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Optica Publishing Group

Twisting light with hyperbolic metamaterials

Open Access Open Access

Abstract

We propose a novel, miniaturized astigmatic optical element based on a single biaxial hyperbolic metamaterial that enables the conversion of Hermite-Gaussian beams into vortex beams carrying an orbital angular momentum and vice versa. As an example, we design a biaxial anisotropic metamaterial that introduces a π/2 phase shift between two orthogonal components of a Hermite–Gaussian beam due to the optical path difference and at the same time astigmatically focuses these orthogonal components such that they recombine in a symmetric Laguerre-Gaussian beam. We design the proposed device using an array of silver nanowires in an MgF2 matrix. The advantages of the proposed approach over the existing bulk optics based techniques include compactness and therefore, compatibility with ultra-compact opto-electronic circuits, potential re-configurability and an increased tolerance to misalignment.

©2013 Optical Society of America

1. Introduction

Hyperbolic metamaterials (HMMs) are a class of strongly anisotropic materials having their principle elements of dielectric permittivity or magnetic permeability tensors of opposite signs. These materials have enabled such novel properties and potential applications as omnidirectional negative refraction, high density of states and, imaging beyond the diffraction limit using a so-called hyperlens [13]. Despite significant theoretical and experimental progress in recent years, no studies of propagation of complex light beams such as those carrying the orbital angular momentum and/or vector beams in HMMs have been reported.

It is noteworthy that hyperbolic metamaterial can be considered as an astigmatic anisotropic element such that rays propagating in two perpendicular planes have different foci. The degree of astigmatism can be designed and even potentially tuned. In particular, we propose a new approach that enables a transformation of uncharged Hermite-Gaussian beams into vortex beams carrying an orbital angular momentum (OAM) of a particular order in a biaxial HMM. Beams with different OAM states have a strong potential to enable a new degree of freedom for increasing capacity through space division multiplexing or for building higher dimensional quantum encryption systems [411]. Very first wirelesses as well as specialty fiber based links, utilizing OAM states, have been recently reported [4, 12].

Previously, several approaches have been developed for generating and manipulating OAM, including fork holograms and spatial light modulators, spiral phase plates, q-plates, and cylindrical lens mode converters [1323]. All these methods developed to date rely on bulky free space optics elements and have drawbacks with respect to alignment, bandwidth, and incompatibility with future ultra-compact opto-electronic circuits. Only recently have first attempts at the realization of miniaturized vortex elements, including those based on metasurfaces [24, 25] and on silicon optical waveguides [28], been reported.

In this paper, we propose an alternative, simple, compact, hyperbolic metamaterial based beam transformer that converts uncharged Hermite-Gaussian modes into a Laguerre-Gaussian vortex beam of a particular order. The basic principle of a proposed device is as follows. A 45° polarized Hermite-Gaussian mode beam can be transformed into a vortex beam of Laguerre-Gaussian mode by the optical path difference along the two perpendicular crystal orientations resulting from the biaxial anisotropy of the HMM. As compared to existing approaches, the proposed device is compact, does not require a precise alignment and is potentially reconfigurable. We discuss basic theory, design, and potential experimental realization of the proposed device.

2. Theory

We consider a Hermite-Gaussian beam (HG) polarized at 45° to the principal axes x and y of the biaxial HMM propagating along the z axis

E(r,z)=E0w0w(z)exp(r2w2(z)ikzikr22R(z)+iς(z))H1(2w0(x+y)),
where w(z)=w01+(zzR)2is the radius of the beam at z, w0 is the waist of the beam, ζ is the Gouy phase, H1(2w0(x+y)) is the first order Hermite polynomial, zR=πw02λis the Raleigh range of the beam, and λ is the wavelength. Such a kind of HG beam can be decomposed into an in-phase HG1, 0 and HG0, 1 modes.

We design an HMM such that when the beam propagates in a biaxial hyperbolic medium, negative refraction resulting from hyperbolic dispersion relation leads to astigmatic focusing inside and outside of the HMM. Simultaneously, due to the anisotropy, a phase difference between the two modes arises after the beam transmitting through the HMM. By optimizing a certain thickness of the HMM so that the phase difference between the two modes is π/2, the output beam turns out to be a vortex beam of LG1,0 mode [15].

