## Systematic approach to complex periodic vortex and helix lattices |

Optics Express, Vol. 19, Issue 10, pp. 9848-9862 (2011)

http://dx.doi.org/10.1364/OE.19.009848

Acrobat PDF (4659 KB)

### 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

## 1. Introduction

1. 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]

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

1. 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]

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]

7. 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]

8. 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]

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

12. 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]

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

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]

14. 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]

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

1. 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]

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

## 2. Periodic interference

*N*plane waves

*E*. We restrict our analysis to

_{n}*N*< ∞ and monochromatic fields. Each of these waves

*E*– propagating in direction

_{n}**k**

*with real amplitude*

_{n}*A*> 0 and phase offset

_{n}*ψ*– can be expressed in scalar approximation as The harmonic time dependence e

_{n}^{i}

*, with*

^{ωt}*ω*= 2

*πc*/

*λ*, is implicitly assumed, but neglected in notation, since only one wavelength

*λ*is considered. This restricts the wave vectors

**k**

*to a sphere of radius*

_{n}*k*:= ||

**k**

*|| = 2*

_{n}*π*/

*λ*∀

*n*in Fourier space.

*I*(

**r**) of the superposition of all

*N*waves is obtained as having introduced

**K**

*=*

_{nm}**k**

*–*

_{n}**k**

*as well as Ψ*

_{m}*=*

_{nm}*ψ*–

_{n}*ψ*.

_{m}*k*with the sites of some periodic (or quasiperiodic) reciprocal lattice. For simplicity, we restrict our considerations to periodic lattices. This criterion, as simple and self-evident as it may sound, is nonetheless not very well known and does not seem to have been exploited to its full extent. Therefore, we will demonstrate its power by giving a general procedure for the construction of periodic nondiffracting beams.

17. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**, 651–654 (1987). [CrossRef]

**a**

_{1}

*,*

**a**

_{2}}. The reciprocal basis ℬ = {

**b**

_{1}

*,*

**b**

_{2}} is then obtained in matrix notation through where the

*n*-th columns of the matrices

*A*and

*B*are the corresponding

*n*-th basis vectors of 𝒜 and ℬ. The transverse wave vectors can be cast into the form with integer vectors

**h**

*, because every*

_{n}**h**

*∈ ℤ*

_{n}^{2}. This choice is what determines the radius of the circle in Fourier space, namely

**h**

*together as the columns of a matrix*

_{n}*H*, we can construct the circle through the three points at

*H*, there may be more than three sites of intersection of the constructed circle with the lattice, i.e. solutions for

*n*> 2, can be added to the overall light field, without the field losing its periodicity in the selected basis.

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

**a**

_{1}=

*a*

**e**

*,*

_{x}**a**

_{2}= 3

*a*

**e**

*(*

_{y}*a*being the lattice parameter), which gives the reciprocal basis as {

**b**

_{1}= (2

*π*/

*a*)

**e**

*,*

_{x}**b**

_{2}= (2

*π*/(3

*a*))

**e**

*}. We choose*

_{y}**h**

_{1}= [2, −8]

*,*

^{T}**h**

_{2}= [2

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

8. 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]

*T*and using Eqs. (4) and (5), we construct the first wave vectors

*A*= 1 and setting all relative phases

_{n}*ψ*= 0, the intensity distribution of Fig. 1(b) is obtained.

_{n}## 3. Symmetry considerations

20. 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]

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

*g,h*∈

*G*, so is

_{E}*gh*∈

*G*, since

_{E}*G*is a group. We can determine the corresponding phase function Φ

_{E}*(*

_{gh}**K**) +

*φ*directly by These relations originate from the associativity

_{gh}*g*(

*h*

**k**) = (

*gh*)

**k**in conjunction with Eq. (7).

**K**) +

*φ*, we turn to the intensity. Fourier transformation of Eq. (2) gives with the three-dimensional Dirac delta distribution

*δ*

^{(3)}(

**K**). Since the intensity

*I*(

**r**) is a real and positive density, the application of RWM-methodology [20

20. 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]

*I*(

**K**) for

*E*(

**k**), leaving out

**k**

_{0}and neglecting the constant phases

*χ*

_{0}and

*φ*, all Eqs. (6)–(11) remain valid.

