Abstract

We present a general comprehensive framework for the description of symmetries of complex light fields, facilitating the construction of sophisticated periodic structures carrying phase dislocations. In particular, we demonstrate the derivation of all three fundamental two-dimensional vortex lattices based on vortices of triangular, quadratic, and hexagonal shape, respectively. We show that these patterns represent the foundation of complex three-dimensional lattices with outstanding helical intensity distributions which suggest valuable applications in holographic lithography. This systematic approach is substantiated by a comparative study of corresponding numerically calculated and experimentally realized complex intensity and phase distributions.

© 2011 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]

2011 (2)

M. Petrović, M. Belić, C. Denz, and Y. Kivshar, “Counterpropagating optical beams and solitons,” Laser Photon. Rev. 5, 214–233 (2011).
[CrossRef]

X. Xiong, X. Chen, M. Wang, R. Peng, and D. Shu, “Optically nonactive assorted helix array with interchangeable magnetic / electric resonance,” Appl. Phys. Lett. 98, 071901 (2011).
[CrossRef]

2010 (6)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[CrossRef]

J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater. 22, 356–360 (2010).
[CrossRef] [PubMed]

B. Terhalle, D. Göries, T. Richter, P. Rose, A. S. Desyatnikov, F. Kaiser, and C. Denz, “Anisotropy-controlled topological stability of discrete vortex solitons in optically induced photonic lattices,” Opt. Lett. 35, 604–606 (2010).
[CrossRef] [PubMed]

M. Decker, R. Zhao, C. M. Soukoulis, S. Linden, and M. Wegener, “Twisted split-ring-resonator photonic meta-material with huge optical activity,” Opt. Lett. 35, 1593–1595 (2010).
[CrossRef] [PubMed]

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).
[CrossRef]

M. Thiel, H. Fischer, G. von Freymann, and M. Wegener, “Three-dimensional chiral photonic superlattices,” Opt. Lett. 35, 166–168 (2010).
[CrossRef] [PubMed]

2009 (6)

C. Lu and R. Lipson, “Interference lithography: a powerful tool for fabricating periodic structures,” Laser Photon. Rev. 4, 568–580 (2009).
[CrossRef]

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Physical Review B 79, 035407 (2009).
[CrossRef]

S. Zhang, Y.-S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative Refractive Index in Chiral Metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. 34, 2501–2503 (2009).
[CrossRef] [PubMed]

M. Mazilu, D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: non-diffracting beams,” Laser Photon. Rev. 4, 529–547 (2009).
[CrossRef]

F. Courvoisier, P.-A. Lacourt, M. Jacquot, M. K. Bhuyan, L. Furfaro, and J. M. Dudley, “Surface nanoprocessing with nondiffracting femtosecond Bessel beams,” Opt. Lett. 34, 3163–3165 (2009).
[CrossRef] [PubMed]

2008 (3)

2007 (5)

M. Thiel, G. von Freymann, and M. Wegener, “Layer-by-layer three-dimensional chiral photonic crystals,” Opt. Lett. 32, 2547–2549 (2007).
[CrossRef] [PubMed]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization Stop Bands in Chiral Polymeric Three-Dimensional Photonic Crystals,” Adv. Mater. 19, 207–210 (2007).
[CrossRef]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization Stop Bands in Chiral Polymeric Three-Dimensional Photonic Crystals,” Adv. Mater. 19, 207–210 (2007).
[CrossRef]

C. Zhang and T. J. Cui, “Negative reflections of electromagnetic waves in a strong chiral medium,” Applied Physics Letters 91, 194101 (2007).
[CrossRef]

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

2006 (3)

C. Guo, Y. Zhang, Y. Han, J. Ding, and H. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259, 449–454 (2006).
[CrossRef]

C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express 14, 6604–6612 (2006).
[CrossRef] [PubMed]

F. Miyamaru and M. Hangyo, “Strong optical activity in chiral metamaterials of metal screw hole arrays,” Appl. Phys. Lett. 89, 211105 (2006).
[CrossRef]

2005 (2)

2004 (1)

J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[CrossRef] [PubMed]

2003 (2)

2002 (1)

J. Curtis, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

2000 (1)

A. König and N. D. Mermin, “Symmetry, extinctions, and band sticking,” Am. J. Phys. 68, 525–530 (2000).
[CrossRef]

1999 (1)

