Letter
Vol. 40, No. 22 / November 15 2015 / Optics Letters
5129
Singular optical lattice generation using light beams with orbital angular momentum WILLAMYS C. SOARES,1,* ANDRÉ L. MOURA,1,2 ASKERY A. CANABARRO,1,3 EMERSON DE LIMA,1 AND JANDIR M. HICKMANN4 1
Grupo de Física da Matéria Condensada, Núcleo de Ciências Exatas—NCEX, Campus Arapiraca, Universidade Federal de Alagoas, 57309-005 Arapiraca, AL, Brazil 2 Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife, PE, Brazil 3 Center for Polymer Studies and Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA 4 Instituto de Física, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, 91501-970 Porto Alegre, RS, Brazil *Corresponding author: [email protected] Received 21 August 2015; revised 5 October 2015; accepted 7 October 2015; posted 7 October 2015 (Doc. ID 248283); published 2 November 2015
In this Letter we numerically and experimentally demonstrated that a lattice with an optical vortex distributed over the entire lattice can be generated in the Fourier space using three higher-order Laguerre–Gauss beams placed at the vertices of an equilateral triangle in real space. In this scheme the optical vortice’s lattice presents a topological defect in its central region. Probing the net topological charge of the whole lattice, we found that it corresponds to the topological charge associated with the orbital angular momentum of each beam in real space. © 2015 Optical Society of America OCIS codes: (080.4865) Optical vortices; (260.6042) Singular optics; (070.2580) Paraxial wave optics. http://dx.doi.org/10.1364/OL.40.005129
Optical vortices [1] have been a subject of investigation in distinct physical phenomena, for example, optical solitons [2] and second harmonic generation [3]. In particular, optical vortices appear in light beams possessing a helical wavefront associated with a singular phase structure expimφ where φ is the azimuthal angle and m is an integer number corresponding to the number of intertwined helical wavefronts, also referred to as the topological charge of the light beam. This phase singularity defines the orbital angular momentum (OAM) of light as pointed out by Allen and co-workers in 1992 [4]. Higher-order Laguerre–Gauss (LG) [4] and higher-order Bessel beams [5] are examples of light beams possessing OAM where the signature of the phase singularity is a dark region at the transverse intensity distribution. In addition, optical vortices are also found in zero-intensity points from destructive interference involving three or more light fields forming an optical vortice’s lattice [6–11]. Optical vortices’ lattices have been observed in the interference of three light fields by amplitude and wavefront division [12], nonlinear propagation of light beams [13], spatial mode pattern 0146-9592/15/225129-03$15/0$15.00 © 2015 Optical Society of America
of vertical-cavity surface-emitting lasers [14,15], and light propagation in multimode waveguides [16]. In this Letter we numerically and experimentally present a novel method to generate optical vortices’ lattices. In our scheme we prepare three high-order LG beams positioned in the vertices of an equilateral triangle. Inserting a lens in the path of the propagating beams, the optical vortices’ lattice is formed in the far-field region, corresponding to the Fourier space. Furthermore, a dark region in its central region of the lattice is observed giving a rising LG-shaped optical lattice. Probing the net topological charge of the whole lattice, we show that it is the same as the LG beams that are used to generate the lattice. Let us start considering one high-order LG beam in the plane z 0. We consider the case where the radial parameter p 0 so that the pattern distribution of the beam is a single ring: l 2 r r Er; ϕ E 0 (1) exp − 2 expil φ; w0 w0 where E 0 , w0 , and l are the amplitude parameter, the waist width, and the topological charge, respectively. The field distribution at the far-field region is obtained using the Fraunhofer diffraction integral, which corresponds to the Fourier transform of Eq. (1). As a result, we get [17] 2 2 k w l Ek ⊥ ; γ Aw0 k ⊥ exp − ⊥ 0 expil γ; (2) 4 where k ⊥ and γ are the transversal component and azimuthal angle of the wave vector, respectively. A is the amplitude parameter. Now, our goal is to determine the Fraunhofer diffraction pattern in the far-field region of a beam shifted r⃗ 0 with respect to the z axis using the two-dimensional shift theorem [18]. The desired quantity is then given by Ψi k⊥ ; γ Ek ⊥ ; γ exp−ik⊥ r 0 cosγ − θn ;
(3)
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Vol. 40, No. 22 / November 15 2015 / Optics Letters
Letter
where r 0 and θn 2πn∕N are the radial displacement and the azimuthal angle of the nth beam, respectively. The index N is the number of the beams. Note that the amplitude spectrum of the Fourier is invariant to translation. Using this approach we prepare three high-order LG beams placed at the vertices of an equilateral triangle at z 0. This spatial configuration of light beams is shown in Fig. 1(a). Performing Eq. (3) for this configuration we have 2 2 k w Ψk⊥ ; γ Aw0 k ⊥ l exp − ⊥ 0 expil γgk ⊥ ; γ; (4) 4 where gk⊥ ; γ
N X n1
2πn ; exp −ik ⊥ r 0 sin γ − N
Fig. 2. (a)–(c) are the phase structure of the whole lattice calculated by the interference of the lattice of Figs. 1(b)–1(d) with the Gaussian profile reference beam. The spiral patterns indicate the presence of an optical vortex in the whole lattice. The topological charge associated with the optical vortex is equal to the topological charge associated with the optical vortex of the beams that are used to generate the lattice.
