Abstract

The dynamics of a single particle moving on a flat plane are examined under the influence of state-dependent potentials. As a result, the electromagnetic Poynting-vector flows are captured for optical vortices driven by gain media. In particular, the signs of the coupling coefficients play a crucial role in establishing vortices. In this respect, the singularities related to vortices are resolved through backward time evolutions. Additional effects of the noncircular nature of quadratic potentials are illustrated through numerical simulations. In addition, the quantum mechanical reduction of our model features multipoles.

© 2014 Optical Society of America

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    [CrossRef]
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2014 (1)

C. He, L. Lin, X.-C. Sun, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Topological photonic states,” Int. J. Mod. Phys. B 28, 1441001 (2014).
[CrossRef]

2013 (10)

L. Tang, H. Shi, H. Gao, J. Du, Z. Zhang, X. Dong, and C. Du, “An eigenvalue method to study the threshold behaviors of plasmonic nano-lasers,” Appl. Phys. B 113, 575–579 (2013).
[CrossRef]

N. R. Cooper and J. Dalibard, “Reaching fractional quantum Hall states with optical flux lattices,” Phys. Rev. Lett. 110, 185301 (2013).
[CrossRef]

M. Liu and X. Zhang, “Plasmon-boosted magneto-optics,” Nat. Photonics 7, 429–430 (2013).
[CrossRef]

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339, 1405–1407 (2013).
[CrossRef]

X. Cheng, J. A. Katine, G. E. Rowlands, and I. N. Krivorotov, “Nonlinear ferromagnetic resonance induced by spin torque in nanoscale magnetic tunnel junctions,” Appl. Phys. Lett. 103, 082402 (2013).
[CrossRef]

J. M. Seoane and M. A. F. Sanjuán, “New developments in classical chaotic scattering,” Rep. Prog. Phys. 76, 016001 (2013).
[CrossRef]

M. Selvanayagam and G. V. Eleftheriades, “Experimental demonstration of active electromagnetic cloaking,” Phys. Rev. X 3, 041011 (2013).
[CrossRef]

H.-I. Lee and J. Mok, “Electromagnetic-energy conservation for media with metallic constituents with special attention to damped waves,” J. Opt. 15, 035002 (2013).
[CrossRef]

H.-I. Lee and J. Mok, “Gain-assisted Poynting-vector vortices realized with a single lossy metal wire,” IEEE J. Sel. Top. Quantum Electron. 19, 4600408 (2013).
[CrossRef]

C. Baesens and R. S. MacKay, “Interaction of two systems with saddle-node bifurcations on invariant circles: I. foundations and the mutualistic case,” Nonlinearity 26, 3043–3076 (2013).
[CrossRef]

2012 (2)

S.-P. Zhu, A. Badran, and X. Lu, “A new exact solution for pricing European options in a two-state regime-switching economy,” Comput. Math. Appl. 64, 2744–2755 (2012).
[CrossRef]

N. M. Litchinitser, “Structured light meets structured matter,” Science 337, 1054–1055 (2012).
[CrossRef]

2011 (3)

G. S. Agarwal, “Engineering non-Gaussian entangled states with vortices by photon subtraction,” New J. Phys. 13, 073008 (2011).
[CrossRef]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. A 369, 3525–3550 (2011).
[CrossRef]

H.-I. Lee and E.-H. Lee, “Complex relaxation rates of the Drude metals and their effects on the lifetime and symmetry of plasmon resonances,” Opt. Express 19, 10410–10422 (2011).
[CrossRef]

2010 (3)

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[CrossRef]

H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. II. isolated particles with lossy plasmas,” Phys. Plasmas 17, 072109 (2010).
[CrossRef]

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

2009 (1)

2008 (4)

R. Bjørk, C. R. H. Bahl, A. Smith, and N. Pryds, “Optimization and improvement of Halbach cylinder design,” J. Appl. Phys. 104, 013910 (2008).
[CrossRef]

G. Sagué, A. Baade, and A. Rauschenbeutel, “Blue-detuned evanescent field surface traps for neutral atoms based on mode interference in ultrathin optical fibres,” New J. Phys. 10, 113008 (2008).
[CrossRef]

G. Ferrari and G. Cuoghi, “Schrödinger equation for a particle on a curved surface in an electric and magnetic field,” Phys. Rev. Lett. 100, 230403 (2008).
[CrossRef]

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef]