A. König and N. D. Mermin, “Screw rotations and glide mirrors: crystallography in Fourier space,” Proc. Natl. Acad. Sci. U.S.A. 96, 3502–3506 (1999).
[CrossRef] [PubMed]

1992 (1)

N. Mermin, “The space groups of icosahedral quasicrystals and cubic, orthorhombic, monoclinic, and triclinic crystals,” Rev. Mod. Phys. 64, 3–49 (1992).
[CrossRef]

1991 (1)

D. Rabson, N. Mermin, D. Rokhsar, and D. Wright, “The space groups of axial crystals and quasicrystals,” Rev. Mod. Phys. 63, 699–733 (1991).
[CrossRef]

1988 (2)

D. S. Rokhsar, D. C. Wright, and N. D. Mermin, “The two-dimensional quasicrystallographic space groups with rotational symmetries less than 23-fold,” Acta Crystallogr. 44, 197–211 (1988).
[CrossRef]

D. Rokhsar, D. Wright, and N. Mermin, “Scale equivalence of quasicrystallographic space groups,” Phys. Rev. B 37, 8145–8149 (1988).
[CrossRef]

1987 (1)

1958 (1)

A. Schinzel, “Sur l’existence d’un cercle passant par un nombre donné de points aux coordonnées entières,” Enseign. Math. 4, 71–72 (1958).

Bai, B.

Barrett, D.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Belic, M.

M. Petrović, M. Belić, C. Denz, and Y. Kivshar, “Counterpropagating optical beams and solitons,” Laser Photon. Rev. 5, 214–233 (2011).
[CrossRef]

Betzig, E.

E. Betzig, “Sparse and composite coherent lattices,” Phys. Rev. A 71, 063406 (2005).
[CrossRef]

Bhuyan, M. K.

Boguslawski, M.

J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater. 22, 356–360 (2010).
[CrossRef] [PubMed]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000), chap. 6, pp. 105–135, 3rd ed.

Chen, X.

X. Xiong, X. Chen, M. Wang, R. Peng, and D. Shu, “Optically nonactive assorted helix array with interchangeable magnetic / electric resonance,” Appl. Phys. Lett. 98, 071901 (2011).
[CrossRef]

Chui, S. T.

S. T. Chui, “Giant wave rotation for small helical structures,” J. Appl. Phys. 104, 013904 (2008).
[CrossRef]

Collett, E.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Courvoisier, F.

Cui, T. J.

C. Zhang and T. J. Cui, “Negative reflections of electromagnetic waves in a strong chiral medium,” Applied Physics Letters 91, 194101 (2007).
[CrossRef]

Curtis, J.

J. Curtis, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

Curtis, J. E.

Decker, M.

M. Decker, R. Zhao, C. M. Soukoulis, S. Linden, and M. Wegener, “Twisted split-ring-resonator photonic meta-material with huge optical activity,” Opt. Lett. 35, 1593–1595 (2010).
[CrossRef] [PubMed]

M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. 34, 2501–2503 (2009).
[CrossRef] [PubMed]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization Stop Bands in Chiral Polymeric Three-Dimensional Photonic Crystals,” Adv. Mater. 19, 207–210 (2007).
[CrossRef]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization Stop Bands in Chiral Polymeric Three-Dimensional Photonic Crystals,” Adv. Mater. 19, 207–210 (2007).
[CrossRef]

Denz, C.

M. Petrović, M. Belić, C. Denz, and Y. Kivshar, “Counterpropagating optical beams and solitons,” Laser Photon. Rev. 5, 214–233 (2011).
[CrossRef]

B. Terhalle, D. Göries, T. Richter, P. Rose, A. S. Desyatnikov, F. Kaiser, and C. Denz, “Anisotropy-controlled topological stability of discrete vortex solitons in optically induced photonic lattices,” Opt. Lett. 35, 604–606 (2010).
[CrossRef] [PubMed]

J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater. 22, 356–360 (2010).
[CrossRef] [PubMed]

Desyatnikov, A. S.

Deubel, M.

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization Stop Bands in Chiral Polymeric Three-Dimensional Photonic Crystals,” Adv. Mater. 19, 207–210 (2007).
[CrossRef]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization Stop Bands in Chiral Polymeric Three-Dimensional Photonic Crystals,” Adv. Mater. 19, 207–210 (2007).
[CrossRef]

Dholakia, K.