As shown in Fig. 1, the incident HG beam from its source at z = 0 is incident into the biaxial HMM which is located at z. Thus the incident angle θi can be obtained by:

θi=arctan(w(z)z),
For an HMM with dielectric tensor:
ε=(εxx000εyy000εzz),
where εxx>0,εyy>0(εxxεyy) and εzz<0, the dispersion relations illustrated by the equal frequency contours (EFCs) in yz-plane and xz-plane shown in Fig. 1(b) and 1(c) can be written as:

 figure: Fig. 1

Fig. 1 Schematic of negative refraction in HMM (a) and the equi-frequency contours of the HMM in xz-plane (b) and yz-plane (c). The black circles are the EFCs of the air and the blue hyperbola are the EFCs of the hyperbolic material. ki and Si are the incident wave vector and Poynting vector, respectively. kr and Sr are the refractive wave vector and Poynting vector, respectively. Due to the indefinite property between xy-plane and z orientation, both the HG1,0 mode and the HG0,1 mode experience a negative refraction. The astigmatic beam is focused inside the indefinite beam.

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kx2εzz+kz2εxx=ω2c2,ky2εzz+kz2εyy=ω2c2.

If the two EFCs are normalized by the EFC of air kx2+kz2=ω2c2, we define the normalized incident wave vector kiand the wave vector of refracted wave kralong different directions as:

ki,x=ki,y=sin(θi)=tan2(θi)1+tan2(θi),ki,z=cos(θi)=11+tan2(θi),
ki,x=ki,y=kr,x=kr,y,
kr,xz=εxx(1kr,x2εzz),kr,yz=εyy(1kr,y2εzz).
Thus, the refractive indices in x and y orientations are:
nkr,xz=kr,xz2+kr,x2,nkr,yz=kr,yz2+kr,y2;
The wave vector refraction angles in xz-plane and yz-plane are:
θkr,zx=arctan(kr,xkr,xz),θkr,yz=arctan(kr,ykr,yz),
The Poynting vector refraction angles in xz-plane and yz-plane are:

θSr,xz=arctan(kr,xεxxkr,zεzz),θSr,yz=arctan(kr,yεyykr,zεzz).

Because the input surface and the output surface of the HMM are parallel to each other, the incident angle θi and the output angle θo are also equal to each other in Fig. 1(a). If we consider the mode of the beam in xy-plane, the optical paths of in yz-plane and xz-plane in the air will be same. As shown in Fig. 1(a), the optical path difference in the HMM between the xz-plane and yz-plane is:

Δlo=dcos(θSr,zx+θkr,zx)cos(θSr,zx)nkr,zxdcos(θSr,yz+θkr,yz)cos(θSr,yz)nkr,yz,
After some algebra deduction:
Δlo=d(Kkr,x2εzzK)Δn
in which K=1kr,x2εzz, a constant for a fixed incident angle, and Δn=εxxεyyis the difference of the refractive indices between x and y orientations.

If Δl0 = λ/4, the phase difference between the two beams in the yz-plane and xz-plane is π/2, which means that the composed beam is a vortex of LG1,0 mode.

3. Design

One of the most common structures used to realize the HMM is the array of metal nanowires embedded in a dielectric matrix [29]. Here, we design a tetragonal silver nanowire array in a MgF2 matrix to realize the biaxial HMM, as shown in Fig. 2. The permittivity of the silver can be described by the Drude model:

εm(ω)=εωp2ω(ω+iγc).
where ε = 6 is the bulk permittivity at infinite frequency, ωp = 1.5 × 1016rad/s and γc = 7.73 × 1013 rad/s are the plasma frequency and the collision frequency, respectively [29, 30]. According to the Maxwell-Garnett theory, we can obtain the permittivity parallel to the nanowire (ε) and vertical to the nanowires (ε) [31, 32]:

 figure: Fig. 2

Fig. 2 Structure of the biaxial HMM.

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ε||=pεm+(1p)εd,
ε=εd+pεd(εmεd)εd+(1p)(εmεd)qeff.

here, qeff is the effective depolarization factor perpendicular to the nanowires and is equal to 1/2 if the wavelength is much larger than the radius of the nanowires [29]. The permittivity of the MgF2 is 1.9 at 532nm. For an HMM with unit size of a = 124nm, b = 117nm, and radius r = 30 nm at 532 nm, filling ratio of the silver along x-, y- and z-axis are px = 0.185, py = 0.205, pz = 0.195, respectively. According to Eq. (14) and (15), the permittivities of the metamaterial are εxx = 2.93 + 0.006i, εyy = 3.08 + 0.007i, and εzz = −0.8 + 0.07i.

