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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 20 — Oct. 7, 2013
  • pp: 23441–23449
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Generation of pseudonondiffracting optical beams with superlattice structures

C. H. Tsou, T. W. Wu, J. C. Tung, H. C. Liang, P. H. Tuan, and Y. F. Chen  »View Author Affiliations


Optics Express, Vol. 21, Issue 20, pp. 23441-23449 (2013)
http://dx.doi.org/10.1364/OE.21.023441


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Abstract

We demonstrate an approach to generate a class of pseudonondiffracting optical beams with the transverse shapes related to the superlattice structures. For constructing the superlattice waves, we consider a coherent superposition of two identical lattice waves with a specific relative angle in the azimuthal direction. We theoretically derive the general conditions of the relative angles for superlattice waves. In the experiment, a mask with multiple apertures which fulfill the conditions for superlattice structures is utilized to generate the pseudonondiffracting superlattice beams. With the analytical wave functions and experimental patterns, the pseudonondiffracting optical beams with a variety of structures can be generated systematically.

© 2013 OSA

1. Introduction

A 2D superlattice pattern is a spatially periodic structure composed of two or more simple planeforms. Since Kudrolli et al. [13

13. A. Kudrolli, B. Pier, and J. P. Gollub, Physica, “Superlattice patterns in surface waves,” Physica D 123(1-4), 99–111 (1998).

] first observed superlattice patterns in a two-frequency forcing Faraday experiment, the superlattice patterns have been widely studied in the experiments of parametrically driven surface waves [14

14. M. Silber and M. R. E. Proctor, “Nonlinear Competition between Small and Large Hexagonal Patterns,” Phys. Rev. Lett. 81(12), 2450–2453 (1998). [CrossRef]

16

16. H. J. Pi, S. Park, J. Lee, and K. J. Lee, “Superlattice, Rhombus, Square, And Hexagonal Standing Waves In Magnetically Driven Ferrofluid Surface,” Phys. Rev. Lett. 84(23), 5316–5319 (2000). [CrossRef] [PubMed]

]. The superlattice patterns observed by Kudrolli et al. [13

13. A. Kudrolli, B. Pier, and J. P. Gollub, Physica, “Superlattice patterns in surface waves,” Physica D 123(1-4), 99–111 (1998).

] are formed by the coherent superposition of two hexagonal lattice waves with a relative angle of 2sin1(1/27)22°. Mathematically, there are numerous relative angles satisfying the condition for generating the superlattice waves from superposing two identical lattice waves. Even so, how to determine the specific relative angles for constructing the superlattice waves has not been explored in detail. Therefore, the determination of relative angles is the first issue for generating pseudonondiffracting optical beams with the superlattice structures.

In this paper, we theoretically derive a general condition for the relative angles to construct the superlattice waves from superposing two identical lattice waves. With the derived formulas for the relative angles, numerous superlattice patterns are numerically demonstrated. To realize the pseudonondiffracting optical beams with superlattice structures, we generate the quasi-plane waves by employing a collimated coherent laser to illuminate a mask with multiple tiny apertures. The positions of the apertures are precisely manufactured with a stencil laser cutting machine to fulfill the condition for generating the superlattice patterns. We also analyze the influence of the aperture size on the formation of the transverse unit cell in the pseudonondiffracting superlattice beam. The experimental results are found to be in a good agreement with the numerical calculations. Furthermore, we manifest the structures of phase singularities for the superlattice patterns. The optical fields with phase singularities, also known as optical vortices, have been studied and generated a lot of interest in recent years [17

17. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974). [CrossRef]

, 18

18. M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt. 42, 219–276 (2001). [CrossRef]

]. We expect that the pseudonondiffracting superlattice beams with phase singularities can be potentially beneficial to future applications for the optical vortex beams.