M. Mazilu, D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: non-diffracting beams,” Laser Photon. Rev. 4, 529–547 (2009).
[CrossRef]

Ding, J.

C. Guo, Y. Zhang, Y. Han, J. Ding, and H. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259, 449–454 (2006).
[CrossRef]

Dong, J.

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Physical Review B 79, 035407 (2009).
[CrossRef]

Dudley, J. M.

Durnin, J.

Fahrbach, F. O.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[CrossRef]

Fedotov, V.

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Physical Review B 79, 035407 (2009).
[CrossRef]

Fischer, H.

Fraher, B.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Furfaro, L.

Garanovich, I. L.

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).
[CrossRef]

Göries, D.

Grier, D. G.

Gunn-Moore, F.

M. Mazilu, D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: non-diffracting beams,” Laser Photon. Rev. 4, 529–547 (2009).
[CrossRef]

Guo, C.

C. Guo, Y. Zhang, Y. Han, J. Ding, and H. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259, 449–454 (2006).
[CrossRef]

Han, Y.

C. Guo, Y. Zhang, Y. Han, J. Ding, and H. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259, 449–454 (2006).
[CrossRef]

Hangyo, M.

F. Miyamaru and M. Hangyo, “Strong optical activity in chiral metamaterials of metal screw hole arrays,” Appl. Phys. Lett. 89, 211105 (2006).
[CrossRef]

Jacquot, M.

Joseph, J.

J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater. 22, 356–360 (2010).
[CrossRef] [PubMed]

Juzeliu¯nas, G.

G. Juzeliu̅nas and P. Öhberg, “Optical Manipulation of Ultracold Atoms,” in Structured Light and Its Applications, D. Andrews, ed., (Elsevier, 2008), chap. 12, pp. 259–333.

Kaiser, F.

Karvinen, P.

Kivshar, Y.

M. Petrović, M. Belić, C. Denz, and Y. Kivshar, “Counterpropagating optical beams and solitons,” Laser Photon. Rev. 5, 214–233 (2011).
[CrossRef]

König, A.

A. König and N. D. Mermin, “Symmetry, extinctions, and band sticking,” Am. J. Phys. 68, 525–530 (2000).
[CrossRef]

A. König and N. D. Mermin, “Screw rotations and glide mirrors: crystallography in Fourier space,” Proc. Natl. Acad. Sci. U.S.A. 96, 3502–3506 (1999).
[CrossRef] [PubMed]

Konishi, K.

Koschny, T.

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Physical Review B 79, 035407 (2009).
[CrossRef]

Kriegler, C. E.

Kuwata-Gonokami, M.

Kwon, D.-H.

Lacourt, P.-A.

Li, J.

S. Zhang, Y.-S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative Refractive Index in Chiral Metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

Lin, J.

Linden, S.

M. Decker, R. Zhao, C. M. Soukoulis, S. Linden, and M. Wegener, “Twisted split-ring-resonator photonic meta-material with huge optical activity,” Opt. Lett. 35, 1593–1595 (2010).
[CrossRef] [PubMed]

M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. 34, 2501–2503 (2009).
[CrossRef] [PubMed]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization Stop Bands in Chiral Polymeric Three-Dimensional Photonic Crystals,” Adv. Mater. 19, 207–210 (2007).
[CrossRef]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization Stop Bands in Chiral Polymeric Three-Dimensional Photonic Crystals,” Adv. Mater. 19, 207–210 (2007).
[CrossRef]

Lipson, R.

C. Lu and R. Lipson, “Interference lithography: a powerful tool for fabricating periodic structures,” Laser Photon. Rev. 4, 568–580 (2009).
[CrossRef]

Lu, C.

C. Lu and R. Lipson, “Interference lithography: a powerful tool for fabricating periodic structures,” Laser Photon. Rev. 4, 568–580 (2009).
[CrossRef]

Lu, X.

S. Zhang, Y.-S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative Refractive Index in Chiral Metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

Matuszewski, M.

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).
[CrossRef]

Mazilu, M.

M. Mazilu, D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: non-diffracting beams,” Laser Photon. Rev. 4, 529–547 (2009).
[CrossRef]

Meng, X.