(5)
gk ⊥ ; γ is set as a lattice function and is responsible by generating a hexagonal lattice pattern in which case N 3. Note that the function g can be used to generate other lattices pattern in Fourier space, such as a square lattice by making N 4. It is found that optical lattices are formed as illustrated in Fig. 1 for l 1 (b), l 2 (c), and l 3 (d). As we can see, hexagonal optical lattices are obtained in Fourier space but a defect in their central region is presented. Note that a Laguerre–Gaussian modulation is imposed over the bright spots forming the reciprocal lattice. This can also be seen by analyzing Eq. (4). In order to investigate the defect presented in the optical lattice of Figs. 1(b)–1(d), we numerically analyzed the phase structure of the whole lattice calculating the interference pattern with the Gaussian profile reference beam. The results are shown in Fig. 2. The observed spiral patterns clearly indicate
Fig. 1. (a) Scheme formed by high-order LG beams placed at the vertices of an equilateral triangle at z 0 in the real space. The diffraction integral for this configuration results in the optical lattices illustrated in (b)–(d) for l 1, l 2, and l 3, respectively. As we can see, hexagonal optical lattices are obtained in the Fourier space but presenting a defect in their central region. Note that an LG modulation is imposed over the bright spots forming the lattice.
the presence of an optical vortex in the whole lattice. It is important to note that the topological charge associated with the optical vortex is equal to the topological charge of the beams that generate the lattice. The same observation can be made by comparing Eqs. (1) and (4). The two equations have the same topological charge. An experiment has been performed to confirm our numerical results. The experimental setup is depicted in Fig. 3. The LG beams are generated using an argon laser, operating at 514 nm to illuminate a computer-generated hologram [19]. Selecting different diffraction orders, we obtain LG with l 1, l 2, and l 3. For each order of the LG beam we exploit a modified Michelson interferometer to generate the optical lattice. Different from previous schemes [20], in the output of the interferometer the beams propagate parallel to their axes corresponding to the vertices of an equilateral triangle and pass through a lens. It is worth pointing out that to analyze the topological charge of the outcome beams the technique proposed by Hickmann et al. was used [21]. The results are shown
Fig. 3. In the experimental setup, the LG beams are produced by an argon laser operating at 514 nm that illuminates a computer-generated hologram. The different l in the LG beams is obtained by selecting different diffraction orders. The modified Michelson interferometer allows us to generate the beams propagating parallel to each other, corresponding to the vertices of an equilateral triangle, as depicted in Fig. 1(a). These beams pass through a lens to generate the lattice in the focal plane of the lens, which coincides with the plane of the CCD camera to register the intensity distribution pattern.
Letter
Fig. 4. Experimental results for the diffraction formed by high-order LG beams by three triangular apertures. Three equilateral triangular apertures were placed in the propagation path of the beams and we measured the far-field diffraction pattern for the topological charge (a) l 2 and (b) l 3.