2. Theoretical analysis for forming superlattice waves

A 2D lattice wave in polar coordinates (ρ,ϕ) which is formed by the superposition of three, four or six plane waves can be expressed as [12

12. Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A 83(5), 053813 (2011). [CrossRef]

]
ψq(ρ,ϕ;Ks)=1qs=0q1eiKsρ,
(1)
where Ks=(Kcos(2πs/q),Ksin(2πs/q)), ρ=(ρcos(ϕ),ρsin(ϕ)), Ks is the wave vector, and q is equal to 3, 4, or 6. Considering the coherent superposition of two identical lattice waves with a relative angle in the azimuthal direction, we can obtain the superposed waves as
Ψq(ρ,ϕ;Ks,Δq)=ψq(ρ,ϕ;Ks)+ψq(ρ,ϕ;Ks),
(2)
where Ks=(Kcos[(2πs/q)+Δq],Ksin[(2πs/q)+Δq]), and Δq is the relative angle between Ks and Ks. Figure 1(a)
Fig. 1 The schematic diagrams of the wave vectors of the superposed waves Ψq(ρ,ϕ;Ks,Δq) with (a) q = 3, (b) q = 4, and (c) q = 6.
-1(c) depicts the wave vectors of the superposed waves with q = 3, 4, and 6, respectively. The solid and dashed vectors represent Ks and Ks, respectively.

The superposed waves are spatially periodic when the wave vectors are located on the reciprocal lattice points of the superposed wave fields. In other words, the wave vectors can be expressed as the linear combinations of reciprocal primitive translation vectors. For instance, since the reciprocal primitive translation vectors are orthogonal for q = 4, the wave vectors shown in Fig. 1(b) must be satisfied the following conditions
{Ks=nsb1+msb2Ks=nsb1+msb2|Ks|=bns2+ms2=bns2+ms2=|Ks|,
(3)
where (ns,ms) and (ns,ms) are two coprime integer pairs, b1 and b2 are the reciprocal primitive translation vectors of the spatially periodic superposed waves, and b=|b1|=|b2|. Combining all conditions of the wave vectors, the most general solutions of ns and ms are given by
{ns=msms=ns.
(4)
Therefore, Ks can be rewritten in terms of ns and ms as
Ks=msb1+nsb2.
(5)
There are some accidental solutions of ns and ms which are not included in Eq. (4). Because these cannot be expressed in an analytic form, we focus on the most general solutions given by Eq. (4). As a result, the specific relative angles Δq for spatially periodic waves are subject to following the condition
Δq=cos1(K0K0|K0||K0|)=cos1(2n0m0n02+m02),forq=4.
(6)
By utilizing the condition of relative angles, we can generate a class of superposed waves for q = 4 with spatial periodicity. With the wave vectors in terms of the reciprocal primitive translation vectors, the reciprocal lattice constant can be given by
b=|b1|=|b2|=Kn02+m02,forq=4.
(7)
Equation (7) indicates that the spatial period becomes longer when the value of n02+m02 gets larger. Following an analogous derivation, the criteria for spatial periodicity of superposed waves with q = 3 and 6 can be obtained. Since the scalar product of the reciprocal primitive translation vectors is −1/2 in the cases of q = 3 and 6, the condition of the relative angles leads to the equation
Δq=cos1(n02m02+4n0m02(n02+m02n0m0)),forq=3or6,
(8)
and the reciprocal lattice constant is given by
b=Kn02+m02n0m0,forq=3or6.
(9)
Consequently, the superlattice waves can be constructed by the superposed waves Ψq(ρ,ϕ;Ks,Δq) with specific relative angles.

Figures 2(a)
Fig. 2 Numerical patterns for the intensity of the superlattice waves |Ψq(ρ,ϕ;Ks,Δq)|2 with q = 4.
-2(c) depict the calculated patterns for the intensity of the superlattice waves |Ψq(ρ,ϕ;Ks,Δq)|2 with q = 4. It can be seen that a rich variety of superlattice wave patterns can be constructed by controlling the specific relative angle. The numerical patterns for the intensity of superlattice waves with q = 3 and 6 are shown in Figs. 3(a)
Fig. 3 Numerical patterns for the intensity of the superlattice waves |Ψq(ρ,ϕ;Ks,Δq)|2 with (a)-(c) q = 3, and (d)-(f) q = 6.
-3(c) and Figs. 3(d)-3(f), respectively. These numerical patterns show that the specific relative angle can turn into the main parameter for generating nondiffracting optical superlattice beams. In the following section we present an approach to realize the pseudonondiffracting optical superlattice beams.