_{g}*g*∈

*G*of the field are at the same time also symmetries of the intensity pattern

_{E}*g*∈

*G*, where

_{I}*G*denotes the point group of the intensity lattice. With the Wiener-Khinchin theorem we find and substituting

_{I}**k**′ =

*g*

**k**″ yields

*g*of the intensity is also a symmetry of the field.

## 4. Periodic phase vortex lattices

26. 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]

27. 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]

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

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

34. J. Lin, X.-C. Yuan, S. H. Tao, X. Peng, and H. B. Niu, “Deterministic approach to the generation of modified helical beams for optical manipulation,” Opt. Express **13**, 3862–3867 (2005). [CrossRef] [PubMed]

35. 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]

36. 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]

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

### 4.1. Lattices of triangular vortices

*V*and

*W*), confined by lines of zero intensity

*M*,

*M*′ and

*M*″.

*m*: if the field’s reflection at the line

*M*(or respectively at

*M*′ or

*M*″) is the equal to the original field phase-shifted by

*π*, we have a

*π*-phase jump at the mirror line

*M*(or

*M*′,

*M*″) and consequently the intensity will vanish there. Placing phase vortices at

*V*and

*W*will result in the desired central intensity dips, but because of the imposed mirror symmetry, the vortices at

*V*must have the sign of their topological charge opposite to those at

*W*, thus restraining rotational symmetry at

*R*,

*V*, or

*W*to a threefold axis

*r*

_{3}. The point group of the field is thus

*e*being the identity).

*r*

_{3}. Different choices are possible (e.g. circles

*A*,

*B*and

*C*in Fig. 2(b)), but one can show that the simplest configuration resulting in the point group

*G*corresponds to the circle

_{E}*C*. In

*A*the mirror

*m*cannot have a

*π*-phase jump, whereas in

*B*, the

*π*-phase jump mirror

*m*is incompatible with the existence of vortices.

*C*yields six points of intersection with the lattice and knowing the directions

**k**

_{0}

*,...,*

**k**

_{5}of the interfering plane waves, their relative phases

*ψ*and amplitudes

_{n}*A*are the only unknowns left. By symmetry, all amplitudes have to be equal. The phases can be constructed by arbitrarily choosing

_{n}*ψ*

_{0}= 0 and applying the mirror symmetry

*m*and rotation

*r*

_{3}in Fourier space. Defining the real space origin to be in the point of highest symmetry

*R*, the phase function of the mirror

*m*is determined by Φ

*(*

_{m}**k**–

**k**

_{0}) = 0 as well as

*φ*= 1/2 and the phase function of the rotation

_{m}*r*

_{3}is zero: Φ

_{r3}(

**k**–

**k**

_{0}) = 0, φ

_{r3}= 0. One obtains Thus, the total field can be expressed in polar coordinates as

### 4.2. Square vortex lattice

*V*and

*W*), separated by lines of zero intensity

*M*and

*M*′. A vortex at

*V*and

*W*will give rise to the local dips, and as before, the lines of zero intensity can be realized by a

*π*-shifting mirror, resulting in a different sign of the topological charge of the vortices at

*V*and

*W*. This means, that the lattice is additionally invariant under the action of the essential glide mirrors (i.e. there is no simple mirror symmetry parallel to it in real space, cf. [38

38. 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]

*G*and

*G*′. Furthermore, one notices a fourfold rotational symmetry about

*V*and

*W*, while only a twofold rotation

*r*

_{2}is possible in

*R*, due to the different signs of the vortices at

*V*and

*W*. The point group of the desired lattice thus has to contain a fourfold rotation

*r*

_{4}(about

*V*or

*W*), a mirror

*m*(along

*M*or

*M*′) and a glide mirror

*g*,

*g*′ (along

*G*or

*G*′ respectively). Fortunately, we will see that a simple mirror

*m*together with a diagonal glide mirror

*g*is sufficient to generate all of the above symmetry elements.

*A*or

*B*with a square lattice. Symmetry dictates that these circles have to be respectively centered at either of the two fourfold symmetry points of the lattice. The configuration using

*A*however does not allow for a

*π*-phase shifting mirror and a glide mirror at 45° to it at the same time. It turns out that a configuration using circle

*B*is the simplest choice of plane waves, compatible with the symmetries we require, and that the

*π*-phase jump mirror

*m*as well as the glide mirror

*g*have to be aligned as indicated in Fig. 3(b). Interchanging them would be incompatible with the fourfold rotational symmetry.