Vol. 40, No. 22 / November 15 2015 / Optics Letters
5131
In summary, we have demonstrated that optical vortices’ lattices can be generated in Fourier space using light beams possessing OAM. We have also verified that the net topological charge of an optical vortice’s lattice is equal to the topological charge of the LG beams forming the lattice. The obtained results can be used as a mechanism of manufacturing periodic structure modulation for the refractive index of a nonlinear media as well as creating two-dimensional photonic crystals. Furthermore, we hope these results shed light on new investigation of recent applications of the angular orbital momentum of light in diverse fields such as micro-manipulation, imaging, communication systems [22], optical tweezers, and many others. Funding. Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) (PDE 207360/2014-6); Fundação de Amparo à Pesquisa do Estado de Alagoas (FAPEAL) (PPP—20110902-011-0025-0069/60030-733/2011); INCT/Informação Quântica. Acknowledgment. We thankfully acknowledge the support from CAPES, MCT/CNPq, PRONEX/FAPEAL, Nanofoton research network, Finep/CTInfra/Proinfra, ANPCTPETRO, and INCT/Informação Quântica.
Fig. 5. Experimental optical lattices generated by the experimental setup presented in Fig. 3 for the topological charge (a) l 1, (b) l 2, and l 3 (c).
in Fig. 4 for the topological charge l 2, 4(a), and l 3, 4(b). In the lens focal plane corresponding to the Fourier space, the intensity distribution pattern is registered by a charge-coupled device (CCD) camera. The generated optical lattices are shown in Fig. 5. Using the Gaussian profile of the argon laser as a reference beam, we obtain the interference pattern of the optical lattices. The results are shown in Fig. 6 demonstrating that the topological defect observed in the whole lattice has the same topological charge of the LG beams used to generate the lattice. Our experimental results are in perfect agreement with our numerical predictions for the spatial structure of an optical lattice as well as the measurement of the net topological charge.
Fig. 6. Experimental interference pattern of the optical lattices presented in Fig. 5 with the Gaussian profile reference beam of the argon laser for the topological charge (a) l 2 and (b) l 3. We can see that the topological defect observed in the whole lattice has the same topological charge as the LG beam used to generate the lattice. These results are in perfect agreement with the numerical predictions presented in Fig. 2.
REFERENCES 1. J. F. Nye and M. V. Berry, Proc. R. Soc. London A 336, 165 (1974). 2. G. A. Swartzlander, Jr. and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992). 3. J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997). 4. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992). 5. K. Volke-Sepulveda, V. Garces-Chez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, J. Opt. B 4, S82 (2002). 6. J. Masajada and B. Dubik, Opt. Commun. 198, 21 (2001). 7. K. O’Holleran, M. J. Padgett, and M. R. Dennis, Opt. Express 14, 3039 (2006). 8. G. Ruben and D. M. Paganin, Phys. Rev. E 75, 066613 (2007). 9. G. Bartal, O. Cohen, H. Buljan, J. W. Fleischer, O. Manela, and M. Segev, Phys. Rev. Lett. 96, 163902 (2010). 10. D. Song, V. Paltoglou, S. Liu, Y. Zhu, D. Gallardo, L. Tang, J. Xu, M. Ablowitz, N. K. Efremidis, and Z. Che, Nat. Commun. 6, 6272 (2015). 11. Y. Zhang, Z. Wu, M. R. Belić, H. Zheng, Z. Wang, M. Xiao, and Y. Zhang, Laser Photon. Rev. 9, 331 (2015). 12. J. Masajada, A. Popiolek-Masajada, and M. Leniec, Opt. Express 15, 5196 (2007). 13. G. Ruben, D. M. Paganin, and M. J. Morgan, Phys. Rev. A 78, 013631 (2008). 14. F. B. De Colstoun, G. Khitrova, A. V. Fedorov, T. R. Nelson, C. Lowery, T. M. Brennan, B. G. Hammons, and P. D. Maker, Chaos Solitons Fractals 4, 1575 (1994). 15. J. Scheuer and M. Orenstein, Science 285, 230 (1999). 16. R. M. Jenkins, J. Banerji, and A. R. Davies, J. Opt. A 3, 527 (2001). 17. Y. Zhang, X. Liu, M. R. Belić, W. Zhong, F. Wen, and Y. Zhang, Opt. Lett. 40, 3786 (2015). 18. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996). 19. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998). 20. S. Vyas and P. Senthilkumaran, Appl. Opt. 46, 2893 (2007). 21. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. ChávezCerda, Phys. Rev. Lett. 105, 053904 (2010). 22. 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, Nat. Photonics 6, 488 (2012).