Mermin, N.

N. Mermin, “The space groups of icosahedral quasicrystals and cubic, orthorhombic, monoclinic, and triclinic crystals,” Rev. Mod. Phys. 64, 3–49 (1992).
[CrossRef]

D. Rabson, N. Mermin, D. Rokhsar, and D. Wright, “The space groups of axial crystals and quasicrystals,” Rev. Mod. Phys. 63, 699–733 (1991).
[CrossRef]

D. Rokhsar, D. Wright, and N. Mermin, “Scale equivalence of quasicrystallographic space groups,” Phys. Rev. B 37, 8145–8149 (1988).
[CrossRef]

Mermin, N. D.

A. König and N. D. Mermin, “Symmetry, extinctions, and band sticking,” Am. J. Phys. 68, 525–530 (2000).
[CrossRef]

A. König and N. D. Mermin, “Screw rotations and glide mirrors: crystallography in Fourier space,” Proc. Natl. Acad. Sci. U.S.A. 96, 3502–3506 (1999).
[CrossRef] [PubMed]

D. S. Rokhsar, D. C. Wright, and N. D. Mermin, “The two-dimensional quasicrystallographic space groups with rotational symmetries less than 23-fold,” Acta Crystallogr. 44, 197–211 (1988).
[CrossRef]

Miyamaru, F.

F. Miyamaru and M. Hangyo, “Strong optical activity in chiral metamaterials of metal screw hole arrays,” Appl. Phys. Lett. 89, 211105 (2006).
[CrossRef]

Niu, H. B.

Öhberg, P.

G. Juzeliu̅nas and P. Öhberg, “Optical Manipulation of Ultracold Atoms,” in Structured Light and Its Applications, D. Andrews, ed., (Elsevier, 2008), chap. 12, pp. 259–333.

Park, Y.-S.

S. Zhang, Y.-S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative Refractive Index in Chiral Metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[CrossRef] [PubMed]

Peng, R.

X. Xiong, X. Chen, M. Wang, R. Peng, and D. Shu, “Optically nonactive assorted helix array with interchangeable magnetic / electric resonance,” Appl. Phys. Lett. 98, 071901 (2011).
[CrossRef]

Peng, X.

Petrovic, M.

M. Petrović, M. Belić, C. Denz, and Y. Kivshar, “Counterpropagating optical beams and solitons,” Laser Photon. Rev. 5, 214–233 (2011).
[CrossRef]

Plum, E.

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Physical Review B 79, 035407 (2009).
[CrossRef]

Rabson, D.

D. Rabson, N. Mermin, D. Rokhsar, and D. Wright, “The space groups of axial crystals and quasicrystals,” Rev. Mod. Phys. 63, 699–733 (1991).
[CrossRef]

Richter, T.

Rohrbach, A.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[CrossRef]

Rokhsar, D.

D. Rabson, N. Mermin, D. Rokhsar, and D. Wright, “The space groups of axial crystals and quasicrystals,” Rev. Mod. Phys. 63, 699–733 (1991).
[CrossRef]

D. Rokhsar, D. Wright, and N. Mermin, “Scale equivalence of quasicrystallographic space groups,” Phys. Rev. B 37, 8145–8149 (1988).
[CrossRef]

Rokhsar, D. S.

D. S. Rokhsar, D. C. Wright, and N. D. Mermin, “The two-dimensional quasicrystallographic space groups with rotational symmetries less than 23-fold,” Acta Crystallogr. 44, 197–211 (1988).
[CrossRef]

Rose, P.

B. Terhalle, D. Göries, T. Richter, P. Rose, A. S. Desyatnikov, F. Kaiser, and C. Denz, “Anisotropy-controlled topological stability of discrete vortex solitons in optically induced photonic lattices,” Opt. Lett. 35, 604–606 (2010).
[CrossRef] [PubMed]

J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater. 22, 356–360 (2010).
[CrossRef] [PubMed]

Ruther, M.

Schaefer, B.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Schinzel, A.

A. Schinzel, “Sur l’existence d’un cercle passant par un nombre donné de points aux coordonnées entières,” Enseign. Math. 4, 71–72 (1958).

Schmitz, C. H. J.

Shu, D.