*ψ*

_{0}

*,...,ψ*

_{7}using the phase functions corresponding to the symmetries

*g*and

*m*. We start again by setting

*ψ*

_{0}= 0 and positioning the origin of real space at point

*S*(cf. Fig. 3(a)), which is invariant with respect to mirror symmetry over

*G*,

*G*′ and

*M*. Consequently, the phase function of the mirror

*m*needs to be constant in this gauge and this constant has to be

*ϕ*= 1/2, since we want a

_{m}*π*-phase jump at the mirror planes of

*M*. The phase functions of the glide mirror

*g*is Here, the value of the constant part

*φ*is obtained by application of Eq. (11) on the identity

_{g}*e*=

*gg*. One finds that either

*φ*= 0 or

_{g}*φ*= 1/2, both of which are valid choices. For simplicity, we opt for

_{g}*φ*= 0. The form of the linear term

_{g}*n*

_{1}/2 is due to the fact, that

*g*is an essential glide mirror (for details see [25

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

*m*and

*g*to the phase

*ψ*

_{0}of the plane wave at

**k**

_{0}we find

*A*evidently need to be equal by symmetry here and thus, the total field can be written as which is gauge equivalent to the phase configuration derived above. An experimental realization of this vortex lattice will be presented in Section 5.

_{n}### 4.3. Hexagonal vortices

*V*, the most basic configuration possible obviously consists of six plane waves along the directions

**k**

_{0}

*,...,*

**k**

_{5}whose transverse wave vectors are on the corners of a regular hexagon in Fourier space (cf. Fig. 4(b)). Rotation of the complex field by an angle of

*π*/3 about one vortex

*V*has to be accompanied with a constant overall phase offset of the same amount of

*π*/3 because we requested a topological charge of 1. In Fourier space, this amounts to phases of

*ψ*=

_{n}*nπ*/3.

*r*

_{6}and comprises no mirror symmetries, making this – in contrast to the two vortex lattices discussed before – a chiral periodic light field. This absence of mirrors is due to the fact, that there are phase vortices on every mirror line of the underlying hexagonal lattice, breaking the mirror symmetry.

## 5. Experimental realization of all three vortex lattices

*E*(

**r**)) of the desired wave field

*E*(

**r**). Another spatial filtering is performed in the Fourier plane of a subsequent demagnifying telescope. The Fourier mask is a metallic plate with small holes drilled in the positions of the desired Fourier components. The size of the holes is chosen such that the sharp foci of the individual spatial frequency components can pass through them without cutoff.

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

*π*-phase jumps at the corresponding positions.

## 6. Three-dimensional helix lattices

*k*-component in the direction of the optical axis (here along the

*z*-axis) [40

40. 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]

### 6.1. Triangular helix lattice

*z*-axis – forms the structure depicted in Fig. 7. The total field now reads with

*A*

_{tri}being the amplitude of the plane wave in arbitrary units.

*A*

_{tri}≈ 4.39. By interference with the additional plane wave, the lattice gains another dimension of periodicity. In real space, the corresponding period is

*c*=

*λ*/(

*k*–

*k*) along the

^{||}*z*-direction, with

*I*

_{max}-isointensity surfaces of the three-dimensional lattice. One notices, that the intensity now forms helices around 3

_{1}-screw axes. The fact that adjacent helices turn in opposite senses corresponds to the vortices of the field

*E*

_{tri}having topological charges with opposite signs. Despite the presence of screw axes, this configuration is still achiral because the added plane wave is on the invariant plane of the mirror symmetry of

*E*

_{tri}. Accordingly, the mirror is retained.

*A*–

*E*along the

*z*-direction, at which in Fig. 7(b) we present a comparison between the corresponding numerically calculated and experimental captured intensities. The obvious conformity between numerics and experiment proves the spiraling character of the experimentally obtained three-dimensional intensity distribution.