Vol. 40, No. 22 / November 15 2015 / Optics Letters
5129
Singular optical lattice generation using light beams with orbital angular momentum WILLAMYS C. SOARES,1,* ANDRÉ L. MOURA,1,2 ASKERY A. CANABARRO,1,3 EMERSON DE LIMA,1 AND JANDIR M. HICKMANN4 1
Grupo de Física da Matéria Condensada, Núcleo de Ciências Exatas—NCEX, Campus Arapiraca, Universidade Federal de Alagoas, 57309-005 Arapiraca, AL, Brazil 2 Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife, PE, Brazil 3 Center for Polymer Studies and Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA 4 Instituto de Física, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, 91501-970 Porto Alegre, RS, Brazil *Corresponding author: [email protected] Received 21 August 2015; revised 5 October 2015; accepted 7 October 2015; posted 7 October 2015 (Doc. ID 248283); published 2 November 2015
In this Letter we numerically and experimentally demonstrated that a lattice with an optical vortex distributed over the entire lattice can be generated in the Fourier space using three higher-order Laguerre–Gauss beams placed at the vertices of an equilateral triangle in real space. In this scheme the optical vortice’s lattice presents a topological defect in its central region. Probing the net topological charge of the whole lattice, we found that it corresponds to the topological charge associated with the orbital angular momentum of each beam in real space. © 2015 Optical Society of America OCIS codes: (080.4865) Optical vortices; (260.6042) Singular optics; (070.2580) Paraxial wave optics. http://dx.doi.org/10.1364/OL.40.005129
Optical vortices [1] have been a subject of investigation in distinct physical phenomena, for example, optical solitons [2] and second harmonic generation [3]. In particular, optical vortices appear in light beams possessing a helical wavefront associated with a singular phase structure expimφ where φ is the azimuthal angle and m is an integer number corresponding to the number of intertwined helical wavefronts, also referred to as the topological charge of the light beam. This phase singularity defines the orbital angular momentum (OAM) of light as pointed out by Allen and co-workers in 1992 [4]. Higher-order Laguerre–Gauss (LG) [4] and higher-order Bessel beams [5] are examples of light beams possessing OAM where the signature of the phase singularity is a dark region at the transverse intensity distribution. In addition, optical vortices are also found in zero-intensity points from destructive interference involving three or more light fields forming an optical vortice’s lattice [6–11]. Optical vortices’ lattices have been observed in the interference of three light fields by amplitude and wavefront division [12], nonlinear propagation of light beams [13], spatial mode pattern 0146-9592/15/225129-03$15/0$15.00 © 2015 Optical Society of America
of vertical-cavity surface-emitting lasers [14,15], and light propagation in multimode waveguides [16]. In this Letter we numerically and experimentally present a novel method to generate optical vortices’ lattices. In our scheme we prepare three high-order LG beams positioned in the vertices of an equilateral triangle. Inserting a lens in the path of the propagating beams, the optical vortices’ lattice is formed in the far-field region, corresponding to the Fourier space. Furthermore, a dark region in its central region of the lattice is observed giving a rising LG-shaped optical lattice. Probing the net topological charge of the whole lattice, we show that it is the same as the LG beams that are used to generate the lattice. Let us start considering one high-order LG beam in the plane z 0. We consider the case where the radial parameter p 0 so that the pattern distribution of the beam is a single ring: l 2 r r Er; ϕ E 0 (1) exp − 2 expil φ; w0 w0 where E 0 , w0 , and l are the amplitude parameter, the waist width, and the topological charge, respectively. The field distribution at the far-field region is obtained using the Fraunhofer diffraction integral, which corresponds to the Fourier transform of Eq. (1). As a result, we get [17] 2 2 k w l Ek ⊥ ; γ Aw0 k ⊥ exp − ⊥ 0 expil γ; (2) 4 where k ⊥ and γ are the transversal component and azimuthal angle of the wave vector, respectively. A is the amplitude parameter. Now, our goal is to determine the Fraunhofer diffraction pattern in the far-field region of a beam shifted r⃗ 0 with respect to the z axis using the two-dimensional shift theorem [18]. The desired quantity is then given by Ψi k⊥ ; γ Ek ⊥ ; γ exp−ik⊥ r 0 cosγ − θn ;
(3)
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Vol. 40, No. 22 / November 15 2015 / Optics Letters
Letter
where r 0 and θn 2πn∕N are the radial displacement and the azimuthal angle of the nth beam, respectively. The index N is the number of the beams. Note that the amplitude spectrum of the Fourier is invariant to translation. Using this approach we prepare three high-order LG beams placed at the vertices of an equilateral triangle at z 0. This spatial configuration of light beams is shown in Fig. 1(a). Performing Eq. (3) for this configuration we have 2 2 k w Ψk⊥ ; γ Aw0 k ⊥ l exp − ⊥ 0 expil γgk ⊥ ; γ; (4) 4 where gk⊥ ; γ
N X n1
2πn ; exp −ik ⊥ r 0 sin γ − N
Fig. 2. (a)–(c) are the phase structure of the whole lattice calculated by the interference of the lattice of Figs. 1(b)–1(d) with the Gaussian profile reference beam. The spiral patterns indicate the presence of an optical vortex in the whole lattice. The topological charge associated with the optical vortex is equal to the topological charge associated with the optical vortex of the beams that are used to generate the lattice.