X. Xiong, X. Chen, M. Wang, R. Peng, and D. Shu, “Optically nonactive assorted helix array with interchangeable magnetic / electric resonance,” Appl. Phys. Lett. 98, 071901 (2011).
[CrossRef]

Simon, P.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[CrossRef]

Smyth, R.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Soukoulis, C.

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Physical Review B 79, 035407 (2009).
[CrossRef]

Soukoulis, C. M.

Spatz, J. P.

Stevenson, D.

M. Mazilu, D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: non-diffracting beams,” Laser Photon. Rev. 4, 529–547 (2009).
[CrossRef]

Sukhorukov, A. A.

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).
[CrossRef]

Svirko, Y. P.

Tao, S. H.

Terhalle, B.

Thiel, M.

M. Thiel, H. Fischer, G. von Freymann, and M. Wegener, “Three-dimensional chiral photonic superlattices,” Opt. Lett. 35, 166–168 (2010).
[CrossRef] [PubMed]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization Stop Bands in Chiral Polymeric Three-Dimensional Photonic Crystals,” Adv. Mater. 19, 207–210 (2007).
[CrossRef]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization Stop Bands in Chiral Polymeric Three-Dimensional Photonic Crystals,” Adv. Mater. 19, 207–210 (2007).
[CrossRef]

M. Thiel, G. von Freymann, and M. Wegener, “Layer-by-layer three-dimensional chiral photonic crystals,” Opt. Lett. 32, 2547–2549 (2007).
[CrossRef] [PubMed]

Turunen, J.

Uhrig, K.

von Freymann, G.

M. Thiel, H. Fischer, G. von Freymann, and M. Wegener, “Three-dimensional chiral photonic superlattices,” Opt. Lett. 35, 166–168 (2010).
[CrossRef] [PubMed]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization Stop Bands in Chiral Polymeric Three-Dimensional Photonic Crystals,” Adv. Mater. 19, 207–210 (2007).
[CrossRef]

M. Thiel, G. von Freymann, and M. Wegener, “Layer-by-layer three-dimensional chiral photonic crystals,” Opt. Lett. 32, 2547–2549 (2007).
[CrossRef] [PubMed]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization Stop Bands in Chiral Polymeric Three-Dimensional Photonic Crystals,” Adv. Mater. 19, 207–210 (2007).
[CrossRef]

Wang, H.

C. Guo, Y. Zhang, Y. Han, J. Ding, and H. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259, 449–454 (2006).
[CrossRef]

Wang, M.

X. Xiong, X. Chen, M. Wang, R. Peng, and D. Shu, “Optically nonactive assorted helix array with interchangeable magnetic / electric resonance,” Appl. Phys. Lett. 98, 071901 (2011).
[CrossRef]

Wegener, M.

Werner, D. H.

Werner, P. L.

Wright, D.

D. Rabson, N. Mermin, D. Rokhsar, and D. Wright, “The space groups of axial crystals and quasicrystals,” Rev. Mod. Phys. 63, 699–733 (1991).
[CrossRef]

D. Rokhsar, D. Wright, and N. Mermin, “Scale equivalence of quasicrystallographic space groups,” Phys. Rev. B 37, 8145–8149 (1988).
[CrossRef]

Wright, D. C.

D. S. Rokhsar, D. C. Wright, and N. D. Mermin, “The two-dimensional quasicrystallographic space groups with rotational symmetries less than 23-fold,” Acta Crystallogr. 44, 197–211 (1988).
[CrossRef]

Xavier, J.

J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater. 22, 356–360 (2010).
[CrossRef] [PubMed]

Xiong, X.

X. Xiong, X. Chen, M. Wang, R. Peng, and D. Shu, “Optically nonactive assorted helix array with interchangeable magnetic / electric resonance,” Appl. Phys. Lett. 98, 071901 (2011).
[CrossRef]

Yuan, X.-C.

Zhang, C.

C. Zhang and T. J. Cui, “Negative reflections of electromagnetic waves in a strong chiral medium,” Applied Physics Letters 91, 194101 (2007).
[CrossRef]

Zhang, S.

S. Zhang, Y.-S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative Refractive Index in Chiral Metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

Zhang, W.

S. Zhang, Y.-S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative Refractive Index in Chiral Metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

Zhang, X.

S. Zhang, Y.-S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative Refractive Index in Chiral Metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

Zhang, Y.