### 6.2. Square helix lattice

*A*

_{square}≈ 4.34 is the optimal amplitude of the added plane wave. The resulting numerically calculated three-dimensional intensity distribution is presented in Fig. 8(a). It consists of helically shaped regions of highest intensity, arranged on a square lattice. Each individual spiral is seen to be cleanly separated from its next neighbors by planes of zero intensity parallel to the

*z*-axis. Again, we find that adjacent helices turn in opposite senses as required by the different signs of the topological charges of the corresponding two-dimensional vortex lattice

*E*

_{square}. The modulation of the intensity along the

*z*-axis is also recognized in the experimentally generated intensity lattice depicted in Fig. 8(b). One observes that the intensity in the plane

*A*matches very well the intensity in

*E*. This is due to the fact that these planes are separated by one longitudinal period

*c*along the

*z*-axis.

41. 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]

### 6.3. Hexagonal helix lattice

*A*

_{hex}≈ 3.52 to be particularly interesting. Its intensity distribution is depicted in Fig. 9(a).

*E*

_{hex}the topological charges of all corresponding vortices have the same sign. Despite the absence of mirrors, each individual helix is still seen to be quite separate from its neighbors (cf. Fig. 9(b)). The origin of this is seen by a closer examination of the phase pattern of

*E*

_{hex}(cf. Fig. 6(c)) revealing additional vortices positioned at the edges and corners of the underlying hexagonal tiling (cf.

*U*and

*W*in Fig. 3(a)). The absence of mirror symmetries in conjunction with the helices all turning in the same sense proves the chirality of this structure, making it at the same time highly attractive for the fabrication of chiral photonic crystals or photonic lattices with fascinating properties.

## 7. Conclusion

## Acknowledgments

## References and links

1. | 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. |

2. | S. T. Chui, “Giant wave rotation for small helical structures,” J. Appl. Phys. |

3. | K. Konishi, B. Bai, X. Meng, P. Karvinen, J. Turunen, Y. P. Svirko, and M. Kuwata-Gonokami, “Observation of extraordinary optical activity in planar chiral photonic crystals,” Opt. Express |

4. | F. Miyamaru and M. Hangyo, “Strong optical activity in chiral metamaterials of metal screw hole arrays,” Appl. Phys. Lett. |

5. | M. Thiel, G. von Freymann, and M. Wegener, “Layer-by-layer three-dimensional chiral photonic crystals,” Opt. Lett. |

6. | D.-H. Kwon, P. L. Werner, and D. H. Werner, “Optical planar chiral metamaterial designs for strong circular dichroism and polarization rotation,” Opt. Express |

7. | 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. |

8. | 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. |

9. | J. B. Pendry, “A chiral route to negative refraction,” Science |

10. | C. Zhang and T. J. Cui, “Negative reflections of electromagnetic waves in a strong chiral medium,” Applied Physics Letters |

11. | 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 |

12. | S. Zhang, Y.-S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative Refractive Index in Chiral Metamaterials,” Phys. Rev. Lett. |

13. | M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A |

14. | 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. |

15. | M. Thiel, H. Fischer, G. von Freymann, and M. Wegener, “Three-dimensional chiral photonic superlattices,” Opt. Lett. |

16. | C. Lu and R. Lipson, “Interference lithography: a powerful tool for fabricating periodic structures,” Laser Photon. Rev. |

17. | J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

18. | E. Betzig, “Sparse and composite coherent lattices,” Phys. Rev. A |

19. | A. Schinzel, “Sur l’existence d’un cercle passant par un nombre donné de points aux coordonnées entières,” Enseign. Math. |

20. | 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. |

21. | D. Rokhsar, D. Wright, and N. Mermin, “Scale equivalence of quasicrystallographic space groups,” Phys. Rev. B |

22. | D. Rabson, N. Mermin, D. Rokhsar, and D. Wright, “The space groups of axial crystals and quasicrystals,” Rev. Mod. Phys. |

23. | N. Mermin, “The space groups of icosahedral quasicrystals and cubic, orthorhombic, monoclinic, and triclinic crystals,” Rev. Mod. Phys. |

24. | R. N. Bracewell, |

25. | A. König and N. D. Mermin, “Symmetry, extinctions, and band sticking,” Am. J. Phys. |

26. | 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. |

27. | 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. |

28. | M. Petrović, M. Belić, C. Denz, and Y. Kivshar, “Counterpropagating optical beams and solitons,” Laser Photon. Rev. |

29. | G. Juzeliu̅nas and P. Öhberg, “Optical Manipulation of Ultracold Atoms,” in |