Considering a 45° polarized HG1,0 Gaussian beam with wavelength of 532nm and waist also 532nm propagating along z axis, the thickness of the HMM with above parameters should be 3.48μm according to Eq. (12).

4. Numerical results

To demonstrate the performance of the vortex beam produced by biaxial HMM, we performed finite-element method based simulations (using Comsol Multiphysics 4.2) for the HG1, 0 beam propagating through a biaxial HMM consisting of silver nanowires array in the MgF2 matrix with the design parameters mentioned above. Results in Fig. 3 show the time averaged energy density distribution and the phase distribution of the output beam at different positions along the z-direction. In the range of 8.5~10.5μm, the shape of the beam exhibits a doughnut shape with a helical phase change in the cross-section, confirming that the 45° polarized HG1,0 beam was transformed into a LG1,0 mode of a good quality in this range. Due to the angular dispersion of the HMM, the thickness d can be varied in the range of 3.2-3.7μm.

 figure: Fig. 3

Fig. 3 The schematic of the device and the results including energy density and the phase distributions of the output beam around the focal length.

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Figure 4 shows the evolution of the beam shape and the phase along the direction of propagation. The first pair on the left is the incident HG1,0 beam with a pattern of two lobes. The second to fourth pairs are inside the HMM. The third pair around the focusing point has a doughnut shape due to the focusing effect of the HMM; however, its phase is not helical. The last six pairs on the right are in the air. Although, the doughnut shape only occurs around the focal range in the air (outside the HMM), the phase of the beam remains helical anywhere after its transmission through the HMM, indicating that the output beam carries an OAM.

 figure: Fig. 4

Fig. 4 The time averaged energy density and phase distributions in the cross section of the beam along the propagation direction.

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7. Conclusion

In conclusion, we proposed and designed a new approach to generating a vortex beam by transforming the HG mode into the LG mode by using a biaxial HMM. We demonstrated that the indefinite dielectric tensor combined with the biaxial anisotropy enables conversion of a 45° polarized HG beam into a vortex beam owing to different focusing effects in two orthogonal planes inside the HMM and therefore, different optical paths. We design the HMMs such that the optical path difference results in a π/2 phase shift between the two orthogonal components, resulting in an output beam carrying an OAM. The proposed approach offers several advantages over the existing techniques, including compactness and therefore, compatibility with ultra-compact opto-electronic circuits, robustness and potentially re-configurability.

Acknowledgments

The authors appreciate discussions with A. N. Cartwright, Zh. A. Kudyshev, M. I. Shalaev, and X. Wang, and acknowledge support of the U.S. Army Research Office under the award W911NF-11-1-0333.

References and links

1. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405–077409 (2003). [CrossRef]   [PubMed]  

2. D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004). [CrossRef]  

3. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef]   [PubMed]  

4. F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012). [CrossRef]  

5. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]  

6. P. Jia, Y. Yang, C. J. Min, H. Fang, and X.-C. Yuan, “Sidelobe-modulated optical vortices for free-space communication,” Opt. Lett. 38(4), 588–590 (2013). [CrossRef]   [PubMed]  

7. M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20(22), 24444–24449 (2012). [CrossRef]   [PubMed]  

8. B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37(17), 3735–3737 (2012). [CrossRef]   [PubMed]  

9. J. Leach, E. Bolduc, D. J. Gauthier, and R. W. Boyd, “Secure information capacity of photons entangled in many dimensions,” Phys. Rev. A 85(6), 060304 (2012). [CrossRef]  

10. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef]   [PubMed]  

11. M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20(12), 13195–13200 (2012). [CrossRef]   [PubMed]  

12. N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, “Control of orbital angular momentum of light with optical fibers,” Opt. Lett. 37(13), 2451–2453 (2012). [CrossRef]   [PubMed]  

13. V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts ,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

14. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]   [PubMed]  

15. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]  

16. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994). [CrossRef]  

17. Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev. 6(4), 339–344 (1999). [CrossRef]  

18. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef]   [PubMed]  

19. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006). [CrossRef]   [PubMed]  

20. L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation,” Appl. Phys. Lett. 88(22), 221102 (2006). [CrossRef]  

21. E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94(23), 231124 (2009). [CrossRef]  

22. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002). [CrossRef]   [PubMed]  

23. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular momentum (Institute of Physics Publishing, 2003), Chap. 3.

