam on the distortion of the multiple optical vortex beams generated by a two-level pure-phase splitter designed for operation with laser radiation at a wavelength of 532 nm. The inset shows a way to measure Iout and I0 used for evaluation.

(Rbeam/R0 = 1.0) to N = 4 (Rbeam/R0 = 0.5). Interestingly, a further decrease of the ratio Rbeam/R0 to 0.3 leads to appearance of a fifth vortex beam, thus, the phenomenon described by us earlier occurs. Note that in this case, with a further decrease of Rbeam/R0 to 0.2, the number of generated vortices does not decrease to 1, as in the case with a splitter with parameters n = 2 and m = 2 as the central part of the splitter has a phase jump of π (see Fig. 4). Thus, if both parameters n and m are even, then with a significant decrease in the radius of the incident first-order vortex beam, the number of generated vortices decreases to 1, but if at least one of these parameters is odd, the number of generated vortices does not decrease to 1 due to the presence of a phase jump of π. In the latter case, if both parameters n and m are odd, then for Rbeam/R0 ≤ 0.3, a light distribution identical to the light field generated by the splitter with n = 1 and m = 1 is shaped, that is, a fifth first-order vortex beam is formed. If only one of the parameters n or m is odd, then a light distribution identical to the light field generated by the splitter with n = 0 and m = 1 is shaped, that is, two first-order vortex beams are formed. It is also of note that when a splitter is used with the parameters n = 2, m = 2 and Rbeam/R0 = 0.3, the initial ring-shaped optical vortex beam incident on the splitter is transformed into an optical vortex beam with a square intensity distribution profile. Such a result can be expected when the phase jump with the square shape is located in the central part of the splitter, as occurs when using splitters with even n and m parameters. Figure 7 shows the simulation results for the effect of changing the wavelength of the initial single first-order vortex beam on the distortion of the multiple optical vortex beams generated by a two-level pure-phase splitter designed for operation with laser radiation at a wavelength of 532 nm. This was evaluated by measuring the ratio of the maximum intensity of one of the generated peripheral vortices Iout to the maximum intensity of the central vortex I0 (see inset on Fig. 7). As can be seen, the highest value of 0.86 is obtained in the ideal case, using laser radiation with the working wavelength of 532 nm. Deviation from this wavelength in the direction of both large and small values leads to a decrease in this ratio, indicating that an acceptable quality of splitting can be achieved if Iout/I0 ≥0.5. Then, in the case of a splitter designed for laser radiation with wavelength of 532 nm, the operating wavelength ranges from 512 to 554 nm. It follows that in relative values, the operating range of the proposed pure-phase two-level splitter is approximately 8% of the value of the central wavelength. This value coincides with the operating range of an S-waveplate (Altechna).

Vol. 25, No. 16 | 7 Aug 2017 | OPTICS EXPRESS 18735

5. Conclusions

In conclusion, we propose a new useful method for splitting first-order vortex beams using a two-level pure-phase splitter. The generation of closed-packed vortices and control of their number, not only by changing the profile of the utilized splitter, but also by changing the radius of an initial single first-order vortex beam incident on it, was demonstrated experimentally. The proposed element also allowed the controlled change of the incident vortex beam intensity profile from the ring-shape into square-shape. In addition, the high working efficiency of the splitter in the wavelength range of approximately 40 nm at a central wavelength of 532 nm (approximately 8% of the central wavelength) was demonstrated numerically. Note, the two-level splitter phase patterns were applied to the liquid crystal display of the multi-level reflective SLM to perform the proof-concept experiments. However, it is obvious that such a splitter can be easily fabricated by electron beam lithography or photolithography to increase its efficiency and the damage threshold, which is especially important for generating multiple optical vortex beams for high-performance parallel laser fabrication. Moreover, low-cost binary refractive or reflective SLMs [59–61] can be used to implement such a splitter. An S-waveplate was used to generate the initial single first-order vortex beam, but the proposed splitting method makes it possible to use other elements as the vortex beam generator, for example, a q-plate, a spiral phase plate, a fork-shaped hologram [62], plasmonic optical vortex emitters and so on. In addition, the proposed splitter should allow splitting of vortex beams generated at the nano- and microscale level [45, 63–67]. We believe the proposed elements provide an efficient way to generate reconfigurable multiple closed-packed optical vortex beams for potential applications in modern optics and photonics, especially in laser fabrication and optical manipulation. Funding

Russian Scientific Foundation (RSF) (16-12-10165).