2007 (1)

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, “Optical bistability in a nonlinear optical coupler with a negative index channel,” Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

2005 (2)

A. V. Krasavin, A. S. Schwanecke, N. I. Zheludev, M. Reichelt, T. Stroucken, S. W. Koch, and E. M. Wright, “Polarization conversion and “focusing” of light propagating through a small chiral hole in a metallic screen,” Appl. Phys. Lett. 86, 201105 (2005).
[CrossRef]

S. L. Prosvirnin and N. I. Zheludev, “Polarization effects in the diffraction of light by a planar chiral structure,” Phys. Rev. E 71, 037603 (2005).
[CrossRef]

2003 (1)

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

1992 (1)

Q. Du, M. D. Gunzburger, and J. S. Peterson, “Analysis and approximation of the Ginzburg–Landau model of superconductivity,” SIAM Rev. 34, 54–81 (1992).
[CrossRef]

1980 (1)

K. Halbach, “Design of permanent multipole magnets with oriented rare earth cobalt material,” Nucl. Instrum. Methods 169, 1–10 (1980).
[CrossRef]

Abramowitz, M.

M. Abramowitz and N. C. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Agarwal, G. S.

G. S. Agarwal, “Engineering non-Gaussian entangled states with vortices by photon subtraction,” New J. Phys. 13, 073008 (2011).
[CrossRef]

Baade, A.

G. Sagué, A. Baade, and A. Rauschenbeutel, “Blue-detuned evanescent field surface traps for neutral atoms based on mode interference in ultrathin optical fibres,” New J. Phys. 10, 113008 (2008).
[CrossRef]

Badran, A.

S.-P. Zhu, A. Badran, and X. Lu, “A new exact solution for pricing European options in a two-state regime-switching economy,” Comput. Math. Appl. 64, 2744–2755 (2012).
[CrossRef]

Baesens, C.

C. Baesens and R. S. MacKay, “Interaction of two systems with saddle-node bifurcations on invariant circles: I. foundations and the mutualistic case,” Nonlinearity 26, 3043–3076 (2013).
[CrossRef]

Bahl, C. R. H.

R. Bjørk, C. R. H. Bahl, A. Smith, and N. Pryds, “Optimization and improvement of Halbach cylinder design,” J. Appl. Phys. 104, 013910 (2008).
[CrossRef]

Bjørk, R.

R. Bjørk, C. R. H. Bahl, A. Smith, and N. Pryds, “Optimization and improvement of Halbach cylinder design,” J. Appl. Phys. 104, 013910 (2008).
[CrossRef]

Bondarenko, O.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

Boyce, W. E.

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 9th ed. (Wiley, 2010).

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimzation (Cambridge University, 2004).

Cheang-Wong, J. C.

Chen, Y.-F.

C. He, L. Lin, X.-C. Sun, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Topological photonic states,” Int. J. Mod. Phys. B 28, 1441001 (2014).
[CrossRef]

Cheng, X.

X. Cheng, J. A. Katine, G. E. Rowlands, and I. N. Krivorotov, “Nonlinear ferromagnetic resonance induced by spin torque in nanoscale magnetic tunnel junctions,” Appl. Phys. Lett. 103, 082402 (2013).
[CrossRef]

Cho, S.-W.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[CrossRef]

Cooper, N. R.

N. R. Cooper and J. Dalibard, “Reaching fractional quantum Hall states with optical flux lattices,” Phys. Rev. Lett. 110, 185301 (2013).
[CrossRef]

Crespo-Sosa, A.

Cuoghi, G.

G. Ferrari and G. Cuoghi, “Schrödinger equation for a particle on a curved surface in an electric and magnetic field,” Phys. Rev. Lett. 100, 230403 (2008).
[CrossRef]

Dalibard, J.

N. R. Cooper and J. Dalibard, “Reaching fractional quantum Hall states with optical flux lattices,” Phys. Rev. Lett. 110, 185301 (2013).
[CrossRef]

DiPrima, R. C.

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 9th ed. (Wiley, 2010).

Dong, X.

L. Tang, H. Shi, H. Gao, J. Du, Z. Zhang, X. Dong, and C. Du, “An eigenvalue method to study the threshold behaviors of plasmonic nano-lasers,” Appl. Phys. B 113, 575–579 (2013).
[CrossRef]

Du, C.