(5)
gk ⊥ ; γ is set as a lattice function and is responsible by generating a hexagonal lattice pattern in which case N 3. Note that the function g can be used to generate other lattices pattern in Fourier space, such as a square lattice by making N 4. It is found that optical lattices are formed as illustrated in Fig. 1 for l 1 (b), l 2 (c), and l 3 (d). As we can see, hexagonal optical lattices are obtained in Fourier space but a defect in their central region is presented. Note that a Laguerre–Gaussian modulation is imposed over the bright spots forming the reciprocal lattice. This can also be seen by analyzing Eq. (4). In order to investigate the defect presented in the optical lattice of Figs. 1(b)–1(d), we numerically analyzed the phase structure of the whole lattice calculating the interference pattern with the Gaussian profile reference beam. The results are shown in Fig. 2. The observed spiral patterns clearly indicate
Fig. 1. (a) Scheme formed by high-order LG beams placed at the vertices of an equilateral triangle at z 0 in the real space. The diffraction integral for this configuration results in the optical lattices illustrated in (b)–(d) for l 1, l 2, and l 3, respectively. As we can see, hexagonal optical lattices are obtained in the Fourier space but presenting a defect in their central region. Note that an LG modulation is imposed over the bright spots forming the lattice.
the presence of an optical vortex in the whole lattice. It is important to note that the topological charge associated with the optical vortex is equal to the topological charge of the beams that generate the lattice. The same observation can be made by comparing Eqs. (1) and (4). The two equations have the same topological charge. An experiment has been performed to confirm our numerical results. The experimental setup is depicted in Fig. 3. The LG beams are generated using an argon laser, operating at 514 nm to illuminate a computer-generated hologram [19]. Selecting different diffraction orders, we obtain LG with l 1, l 2, and l 3. For each order of the LG beam we exploit a modified Michelson interferometer to generate the optical lattice. Different from previous schemes [20], in the output of the interferometer the beams propagate parallel to their axes corresponding to the vertices of an equilateral triangle and pass through a lens. It is worth pointing out that to analyze the topological charge of the outcome beams the technique proposed by Hickmann et al. was used [21]. The results are shown
Fig. 3. In the experimental setup, the LG beams are produced by an argon laser operating at 514 nm that illuminates a computer-generated hologram. The different l in the LG beams is obtained by selecting different diffraction orders. The modified Michelson interferometer allows us to generate the beams propagating parallel to each other, corresponding to the vertices of an equilateral triangle, as depicted in Fig. 1(a). These beams pass through a lens to generate the lattice in the focal plane of the lens, which coincides with the plane of the CCD camera to register the intensity distribution pattern.
Letter
Fig. 4. Experimental results for the diffraction formed by high-order LG beams by three triangular apertures. Three equilateral triangular apertures were placed in the propagation path of the beams and we measured the far-field diffraction pattern for the topological charge (a) l 2 and (b) l 3.