3. Generation of the pseudonondiffracting optical superlattice beams

A pseudonondiffracting Bessel beam can be generated by an annular slit illuminated with a collimated light and placed in the focal plane in front of a lens [2

2. J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]

]. Based on Fourier optics, the relation between the input field Ei(ρ,ϕ) in the focal plane in front of a lens and the output field Eo(ρ,ϕ,z) behind the lens at a distance z can be expressed as
Eo(ρ,ϕ,z)=iei2πλ(f+z)λfEi(ρ,ϕ)ei2πλρ22f(1zf)ei2πρρλfcos(ϕϕ)ρdρdϕ,
(10)
where λ is the wavelength of coherent light source, and f is the focal length of the lens. For a pseudonondiffracting Bessel beam, the input field is determined by an infinitesimally thin annulus at ρ=R expressed as
Ei(ρ,ϕ)=δ(ρR),
(11)
where δ() is the Dirac delta function. A finite-energy pseudonondiffracting Bessel beam requires an annular ring of finite thickness at the input plane. The pseudonondiffracting beams with crystalline and quasicrystalline structures can be generated with a collimated light illuminating a mask with multiple apertures regularly distributed on a ring [12

12. Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A 83(5), 053813 (2011). [CrossRef]

]. Therefore, the input field just after the mask for the pseudonondiffracting beams with crystalline and quasicrystalline structures can be approximately given by
Ei(ρ,ϕ)=1qδ(ρR)s=0q1δ(ϕ2πsq),
(12)
where R is the radius of the ring where the apertures are located on. Equation (12) implies that the aperture size must be infinitesimal for generating ideal nondiffracting beams with crystalline structures. For generating nondiffracting superlattice beams, the input field is given by
Ei(ρ,ϕ)=δ(ρR)qs=0q1[δ(ϕ2πsq)+δ(ϕ2πsqΔq)],
(13)
where Δq satisfies the criteria of superlattice patterns in Eq. (6) or (8). However, an infinitesimal aperture is not realistic. Since the aperture sizes cannot be infinitesimal, the generated beams are called the pseudo-nondiffracting beams. Furthermore, the selection of the aperture size determines how many spatial periods can be included in the pseudonondiffracting superlattice beam. Therefore, the analysis for determining the aperture size is of crucial importance. For considering the effect of the aperture size with finite energy, we exploit the multiple Gaussian beams to model the input field just after the apertures. Based on the locations of the pinholes in Eq. (13), the multiple Gaussian waves for describing the input field is given by
Ei(ρ,ϕ)=(2πa2)12s=0q1{exp[ρ2+R22ρRcos(ϕ(2πs/q))a2]+exp[ρ2+R22ρRcos(ϕ(2πs/q)Δq)a2]},
(14)
where the beam waist of the Gaussian beam is set to be the radius of the aperture a. Since the output field in the focal plane behind the lens can be found by the Fourier transform of the input field, the substitution of Eq. (14) into Eq. (10) and considering z = f lead to an equation for the output field
Eo(ρ,ϕ;f)=(2πa2)12ei2πλ(2f)eR2a2iλf×eρ2a2e2Rρa2cos(ϕϕ)ei2πρρλfcos(ϕϕ)ρdρdϕ.
(15)
With transformation of coordinates from polar coordinates to Cartesian coordinates, Eq. (15) can be an analytical integration by utilizing Gaussian integral:
eαx22βxdx=παeβ2/α.
(16)
By some algebraic operation, the output field in polar coordinates can be derived as
Eo(ρ,ϕ;f)=πa2ei2πλ(2f)eπa2λfρ2iλfs=0q1[ei2πRλfρcos(ϕ2πsq)+ei2πRλfρcos(ϕ2πsqΔq)].
(17)
It can be seen that the terms in the summation represent the superlattice waves with K=2πR/λf. The factor of exp(πa2ρ2/λf) shows that the larger the aperture size, the smaller visible region, namely the less energy in the final beam. Figures 4(a)
Fig. 4 (a)-(c) Numerically patterns for the output intensity profiles of pseudonondiffracting optical superlattice patterns with different radii of apertures.
-4(c) illustrate the numerical patterns for the intensity of the output fields Eo(ρ,ϕ;f). It can be seen that more spatial periods can be observed with smaller size of apertures, but it is arduous to generate visible patterns with too small size of apertures in experiments. Thus, the radius of aperture is selected as 85μm for generating clear pseudonondiffracting optical superlattice patterns.