24. E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009). [CrossRef]   [PubMed]  

25. T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE 7400, 740020 (2009). [CrossRef]  

26. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011). [CrossRef]   [PubMed]  

27. P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012). [CrossRef]  

28. X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012). [CrossRef]   [PubMed]  

29. Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express 16(20), 15439–15448 (2008). [CrossRef]   [PubMed]  

30. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]  

31. A. Sihvola, Electromagnetic Mixing Formulas and Applications, Institution of Electrical Engineers (1999).

32. C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template synthesized nanoscopic gold particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem. 98(11), 2963–2971 (1994). [CrossRef]  

References

  • View by:

  1. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405–077409 (2003).
    [Crossref] [PubMed]
  2. D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
    [Crossref]
  3. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
    [Crossref] [PubMed]
  4. F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
    [Crossref]
  5. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
    [Crossref]
  6. P. Jia, Y. Yang, C. J. Min, H. Fang, and X.-C. Yuan, “Sidelobe-modulated optical vortices for free-space communication,” Opt. Lett. 38(4), 588–590 (2013).
    [Crossref] [PubMed]
  7. M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20(22), 24444–24449 (2012).
    [Crossref] [PubMed]
  8. B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37(17), 3735–3737 (2012).
    [Crossref] [PubMed]
  9. J. Leach, E. Bolduc, D. J. Gauthier, and R. W. Boyd, “Secure information capacity of photons entangled in many dimensions,” Phys. Rev. A 85(6), 060304 (2012).
    [Crossref]
  10. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
    [Crossref] [PubMed]
  11. M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20(12), 13195–13200 (2012).
    [Crossref] [PubMed]
  12. N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, “Control of orbital angular momentum of light with optical fibers,” Opt. Lett. 37(13), 2451–2453 (2012).
    [Crossref] [PubMed]
  13. V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].
  14. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref] [PubMed]
  15. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
    [Crossref]
  16. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994).
    [Crossref]
  17. Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev. 6(4), 339–344 (1999).
    [Crossref]
  18. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
    [Crossref] [PubMed]
  19. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
    [Crossref] [PubMed]
  20. L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation,” Appl. Phys. Lett. 88(22), 221102 (2006).
    [Crossref]
  21. E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94(23), 231124 (2009).
    [Crossref]
  22. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002).
    [Crossref] [PubMed]
  23. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular momentum (Institute of Physics Publishing, 2003), Chap. 3.
  24. E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
    [Crossref] [PubMed]
  25. T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE 7400, 740020 (2009).
    [Crossref]
  26. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
    [Crossref] [PubMed]
  27. P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
    [Crossref]
  28. X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
    [Crossref] [PubMed]
  29. Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express 16(20), 15439–15448 (2008).
    [Crossref] [PubMed]
  30. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
    [Crossref]
  31. A. Sihvola, Electromagnetic Mixing Formulas and Applications, Institution of Electrical Engineers (1999).
  32. C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template synthesized nanoscopic gold particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem. 98(11), 2963–2971 (1994).
    [Crossref]

2013 (1)

2012 (9)

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20(22), 24444–24449 (2012).
[Crossref] [PubMed]

B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37(17), 3735–3737 (2012).
[Crossref] [PubMed]

J. Leach, E. Bolduc, D. J. Gauthier, and R. W. Boyd, “Secure information capacity of photons entangled in many dimensions,” Phys. Rev. A 85(6), 060304 (2012).
[Crossref]

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20(12), 13195–13200 (2012).
[Crossref] [PubMed]

N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, “Control of orbital angular momentum of light with optical fibers,” Opt. Lett. 37(13), 2451–2453 (2012).
[Crossref] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

2011 (1)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

2009 (3)

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[Crossref] [PubMed]

T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE 7400, 740020 (2009).
[Crossref]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94(23), 231124 (2009).
[Crossref]

2008 (1)

2006 (3)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation,” Appl. Phys. Lett. 88(22), 221102 (2006).
[Crossref]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[Crossref] [PubMed]

2004 (2)

2003 (1)

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405–077409 (2003).
[Crossref] [PubMed]

2002 (1)

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002).
[Crossref] [PubMed]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref] [PubMed]

1999 (1)

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev. 6(4), 339–344 (1999).
[Crossref]

1994 (2)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template synthesized nanoscopic gold particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem. 98(11), 2963–2971 (1994).
[Crossref]

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1990 (1)

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Ahmed, N.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Aieta, F.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Alekseyev, L. V.

Allen, L.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002).
[Crossref] [PubMed]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Asavei, T.

T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE 7400, 740020 (2009).
[Crossref]

Barnett, S. M.

Bartal, G.

Bazhenov, V. Y.