Vol. 40, No. 22 / November 15 2015 / Optics Letters
5131
In summary, we have demonstrated that optical vortices’ lattices can be generated in Fourier space using light beams possessing OAM. We have also verified that the net topological charge of an optical vortice’s lattice is equal to the topological charge of the LG beams forming the lattice. The obtained results can be used as a mechanism of manufacturing periodic structure modulation for the refractive index of a nonlinear media as well as creating two-dimensional photonic crystals. Furthermore, we hope these results shed light on new investigation of recent applications of the angular orbital momentum of light in diverse fields such as micro-manipulation, imaging, communication systems [22], optical tweezers, and many others. Funding. Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) (PDE 207360/2014-6); Fundação de Amparo à Pesquisa do Estado de Alagoas (FAPEAL) (PPP—20110902-011-0025-0069/60030-733/2011); INCT/Informação Quântica. Acknowledgment. We thankfully acknowledge the support from CAPES, MCT/CNPq, PRONEX/FAPEAL, Nanofoton research network, Finep/CTInfra/Proinfra, ANPCTPETRO, and INCT/Informação Quântica.
Fig. 5. Experimental optical lattices generated by the experimental setup presented in Fig. 3 for the topological charge (a) l 1, (b) l 2, and l 3 (c).
in Fig. 4 for the topological charge l 2, 4(a), and l 3, 4(b). In the lens focal plane corresponding to the Fourier space, the intensity distribution pattern is registered by a charge-coupled device (CCD) camera. The generated optical lattices are shown in Fig. 5. Using the Gaussian profile of the argon laser as a reference beam, we obtain the interference pattern of the optical lattices. The results are shown in Fig. 6 demonstrating that the topological defect observed in the whole lattice has the same topological charge of the LG beams used to generate the lattice. Our experimental results are in perfect agreement with our numerical predictions for the spatial structure of an optical lattice as well as the measurement of the net topological charge.
Fig. 6. Experimental interference pattern of the optical lattices presented in Fig. 5 with the Gaussian profile reference beam of the argon laser for the topological charge (a) l 2 and (b) l 3. We can see that the topological defect observed in the whole lattice has the same topological charge as the LG beam used to generate the lattice. These results are in perfect agreement with the numerical predictions presented in Fig. 2.
REFERENCES 1. J. F. Nye and M. V. Berry, Proc. R. Soc. London A 336, 165 (1974). 2. G. A. Swartzlander, Jr. and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992). 3. J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997). 4. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992). 5. K. Volke-Sepulveda, V. Garces-Chez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, J. Opt. B 4, S82 (2002). 6. J. Masajada and B. Dubik, Opt. Commun. 198, 21 (2001). 7. K. O’Holleran, M. J. Padgett, and M. R. Dennis, Opt. Express 14, 3039 (2006). 8. G. Ruben and D. M. Paganin, Phys. Rev. E 75, 066613 (2007). 9. G. Bartal, O. Cohen, H. Buljan, J. W. Fleischer, O. Manela, and M. Segev, Phys. Rev. Lett. 96, 163902 (2010). 10. D. Song, V. Paltoglou, S. Liu, Y. Zhu, D. Gallardo, L. Tang, J. Xu, M. Ablowitz, N. K. Efremidis, and Z. Che, Nat. Commun. 6, 6272 (2015). 11. Y. Zhang, Z. Wu, M. R. Belić, H. Zheng, Z. Wang, M. Xiao, and Y. Zhang, Laser Photon. Rev. 9, 331 (2015). 12. J. Masajada, A. Popiolek-Masajada, and M. Leniec, Opt. Express 15, 5196 (2007). 13. G. Ruben, D. M. Paganin, and M. J. Morgan, Phys. Rev. A 78, 013631 (2008). 14. F. B. De Colstoun, G. Khitrova, A. V. Fedorov, T. R. Nelson, C. Lowery, T. M. Brennan, B. G. Hammons, and P. D. Maker, Chaos Solitons Fractals 4, 1575 (1994). 15. J. Scheuer and M. Orenstein, Science 285, 230 (1999). 16. R. M. Jenkins, J. Banerji, and A. R. Davies, J. Opt. A 3, 527 (2001). 17. Y. Zhang, X. Liu, M. R. Belić, W. Zhong, F. Wen, and Y. Zhang, Opt. Lett. 40, 3786 (2015). 18. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996). 19. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998). 20. S. Vyas and P. Senthilkumaran, Appl. Opt. 46, 2893 (2007). 21. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. ChávezCerda, Phys. Rev. Lett. 105, 053904 (2010). 22. 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, Nat. Photonics 6, 488 (2012).