L. Tang, H. Shi, H. Gao, J. Du, Z. Zhang, X. Dong, and C. Du, “An eigenvalue method to study the threshold behaviors of plasmonic nano-lasers,” Appl. Phys. B 113, 575–579 (2013).
[CrossRef]

Du, J.

L. Tang, H. Shi, H. Gao, J. Du, Z. Zhang, X. Dong, and C. Du, “An eigenvalue method to study the threshold behaviors of plasmonic nano-lasers,” Appl. Phys. B 113, 575–579 (2013).
[CrossRef]

Du, Q.

Q. Du, M. D. Gunzburger, and J. S. Peterson, “Analysis and approximation of the Ginzburg–Landau model of superconductivity,” SIAM Rev. 34, 54–81 (1992).
[CrossRef]

Eleftheriades, G. V.

M. Selvanayagam and G. V. Eleftheriades, “Experimental demonstration of active electromagnetic cloaking,” Phys. Rev. X 3, 041011 (2013).
[CrossRef]

Fainman, Y.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

Feng, L.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

Ferrari, G.

G. Ferrari and G. Cuoghi, “Schrödinger equation for a particle on a curved surface in an electric and magnetic field,” Phys. Rev. Lett. 100, 230403 (2008).
[CrossRef]

Gabitov, I. R.

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, “Optical bistability in a nonlinear optical coupler with a negative index channel,” Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

Gao, H.

L. Tang, H. Shi, H. Gao, J. Du, Z. Zhang, X. Dong, and C. Du, “An eigenvalue method to study the threshold behaviors of plasmonic nano-lasers,” Appl. Phys. B 113, 575–579 (2013).
[CrossRef]

Gorodetski, Y.

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef]

Gunzburger, M. D.

Q. Du, M. D. Gunzburger, and J. S. Peterson, “Analysis and approximation of the Ginzburg–Landau model of superconductivity,” SIAM Rev. 34, 54–81 (1992).
[CrossRef]

Halbach, K.

K. Halbach, “Design of permanent multipole magnets with oriented rare earth cobalt material,” Nucl. Instrum. Methods 169, 1–10 (1980).
[CrossRef]

Hamm, J. M.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. A 369, 3525–3550 (2011).
[CrossRef]

Hasman, E.

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef]

He, C.

C. He, L. Lin, X.-C. Sun, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Topological photonic states,” Int. J. Mod. Phys. B 28, 1441001 (2014).
[CrossRef]

Hess, O.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. A 369, 3525–3550 (2011).
[CrossRef]

Jakubowski, M. H.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

Kang, M.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[CrossRef]

Katine, J. A.

X. Cheng, J. A. Katine, G. E. Rowlands, and I. N. Krivorotov, “Nonlinear ferromagnetic resonance induced by spin torque in nanoscale magnetic tunnel junctions,” Appl. Phys. Lett. 103, 082402 (2013).
[CrossRef]

Kim, H.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[CrossRef]

Kleiner, V.

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef]

Koch, S. W.

A. V. Krasavin, A. S. Schwanecke, N. I. Zheludev, M. Reichelt, T. Stroucken, S. W. Koch, and E. M. Wright, “Polarization conversion and “focusing” of light propagating through a small chiral hole in a metallic screen,” Appl. Phys. Lett. 86, 201105 (2005).
[CrossRef]

Krasavin, A. V.

A. V. Krasavin, A. S. Schwanecke, N. I. Zheludev, M. Reichelt, T. Stroucken, S. W. Koch, and E. M. Wright, “Polarization conversion and “focusing” of light propagating through a small chiral hole in a metallic screen,” Appl. Phys. Lett. 86, 201105 (2005).
[CrossRef]

Krivorotov, I. N.

X. Cheng, J. A. Katine, G. E. Rowlands, and I. N. Krivorotov, “Nonlinear ferromagnetic resonance induced by spin torque in nanoscale magnetic tunnel junctions,” Appl. Phys. Lett. 103, 082402 (2013).
[CrossRef]

Lee, B.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[CrossRef]

Lee, E.-H.

Lee, H.-I.