30. | J. Curtis, “Dynamic holographic optical tweezers,” Opt. Commun. |

31. | D. G. Grier, “A revolution in optical manipulation,” Nature |

32. | C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express |

33. | J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. |

34. | J. Lin, X.-C. Yuan, S. H. Tao, X. Peng, and H. B. Niu, “Deterministic approach to the generation of modified helical beams for optical manipulation,” Opt. Express |

35. | M. Mazilu, D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: non-diffracting beams,” Laser Photon. Rev. |

36. | 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. |

37. | F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics |

38. | A. König and N. D. Mermin, “Screw rotations and glide mirrors: crystallography in Fourier space,” Proc. Natl. Acad. Sci. U.S.A. |

39. | B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. |

40. | J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater. |

41. | X. Xiong, X. Chen, M. Wang, R. Peng, and D. Shu, “Optically nonactive assorted helix array with interchangeable magnetic / electric resonance,” Appl. Phys. Lett. |

**OCIS Codes**

(090.0090) Holography : Holography

(220.4000) Optical design and fabrication : Microstructure fabrication

(160.1585) Materials : Chiral media

(050.4865) Diffraction and gratings : Optical vortices

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 22, 2011

Revised Manuscript: March 29, 2011

Manuscript Accepted: March 29, 2011

Published: May 5, 2011

**Citation**

Julian Becker, Patrick Rose, Martin Boguslawski, and Cornelia Denz, "Systematic approach to complex periodic vortex and helix lattices," Opt. Express **19**, 9848-9862 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9848

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### References

- 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]
- S. T. Chui, “Giant wave rotation for small helical structures,” J. Appl. Phys. 104, 013904 (2008). [CrossRef]
- K. Konishi, B. Bai, X. Meng, P. Karvinen, J. Turunen, Y. P. Svirko, and M. Kuwata-Gonokami, “Observation of extraordinary optical activity in planar chiral photonic crystals,” Opt. Express 16, 7189–7196 (2008). [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]
- M. Thiel, G. von Freymann, and M. Wegener, “Layer-by-layer three-dimensional chiral photonic crystals,” Opt. Lett. 32, 2547–2549 (2007). [CrossRef] [PubMed]
- D.-H. Kwon, P. L. Werner, and D. H. Werner, “Optical planar chiral metamaterial designs for strong circular dichroism and polarization rotation,” Opt. Express 16, 11802–11807 (2008). [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. 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. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004). [CrossRef] [PubMed]
- C. Zhang and T. J. Cui, “Negative reflections of electromagnetic waves in a strong chiral medium,” Applied Physics Letters 91, 194101 (2007). [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. 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, 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, H. Fischer, G. von Freymann, and M. Wegener, “Three-dimensional chiral photonic superlattices,” Opt. Lett. 35, 166–168 (2010). [CrossRef] [PubMed]
- C. Lu and R. Lipson, “Interference lithography: a powerful tool for fabricating periodic structures,” Laser Photon. Rev. 4, 568–580 (2009). [CrossRef]
- J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]
- E. Betzig, “Sparse and composite coherent lattices,” Phys. Rev. A 71, 063406 (2005). [CrossRef]
- A. Schinzel, “Sur l’existence d’un cercle passant par un nombre donné de points aux coordonnées entières,” Enseign. Math. 4, 71–72 (1958).
- 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]
- 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]
- R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000), chap. 6, pp. 105–135, 3rd ed.
- A. König and N. D. Mermin, “Symmetry, extinctions, and band sticking,” Am. J. Phys. 68, 525–530 (2000). [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]
- 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. Petrović, M. Belić, C. Denz, and Y. Kivshar, “Counterpropagating optical beams and solitons,” Laser Photon. Rev. 5, 214–233 (2011). [CrossRef]
- 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.
- J. Curtis, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002). [CrossRef]
- D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef] [PubMed]
- 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]
- J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28, 872–874 (2003). [CrossRef] [PubMed]
- J. Lin, X.-C. Yuan, S. H. Tao, X. Peng, and H. B. Niu, “Deterministic approach to the generation of modified helical beams for optical manipulation,” Opt. Express 13, 3862–3867 (2005). [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]
- F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010). [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]
- B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (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]
- 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]

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