Based on the theoretical analysis, an optical configuration is set up to realize pseudonondiffracting optical superlattice patterns, as shown in Fig. 5
Fig. 5 Experimental setup for generating pseudonondiffracting optical beams with superlattice structures.
. The light source was a linearly polarized 20-mW He-Ne laser with central wavelength at 632.8 nm. A beam expander was employed to generate a collimated light and reduce the beam divergence less than 0.1 mrad. By using a laser stencil-cutting machine, we fabricate the steel masks with high precision. The radius of the ring and aperture are 3 mm and 85μm, respectively. The focal length of the lens is 1000 mm. Interference patterns formed in the region behind the focal lens were imaged by a CCD camera.

Figures 6(a)
Fig. 6 Experimental patterns observed for pseudonondiffracting optical superlattice beams with q = 4 under the optimal alignment.
-6(c) display the interference patterns for pseudonondiffracting optical superlattice structures with q = 4 observed in the experiment under the condition of the optimal alignment. It can be seen that the experimental observations agree very well with the numerical patterns shown in Fig. 2(a)-2(c). The experimental patterns reveal that the relation between the reciprocal lattice constant and the spatial period is satisfied in our theory. Figures 7(a)
Fig. 7 Experimental patterns observed for pseudonondiffracting optical superlattice beams with (a)-(c) q = 3, and (d)-(f) q = 6.
-7(c) and Figs. 7(d)-7(f) illustrate the experimental results for pseudonondiffracting optical superlattice patterns with q = 3 and 6, respectively. Once again, the experimental results match very well the numerical calculations depicted in Figs. 3(a)-3(c) and Figs. 3(d)-2(f), respectively. It can be experimentally observed that there are honeycomb strucutres in some optical superlattice pattern with q = 3, as shown in Figs. 7(b) and 7(c). Moreover, in Fig. 7(f), the optical superlattice pattern with q = 6 displays the exotic kaleidoscopic structure. The excellent agreement validates the theoretical analysis of superlattice waves and confirms the experimental approach. The experimental patterns also confirm our analysis of the influence of the aperture size on the transverse unit cell.

The reliable generation of optical beams with complex structures has become increasingly important in the studies of optical manipulations. The complex optical fields with the phase singularities, so-called the optical vortex beams, have been extensively employed. For superlattice waves, the phase singularities are the undefined locations in the phase angle fields which are given by
Θq(ρ,ϕ;Ks,Δq)=tan1{Im[Ψq(ρ,ϕ;Ks,Δq)]/Re[Ψq(ρ,ϕ;Ks,Δq)]},
(18)
where Im[Ψq(ρ,ϕ;Ks,Δq)] and Re[Ψq(ρ,ϕ;Ks,Δq)] are the imaginary and real parts of the superlattice waves, respectively. Figures 8(a)
Fig. 8 (a)-(c) Contour plots of phase fields Θq(ρ,ϕ;Ks,Δq) for the boxed regions shown in Fig. 3(a)-3(c), respectively.
-8(c) illustrate the contour plots of phase fields Θq(ρ,ϕ;Ks,Δq) for Figs. 3(a)-3(c) to display the feature of the phase singularities, respectively. The experimental results verify that the various vortex-lattice structures can be generated by the pseudonondiffracting optical superlattice patterns with q = 3.

4. Conclusions

Acknowledgments

The authors thank the National Science Council for their financial support of this research under Contract No. NSC-100-2628-M-009-001-MY3.

References and links

1.

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

2.

J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]

3.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002). [CrossRef] [PubMed]

4.

D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28(8), 657–659 (2003). [CrossRef] [PubMed]

5.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001). [CrossRef]

6.

Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. 27(4), 243–245 (2002). [CrossRef] [PubMed]

7.

C. Yu, M. R. Wang, A. J. Varela, and B. Chen, “High-density non-diffracting beam array for optical interconnection,” Opt. Commun. 177(1-6), 369–376 (2000). [CrossRef]

8.

Z. Bouchal, “Nondiffracting optical beams-physical properties, experiments, and applications,” Czech. J. Phys. 53(7), 537–578 (2003). [CrossRef]

9.