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Bianchini, A.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
[Crossref]

Blanchard, R.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

Bolduc, E.

J. Leach, E. Bolduc, D. J. Gauthier, and R. W. Boyd, “Secure information capacity of photons entangled in many dimensions,” Phys. Rev. A 85(6), 060304 (2012).
[Crossref]

Boyd, R. W.

Bozinovic, N.

Brasselet, E.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[Crossref] [PubMed]

Cai, X.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Capasso, F.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

Courtial, J.

Dolinar, S.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Fang, H.

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Foss, C. A.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template synthesized nanoscopic gold particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem. 98(11), 2963–2971 (1994).
[Crossref]

Franke-Arnold, S.

Gaburro, Z.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Gauthier, D. J.

J. Leach, E. Bolduc, D. J. Gauthier, and R. W. Boyd, “Secure information capacity of photons entangled in many dimensions,” Phys. Rev. A 85(6), 060304 (2012).
[Crossref]

Genevet, P.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Gibson, G.

Golowich, S.

Hara, T.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev. 6(4), 339–344 (1999).
[Crossref]

Heckenberg, N. R.

T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE 7400, 740020 (2009).
[Crossref]

Hornyak, G. L.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template synthesized nanoscopic gold particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem. 98(11), 2963–2971 (1994).
[Crossref]

Huang, H.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Igasaki, Y.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev. 6(4), 339–344 (1999).
[Crossref]

Inoue, T.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev. 6(4), 339–344 (1999).
[Crossref]

Jacob, Z.

Jia, P.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Johnson-Morris, B.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Juodkazis, S.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[Crossref] [PubMed]

Karimi, E.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94(23), 231124 (2009).
[Crossref]

Kats, M. A.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Kobayashi, Y.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev. 6(4), 339–344 (1999).
[Crossref]

Kolinko, P.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

Kristensen, P.

Lavery, M. P. J.

Leach, J.

Li, F.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev. 6(4), 339–344 (1999).
[Crossref]

Lin, J.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

Liu, Y.

Loke, V. L. Y.

T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE 7400, 740020 (2009).
[Crossref]

MacVicar, I.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002).
[Crossref] [PubMed]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref] [PubMed]

Malik, M.

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation,” Appl. Phys. Lett. 88(22), 221102 (2006).
[Crossref]

Mari, E.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
[Crossref]

Marrucci, L.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94(23), 231124 (2009).
[Crossref]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation,” Appl. Phys. Lett. 88(22), 221102 (2006).
[Crossref]

Martin, C. R.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template synthesized nanoscopic gold particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem. 98(11), 2963–2971 (1994).
[Crossref]

Min, C. J.

Mirhosseini, M.

Misawa, H.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[Crossref] [PubMed]

Mock, J. J.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

Mukohzaka, N.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev. 6(4), 339–344 (1999).
[Crossref]

Murazawa, N.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[Crossref] [PubMed]

Nagali, E.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94(23), 231124 (2009).
[Crossref]

Narimanov, E.

Nieminen, T. A.

T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE 7400, 740020 (2009).
[Crossref]

O’Brien, J. L.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

O’Neil, A. T.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002).
[Crossref] [PubMed]

O’Sullivan, M. N.

Padgett, M. J.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation,” Appl. Phys. Lett. 88(22), 221102 (2006).
[Crossref]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

Pas’ko, V.

Piccirillo, B.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94(23), 231124 (2009).
[Crossref]

Ramachandran, S.

Ren, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Robertson, D. J.

Rodenburg, B.

Romanato, F.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
[Crossref]

Rubinsztein-Dunlop, H.

T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE 7400, 740020 (2009).
[Crossref]

Rye, P.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

Santamato, E.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94(23), 231124 (2009).
[Crossref]

Schurig, D.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405–077409 (2003).
[Crossref] [PubMed]

Scully, M. O.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

Smith, D. R.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405–077409 (2003).
[Crossref] [PubMed]

Sorel, M.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Soskin, M. S.

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

Sponselli, A.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Stockert, J. A.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template synthesized nanoscopic gold particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem. 98(11), 2963–2971 (1994).
[Crossref]

Strain, M. J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Tamburini, F.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
[Crossref]

Tetienne, J.-P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Thidé, B.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
[Crossref]

Thompson, M. G.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Toyoda, H.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev. 6(4), 339–344 (1999).
[Crossref]

Tur, M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

Vasnetsov, M.

Vasnetsov, M. V.