C. Guo, Y. Zhang, Y. Han, J. Ding, and H. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259, 449–454 (2006).
[CrossRef]

Zhao, R.

Zheludev, N.

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Physical Review B 79, 035407 (2009).
[CrossRef]

Zhou, J.

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Physical Review B 79, 035407 (2009).
[CrossRef]

M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. 34, 2501–2503 (2009).
[CrossRef] [PubMed]

Acta Crystallogr. (1)

D. S. Rokhsar, D. C. Wright, and N. D. Mermin, “The two-dimensional quasicrystallographic space groups with rotational symmetries less than 23-fold,” Acta Crystallogr. 44, 197–211 (1988).
[CrossRef]

Adv. Mater. (3)

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization Stop Bands in Chiral Polymeric Three-Dimensional Photonic Crystals,” Adv. Mater. 19, 207–210 (2007).
[CrossRef]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization Stop Bands in Chiral Polymeric Three-Dimensional Photonic Crystals,” Adv. Mater. 19, 207–210 (2007).
[CrossRef]

J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater. 22, 356–360 (2010).
[CrossRef] [PubMed]

Am. J. Phys. (2)

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

A. König and N. D. Mermin, “Symmetry, extinctions, and band sticking,” Am. J. Phys. 68, 525–530 (2000).
[CrossRef]

Appl. Phys. Lett. (2)

F. Miyamaru and M. Hangyo, “Strong optical activity in chiral metamaterials of metal screw hole arrays,” Appl. Phys. Lett. 89, 211105 (2006).
[CrossRef]

X. Xiong, X. Chen, M. Wang, R. Peng, and D. Shu, “Optically nonactive assorted helix array with interchangeable magnetic / electric resonance,” Appl. Phys. Lett. 98, 071901 (2011).
[CrossRef]

Applied Physics Letters (1)

C. Zhang and T. J. Cui, “Negative reflections of electromagnetic waves in a strong chiral medium,” Applied Physics Letters 91, 194101 (2007).
[CrossRef]

Enseign. Math. (1)

A. Schinzel, “Sur l’existence d’un cercle passant par un nombre donné de points aux coordonnées entières,” Enseign. Math. 4, 71–72 (1958).

J. Appl. Phys. (1)

S. T. Chui, “Giant wave rotation for small helical structures,” J. Appl. Phys. 104, 013904 (2008).
[CrossRef]

J. Opt. Soc. Am. A (1)

Laser Photon. Rev. (3)

M. Petrović, M. Belić, C. Denz, and Y. Kivshar, “Counterpropagating optical beams and solitons,” Laser Photon. Rev. 5, 214–233 (2011).
[CrossRef]

M. Mazilu, D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: non-diffracting beams,” Laser Photon. Rev. 4, 529–547 (2009).
[CrossRef]

C. Lu and R. Lipson, “Interference lithography: a powerful tool for fabricating periodic structures,” Laser Photon. Rev. 4, 568–580 (2009).
[CrossRef]

Nat. Photonics (1)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[CrossRef]

Nature (1)

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

Opt. Commun. (2)

J. Curtis, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

C. Guo, Y. Zhang, Y. Han, J. Ding, and H. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Opt. Commun. 259, 449–454 (2006).
[CrossRef]

Opt. Express (4)

Opt. Lett. (7)

Phys. Rev. A (2)

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).
[CrossRef]

E. Betzig, “Sparse and composite coherent lattices,” Phys. Rev. A 71, 063406 (2005).
[CrossRef]

Phys. Rev. B (1)

D. Rokhsar, D. Wright, and N. Mermin, “Scale equivalence of quasicrystallographic space groups,” Phys. Rev. B 37, 8145–8149 (1988).
[CrossRef]

Phys. Rev. Lett. (1)

S. Zhang, Y.-S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative Refractive Index in Chiral Metamaterials,” Phys. Rev. Lett. 102, 023901 (2009).
[CrossRef] [PubMed]

Physical Review B (1)

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Physical Review B 79, 035407 (2009).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

A. König and N. D. Mermin, “Screw rotations and glide mirrors: crystallography in Fourier space,” Proc. Natl. Acad. Sci. U.S.A. 96, 3502–3506 (1999).
[CrossRef] [PubMed]

Rev. Mod. Phys. (2)

D. Rabson, N. Mermin, D. Rokhsar, and D. Wright, “The space groups of axial crystals and quasicrystals,” Rev. Mod. Phys. 63, 699–733 (1991).
[CrossRef]

N. Mermin, “The space groups of icosahedral quasicrystals and cubic, orthorhombic, monoclinic, and triclinic crystals,” Rev. Mod. Phys. 64, 3–49 (1992).
[CrossRef]

Science (1)

J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[CrossRef] [PubMed]

Other (2)

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000), chap. 6, pp. 105–135, 3rd ed.