H.-I. Lee and J. Mok, “Gain-assisted Poynting-vector vortices realized with a single lossy metal wire,” IEEE J. Sel. Top. Quantum Electron. 19, 4600408 (2013).
[CrossRef]

H.-I. Lee and J. Mok, “Electromagnetic-energy conservation for media with metallic constituents with special attention to damped waves,” J. Opt. 15, 035002 (2013).
[CrossRef]

H.-I. Lee and E.-H. Lee, “Complex relaxation rates of the Drude metals and their effects on the lifetime and symmetry of plasmon resonances,” Opt. Express 19, 10410–10422 (2011).
[CrossRef]

H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. II. isolated particles with lossy plasmas,” Phys. Plasmas 17, 072109 (2010).
[CrossRef]

Lee, S.-Y.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[CrossRef]

Lin, L.

C. He, L. Lin, X.-C. Sun, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Topological photonic states,” Int. J. Mod. Phys. B 28, 1441001 (2014).
[CrossRef]

Litchinitser, N. M.

N. M. Litchinitser, “Structured light meets structured matter,” Science 337, 1054–1055 (2012).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, “Optical bistability in a nonlinear optical coupler with a negative index channel,” Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

Liu, M.

M. Liu and X. Zhang, “Plasmon-boosted magneto-optics,” Nat. Photonics 7, 429–430 (2013).
[CrossRef]

Liu, X.-P.

C. He, L. Lin, X.-C. Sun, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Topological photonic states,” Int. J. Mod. Phys. B 28, 1441001 (2014).
[CrossRef]

Lomakin, V.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

López-Suárez, A.

Lu, M.-H.

C. He, L. Lin, X.-C. Sun, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Topological photonic states,” Int. J. Mod. Phys. B 28, 1441001 (2014).
[CrossRef]

Lu, X.

S.-P. Zhu, A. Badran, and X. Lu, “A new exact solution for pricing European options in a two-state regime-switching economy,” Comput. Math. Appl. 64, 2744–2755 (2012).
[CrossRef]

MacKay, R. S.

C. Baesens and R. S. MacKay, “Interaction of two systems with saddle-node bifurcations on invariant circles: I. foundations and the mutualistic case,” Nonlinearity 26, 3043–3076 (2013).
[CrossRef]

Maimistov, A. I.

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, “Optical bistability in a nonlinear optical coupler with a negative index channel,” Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

Mizrahi, A.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

Mok, J.

H.-I. Lee and J. Mok, “Gain-assisted Poynting-vector vortices realized with a single lossy metal wire,” IEEE J. Sel. Top. Quantum Electron. 19, 4600408 (2013).
[CrossRef]

H.-I. Lee and J. Mok, “Electromagnetic-energy conservation for media with metallic constituents with special attention to damped waves,” J. Opt. 15, 035002 (2013).
[CrossRef]

H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. II. isolated particles with lossy plasmas,” Phys. Plasmas 17, 072109 (2010).
[CrossRef]

Nezhad, M. P.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

Niv, A.

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef]

Oliver, A.

Park, J.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[CrossRef]

Peterson, J. S.

Q. Du, M. D. Gunzburger, and J. S. Peterson, “Analysis and approximation of the Ginzburg–Landau model of superconductivity,” SIAM Rev. 34, 54–81 (1992).
[CrossRef]

Prosvirnin, S. L.

S. L. Prosvirnin and N. I. Zheludev, “Polarization effects in the diffraction of light by a planar chiral structure,” Phys. Rev. E 71, 037603 (2005).
[CrossRef]

Pryds, N.

R. Bjørk, C. R. H. Bahl, A. Smith, and N. Pryds, “Optimization and improvement of Halbach cylinder design,” J. Appl. Phys. 104, 013910 (2008).
[CrossRef]

Pusch, A.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. A 369, 3525–3550 (2011).
[CrossRef]

Rauschenbeutel, A.

G. Sagué, A. Baade, and A. Rauschenbeutel, “Blue-detuned evanescent field surface traps for neutral atoms based on mode interference in ultrathin optical fibres,” New J. Phys. 10, 113008 (2008).
[CrossRef]

Reichelt, M.

A. V. Krasavin, A. S. Schwanecke, N. I. Zheludev, M. Reichelt, T. Stroucken, S. W. Koch, and E. M. Wright, “Polarization conversion and “focusing” of light propagating through a small chiral hole in a metallic screen,” Appl. Phys. Lett. 86, 201105 (2005).
[CrossRef]

Reyes-Esqueda, J. A.

Rho, J.

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339, 1405–1407 (2013).
[CrossRef]

Rodríguez-Fernández, L.

Rodríguez-Iglesias, V.

Rowlands, G. E.