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref] [PubMed]

Wang, J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref] [PubMed]

Willner, A. E.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Yan, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yang, J.-Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yang, Y.

Yoshida, N.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev. 6(4), 339–344 (1999).
[Crossref]

Yu, N.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Yu, S.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Yuan, X.-C.

Yue, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref] [PubMed]

Zhang, X.

Zhu, J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Appl. Phys. Lett. (4)

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation,” Appl. Phys. Lett. 88(22), 221102 (2006).
[Crossref]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94(23), 231124 (2009).
[Crossref]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

J. Phys. Chem. (1)

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template synthesized nanoscopic gold particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem. 98(11), 2963–2971 (1994).
[Crossref]

Laser-beams with screw dislocations in their wavefronts (1)

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

Nat. Photonics (1)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref] [PubMed]

New J. Phys. (1)

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012).
[Crossref]

Opt. Commun. (2)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

Opt. Express (5)

Opt. Lett. (3)

Opt. Rev. (1)

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev. 6(4), 339–344 (1999).
[Crossref]

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

J. Leach, E. Bolduc, D. J. Gauthier, and R. W. Boyd, “Secure information capacity of photons entangled in many dimensions,” Phys. Rev. A 85(6), 060304 (2012).
[Crossref]

Phys. Rev. B (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Phys. Rev. Lett. (4)

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[Crossref] [PubMed]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002).
[Crossref] [PubMed]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405–077409 (2003).
[Crossref] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

Proc. SPIE (1)

T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE 7400, 740020 (2009).
[Crossref]

Science (2)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Other (2)

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular momentum (Institute of Physics Publishing, 2003), Chap. 3.

A. Sihvola, Electromagnetic Mixing Formulas and Applications, Institution of Electrical Engineers (1999).

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Figures (4)

Fig. 1
Fig. 1 Schematic of negative refraction in HMM (a) and the equi-frequency contours of the HMM in xz-plane (b) and yz-plane (c). The black circles are the EFCs of the air and the blue hyperbola are the EFCs of the hyperbolic material. ki and Si are the incident wave vector and Poynting vector, respectively. kr and Sr are the refractive wave vector and Poynting vector, respectively. Due to the indefinite property between xy-plane and z orientation, both the HG1,0 mode and the HG0,1 mode experience a negative refraction. The astigmatic beam is focused inside the indefinite beam.
Fig. 2
Fig. 2 Structure of the biaxial HMM.
Fig. 3
Fig. 3 The schematic of the device and the results including energy density and the phase distributions of the output beam around the focal length.
Fig. 4
Fig. 4 The time averaged energy density and phase distributions in the cross section of the beam along the propagation direction.

Equations (15)

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

E(r,z)= E 0 w 0 w(z) exp( r 2 w 2 (z) ikzik r 2 2R(z) +iς(z) ) H 1 ( 2 w 0 (x+y) ),
θ i =arctan( w(z) z ),
ε =( ε xx 0 0 0 ε yy 0 0 0 ε zz ),
k x 2 ε zz + k z 2 ε xx = ω 2 c 2 , k y 2 ε zz + k z 2 ε yy = ω 2 c 2 .
k i, x = k i, y =sin( θ i )= tan 2 ( θ i ) 1+ tan 2 ( θ i ) , k i, z =cos( θ i )= 1 1+ tan 2 ( θ i ) ,
k i, x = k i, y = k r, x = k r, y ,
k r,xz = ε xx ( 1 k r,x 2 ε zz ) , k r,yz = ε yy ( 1 k r,y 2 ε zz ) .
n kr, xz = k r,xz 2 + k r,x 2 , n kr, yz = k r,yz 2 + k r,y 2 ;
θ kr,zx =arctan( k r, x k r, xz ) , θ kr,yz =arctan( k r, y k r,yz ),
θ Sr,xz =arctan( k r, x ε xx k r, z ε zz ) , θ Sr,yz =arctan( k r, y ε yy k r, z ε zz ).
Δ l o = dcos( θ Sr,zx + θ kr,zx ) cos( θ Sr,zx ) n kr, zx dcos( θ Sr,yz + θ kr,yz ) cos( θ Sr,yz ) n kr, yz ,
Δ l o =d( K k r,x 2 ε zz K )Δn
ε m ( ω )= ε ω p 2 ω( ω+i γ c ) .
ε || =p ε m +( 1p ) ε d ,
ε = ε d + p ε d ( ε m ε d ) ε d +( 1p )( ε m ε d ) q eff .

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