M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A 84(1), 013832 (2011). [CrossRef]

10.

M. Boguslawski, P. Rose, and C. Denz, “Nondiffracting kagome lattice,” Appl. Phys. Lett. 98(6), 061111 (2011). [CrossRef]

11.

P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. 14(3), 033018 (2012). [CrossRef]

12.

Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A 83(5), 053813 (2011). [CrossRef]

13.

A. Kudrolli, B. Pier, and J. P. Gollub, Physica, “Superlattice patterns in surface waves,” Physica D 123(1-4), 99–111 (1998).

14.

M. Silber and M. R. E. Proctor, “Nonlinear Competition between Small and Large Hexagonal Patterns,” Phys. Rev. Lett. 81(12), 2450–2453 (1998). [CrossRef]

15.

H. Arbell and J. Fineberg, “Spatial and Temporal Dynamics of Two Interacting Modesin Parametrically Driven Surface Waves,” Phys. Rev. Lett. 81(20), 4384–4387 (1998). [CrossRef]

16.

H. J. Pi, S. Park, J. Lee, and K. J. Lee, “Superlattice, Rhombus, Square, And Hexagonal Standing Waves In Magnetically Driven Ferrofluid Surface,” Phys. Rev. Lett. 84(23), 5316–5319 (2000). [CrossRef] [PubMed]

17.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974). [CrossRef]

18.

M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt. 42, 219–276 (2001). [CrossRef]

OCIS Codes
(070.3185) Fourier optics and signal processing : Invariant optical fields
(050.4865) Diffraction and gratings : Optical vortices
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Diffraction and Gratings

History
Original Manuscript: August 8, 2013
Revised Manuscript: September 16, 2013
Manuscript Accepted: September 17, 2013
Published: September 25, 2013

Citation
C. H. Tsou, T. W. Wu, J. C. Tung, H. C. Liang, P. H. Tuan, and Y. F. Chen, "Generation of pseudonondiffracting optical beams with superlattice structures," Opt. Express 21, 23441-23449 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23441


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References

  1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A4(4), 651–654 (1987). [CrossRef]
  2. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987). [CrossRef] [PubMed]
  3. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002). [CrossRef] [PubMed]
  4. D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett.28(8), 657–659 (2003). [CrossRef] [PubMed]
  5. J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.197(4-6), 239–245 (2001). [CrossRef]
  6. Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett.27(4), 243–245 (2002). [CrossRef] [PubMed]
  7. C. Yu, M. R. Wang, A. J. Varela, and B. Chen, “High-density non-diffracting beam array for optical interconnection,” Opt. Commun.177(1-6), 369–376 (2000). [CrossRef]
  8. Z. Bouchal, “Nondiffracting optical beams-physical properties, experiments, and applications,” Czech. J. Phys.53(7), 537–578 (2003). [CrossRef]
  9. M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A84(1), 013832 (2011). [CrossRef]
  10. M. Boguslawski, P. Rose, and C. Denz, “Nondiffracting kagome lattice,” Appl. Phys. Lett.98(6), 061111 (2011). [CrossRef]
  11. P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys.14(3), 033018 (2012). [CrossRef]
  12. Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A83(5), 053813 (2011). [CrossRef]
  13. A. Kudrolli, B. Pier, and J. P. Gollub, Physica, “Superlattice patterns in surface waves,” Physica D123(1-4), 99–111 (1998).
  14. M. Silber and M. R. E. Proctor, “Nonlinear Competition between Small and Large Hexagonal Patterns,” Phys. Rev. Lett.81(12), 2450–2453 (1998). [CrossRef]
  15. H. Arbell and J. Fineberg, “Spatial and Temporal Dynamics of Two Interacting Modesin Parametrically Driven Surface Waves,” Phys. Rev. Lett.81(20), 4384–4387 (1998). [CrossRef]
  16. H. J. Pi, S. Park, J. Lee, and K. J. Lee, “Superlattice, Rhombus, Square, And Hexagonal Standing Waves In Magnetically Driven Ferrofluid Surface,” Phys. Rev. Lett.84(23), 5316–5319 (2000). [CrossRef] [PubMed]
  17. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci.336(1605), 165–190 (1974). [CrossRef]
  18. M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt.42, 219–276 (2001). [CrossRef]

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