X. Cheng, J. A. Katine, G. E. Rowlands, and I. N. Krivorotov, “Nonlinear ferromagnetic resonance induced by spin torque in nanoscale magnetic tunnel junctions,” Appl. Phys. Lett. 103, 082402 (2013).
[CrossRef]

Sagué, G.

G. Sagué, A. Baade, and A. Rauschenbeutel, “Blue-detuned evanescent field surface traps for neutral atoms based on mode interference in ultrathin optical fibres,” New J. Phys. 10, 113008 (2008).
[CrossRef]

Sanjuán, M. A. F.

J. M. Seoane and M. A. F. Sanjuán, “New developments in classical chaotic scattering,” Rep. Prog. Phys. 76, 016001 (2013).
[CrossRef]

Santiago-Ramírez, A.-L.

Schwanecke, A. S.

A. V. Krasavin, A. S. Schwanecke, N. I. Zheludev, M. Reichelt, T. Stroucken, S. W. Koch, and E. M. Wright, “Polarization conversion and “focusing” of light propagating through a small chiral hole in a metallic screen,” Appl. Phys. Lett. 86, 201105 (2005).
[CrossRef]

Sears, S. M.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

Segev, M.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

Selvanayagam, M.

M. Selvanayagam and G. V. Eleftheriades, “Experimental demonstration of active electromagnetic cloaking,” Phys. Rev. X 3, 041011 (2013).
[CrossRef]

Seoane, J. M.

J. M. Seoane and M. A. F. Sanjuán, “New developments in classical chaotic scattering,” Rep. Prog. Phys. 76, 016001 (2013).
[CrossRef]

Shi, H.

L. Tang, H. Shi, H. Gao, J. Du, Z. Zhang, X. Dong, and C. Du, “An eigenvalue method to study the threshold behaviors of plasmonic nano-lasers,” Appl. Phys. B 113, 575–579 (2013).
[CrossRef]

Silva-Pereyra, H.-G.

Simic, A.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

Slutsky, B.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

Smith, A.

R. Bjørk, C. R. H. Bahl, A. Smith, and N. Pryds, “Optimization and improvement of Halbach cylinder design,” J. Appl. Phys. 104, 013910 (2008).
[CrossRef]

Soljacic, M.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

Squier, R.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

Stegun, N. C.

M. Abramowitz and N. C. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Steiglitz, K.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

Stroucken, T.

A. V. Krasavin, A. S. Schwanecke, N. I. Zheludev, M. Reichelt, T. Stroucken, S. W. Koch, and E. M. Wright, “Polarization conversion and “focusing” of light propagating through a small chiral hole in a metallic screen,” Appl. Phys. Lett. 86, 201105 (2005).
[CrossRef]

Sun, X.-C.

C. He, L. Lin, X.-C. Sun, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Topological photonic states,” Int. J. Mod. Phys. B 28, 1441001 (2014).
[CrossRef]

Tang, L.

L. Tang, H. Shi, H. Gao, J. Du, Z. Zhang, X. Dong, and C. Du, “An eigenvalue method to study the threshold behaviors of plasmonic nano-lasers,” Appl. Phys. B 113, 575–579 (2013).
[CrossRef]

Torres-Torres, C.

Tsakmakidis, K. L.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. A 369, 3525–3550 (2011).
[CrossRef]

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimzation (Cambridge University, 2004).

Wang, Y.

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339, 1405–1407 (2013).
[CrossRef]

Wright, E. M.

A. V. Krasavin, A. S. Schwanecke, N. I. Zheludev, M. Reichelt, T. Stroucken, S. W. Koch, and E. M. Wright, “Polarization conversion and “focusing” of light propagating through a small chiral hole in a metallic screen,” Appl. Phys. Lett. 86, 201105 (2005).
[CrossRef]

Wuestner, S.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. A 369, 3525–3550 (2011).
[CrossRef]

Ye, Z.

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339, 1405–1407 (2013).
[CrossRef]

Yin, X.

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339, 1405–1407 (2013).
[CrossRef]

Zhang, X.

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339, 1405–1407 (2013).
[CrossRef]

M. Liu and X. Zhang, “Plasmon-boosted magneto-optics,” Nat. Photonics 7, 429–430 (2013).
[CrossRef]

Zhang, Z.

L. Tang, H. Shi, H. Gao, J. Du, Z. Zhang, X. Dong, and C. Du, “An eigenvalue method to study the threshold behaviors of plasmonic nano-lasers,” Appl. Phys. B 113, 575–579 (2013).
[CrossRef]

Zheludev, N. I.