G. Juzeliu̅nas and P. Öhberg, “Optical Manipulation of Ultracold Atoms,” in Structured Light and Its Applications, D. Andrews, ed., (Elsevier, 2008), chap. 12, pp. 259–333.

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Figures (9)

Fig. 1
Fig. 1

Lattice given by the interference of five plane waves. (a) Fourier components of the field as points of intersection of a circle with the reciprocal lattice. (b) Numerically calculated intensity distribution in the transverse plane with the desired periodicity.

Fig. 2
Fig. 2

(a) Desired schematic real space field distribution. (b) Basic field configurations resulting in a threefold rotational symmetry of the complex field.

Fig. 3
Fig. 3

(a) Desired schematic real space field distribution. (b) Basic field configurations resulting in a fourfold rotational symmetry of the complex field.

Fig. 4
Fig. 4

(a) Desired schematic real space field distribution. (b) Most basic field configuration resulting in a sixfold rotational symmetry of the complex field.

Fig. 5
Fig. 5

Setup for the generation of the discussed lattices. BS: beam splitter, CCD: camera, L: lens, FF: Fourier filter, HWP: half-wave plate, M: mirror, MO: microscope objective, P: polarizer, PBS: polarizing beam splitter, PH: pinhole, QWP: quarter-wave plate, S: shutter, SLM: spatial light modulator, TS: translational stage.

Fig. 6
Fig. 6

Periodic nondiffracting vortex beams. First column: Transverse Fourier components of the field. Second and third column: Numerically calculated and experimentally measured intensity distributions. Fourth and fifth column: Numerically calculated and experimentally measured phase distributions.

Fig. 7
Fig. 7

Three-dimensional triangular helix lattice with longitudinal periodicity c. (a) Numerically calculated 0.6I max-isointensity surfaces with the slices AE corresponding to different z-planes. (b) Comparison between numerical and experimental intensity distributions at different z.

Fig. 8
Fig. 8

Three-dimensional square helix lattice with longitudinal periodicity c. (a) Numerically calculated 0.6I max-isointensity surfaces with the slices AE corresponding to different z-planes. (b) Comparison between numerical and experimental intensity distributions at different z.

Fig. 9
Fig. 9

Three-dimensional hexagonal helix lattice with longitudinal periodicity c. (a) Numerically calculated 0.6I max-isointensity surfaces with the slices AE corresponding to different z-planes. (b) Comparison between numerical and experimental intensity distributions at different z.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

E n = A n e i k n r + i ψ n , n = 1 , , N .
I ( r ) = n = 1 N A n 2 + 2 n = 2 N m = 1 n 1 A n A m cos ( K n m r + Ψ n m ) ,
B = 2 π ( A 1 ) T ,
k n = k 0 + B h n , n = 1 , 2
k 0 = 1 4 π A ( H 1 ) T β , β n : = B h n 2 .
k 1 = k 0 + 2 b 1 8 b 2 , k 2 = k 0 + 2 b 1 + 8 b 2 , k 3 = k 0 + 5 b 1 5 b 2 , k 4 = k 0 + 5 b 1 + 5 b 2 .
k 0 = 50 π 9 a e x , k 1 = k 0 + 2 b 1 8 b 2 = π 9 a [ 14 e x + 48 e y ] , k 2 = k 0 + 2 b 1 + 8 b 2 = π 9 a [ 14 e x 48 e y ] .
E ( k ) = e i 2 π ( χ ( k k 0 ) + χ 0 ) E ( k ) ,
χ ( i h i b i ) = i h i χ ( b i ) h i .
E ( g k ) = e i 2 π [ Φ g ( k k 0 ) + φ g ] E ( k ) .
Φ g ( k k 0 ) + φ g Φ g ( k k 0 ) + φ g + χ ( g k k ) ,
Φ g ( k k 0 ) Φ g ( k k 0 ) + χ ( [ g 1 ] ( k k 0 ) )
φ g φ g + χ ( g k 0 k 0 ) .
Φ g h ( k k 0 ) Φ g ( h ( k k 0 ) ) + Φ h ( k k 0 ) φ g h φ g + φ h + Φ g ( h k 0 k 0 ) ,
I ( K ) = n = 1 N A n 2 δ ( 3 ) ( K ) + n = 2 N m = 1 n 1 A n A m [ e i Ψ n m δ ( 3 ) ( K K n m ) + e i Ψ n m δ ( 3 ) ( K + K n m ) ] ,
I ( K ) = d 3 k E ( k ) E * ( k K ) I ( g K ) = d 3 k E ( k ) E * ( k g K )
I ( g K ) = d 3 k E ( g k ) E * ( g [ k K ] ) = ( 6 ) d 3 k e i 2 π [ Φ g ( k k 0 ) Φ g ( k K k 0 ) ] E ( k ) E * ( k K ) = e i 2 π Φ g ( K ) d 3 k E ( k ) E * ( k K ) I ( g K ) = ( 13 ) e i 2 π Φ g ( k ) I ( K ) .
ψ 1 = ψ 0 = 0 + 2 π ( Φ m ( k 1 k 0 ) = 0 + φ m = 1 / 2 ) = π , ψ 2 = ψ 0 = 0 + 2 π ( Φ r 3 ( k 2 k 0 ) = 0 + φ r 3 = 0 ) = 0 , ψ 3 = ψ 1 = π + 2 π ( Φ r 3 ( k 3 k 0 ) = 0 + φ r 3 = 0 ) = π , ψ 4 = ψ 2 = 0 + 2 π ( Φ r 3 ( k 4 k 0 ) = 0 + φ r 3 = 0 ) = 0 , ψ 5 = ψ 3 = π + 2 π ( Φ r 3 ( k 5 k 0 ) = 0 + φ r 3 = 0 ) = π .
E tri ( θ , r ) = n = 0 2 [ e i k r cos ( θ + 2 π n 3 + arctan ( 3 3 ) ) + e i k r cos ( θ + 2 π n 3 arctan ( 3 3 ) ) + i π ] .
Φ g ( n 1 b 1 + n 2 b 2 ) + φ g = n 1 2 + 0 = n 1 2 .
ψ 7 = ψ 0 = 0 + 2 π ( Φ g ( b 2 ) = 0 + φ g = 0 ) = 0 , ψ 6 = ψ 7 = 0 + 2 π ( Φ m ( b 1 + 2 b 2 ) = 0 + φ m = 1 / 2 ) = π , ψ 1 = ψ 6 = π + 2 π ( Φ g ( b 1 b 2 ) = 1 / 2 + φ g = 0 ) = 2 π 0 mod 2 π , ψ 4 = ψ 1 = 0 + 2 π ( Φ m ( 3 b 1 + b 2 ) = 0 + φ m = 1 / 2 ) = π , ψ 3 = ψ 4 = π + 2 π ( Φ g ( 3 b 1 ) = 1 / 2 + φ g = 0 ) = 2 π 0 mod 2 π , ψ 2 = ψ 3 = 0 + 2 π ( Φ m ( 2 b 1 b 2 ) = 0 + φ m = 1 / 2 ) = π , ψ 5 = ψ 2 = π + 2 π ( Φ g ( 2 b 1 + 2 b 2 ) = 0 + φ g = 0 ) = 0.
E square ( θ , r ) = n = 0 3 [ e i k r cos ( θ + n π 2 + arctan 2 ) + i n π 2 + e i k r cos ( θ + n π 2 arctan 2 ) + i ( n + 2 ) π 2 ] ,
E hex ( θ , r ) = n = 0 5 e i k r cos ( θ π n 3 ) + i π n 3 .
E tri , 3 D ( θ , r , z ) = E tri ( θ , r ) e i k z + A tri e i k z ,
A tri = max r 3 I t r i ( r ) .
E square , 3 D ( θ , r , z ) = E square ( θ , r ) e i k z + A square e i k z
E hex , 3 D ( θ , r , z ) = E hex ( θ , r ) e i k z + A hex e i k z ,

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