A. V. Krasavin, A. S. Schwanecke, N. I. Zheludev, M. Reichelt, T. Stroucken, S. W. Koch, and E. M. Wright, “Polarization conversion and “focusing” of light propagating through a small chiral hole in a metallic screen,” Appl. Phys. Lett. 86, 201105 (2005).
[CrossRef]

S. L. Prosvirnin and N. I. Zheludev, “Polarization effects in the diffraction of light by a planar chiral structure,” Phys. Rev. E 71, 037603 (2005).
[CrossRef]

Zhu, S.-P.

S.-P. Zhu, A. Badran, and X. Lu, “A new exact solution for pricing European options in a two-state regime-switching economy,” Comput. Math. Appl. 64, 2744–2755 (2012).
[CrossRef]

Appl. Phys. B (1)

L. Tang, H. Shi, H. Gao, J. Du, Z. Zhang, X. Dong, and C. Du, “An eigenvalue method to study the threshold behaviors of plasmonic nano-lasers,” Appl. Phys. B 113, 575–579 (2013).
[CrossRef]

Appl. Phys. Lett. (2)

A. V. Krasavin, A. S. Schwanecke, N. I. Zheludev, M. Reichelt, T. Stroucken, S. W. Koch, and E. M. Wright, “Polarization conversion and “focusing” of light propagating through a small chiral hole in a metallic screen,” Appl. Phys. Lett. 86, 201105 (2005).
[CrossRef]

X. Cheng, J. A. Katine, G. E. Rowlands, and I. N. Krivorotov, “Nonlinear ferromagnetic resonance induced by spin torque in nanoscale magnetic tunnel junctions,” Appl. Phys. Lett. 103, 082402 (2013).
[CrossRef]

Comput. Math. Appl. (1)

S.-P. Zhu, A. Badran, and X. Lu, “A new exact solution for pricing European options in a two-state regime-switching economy,” Comput. Math. Appl. 64, 2744–2755 (2012).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

H.-I. Lee and J. Mok, “Gain-assisted Poynting-vector vortices realized with a single lossy metal wire,” IEEE J. Sel. Top. Quantum Electron. 19, 4600408 (2013).
[CrossRef]

Int. J. Mod. Phys. B (1)

C. He, L. Lin, X.-C. Sun, X.-P. Liu, M.-H. Lu, and Y.-F. Chen, “Topological photonic states,” Int. J. Mod. Phys. B 28, 1441001 (2014).
[CrossRef]

J. Appl. Phys. (1)

R. Bjørk, C. R. H. Bahl, A. Smith, and N. Pryds, “Optimization and improvement of Halbach cylinder design,” J. Appl. Phys. 104, 013910 (2008).
[CrossRef]

J. Opt. (1)

H.-I. Lee and J. Mok, “Electromagnetic-energy conservation for media with metallic constituents with special attention to damped waves,” J. Opt. 15, 035002 (2013).
[CrossRef]

Nano Lett. (1)

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[CrossRef]

Nat. Photonics (2)

M. Liu and X. Zhang, “Plasmon-boosted magneto-optics,” Nat. Photonics 7, 429–430 (2013).
[CrossRef]

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

New J. Phys. (2)

G. Sagué, A. Baade, and A. Rauschenbeutel, “Blue-detuned evanescent field surface traps for neutral atoms based on mode interference in ultrathin optical fibres,” New J. Phys. 10, 113008 (2008).
[CrossRef]

G. S. Agarwal, “Engineering non-Gaussian entangled states with vortices by photon subtraction,” New J. Phys. 13, 073008 (2011).
[CrossRef]

Nonlinearity (1)

C. Baesens and R. S. MacKay, “Interaction of two systems with saddle-node bifurcations on invariant circles: I. foundations and the mutualistic case,” Nonlinearity 26, 3043–3076 (2013).
[CrossRef]

Nucl. Instrum. Methods (1)

K. Halbach, “Design of permanent multipole magnets with oriented rare earth cobalt material,” Nucl. Instrum. Methods 169, 1–10 (1980).
[CrossRef]

Opt. Express (2)

Phil. Trans. R. Soc. A (1)

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Gain and plasmon dynamics in active negative-index metamaterials,” Phil. Trans. R. Soc. A 369, 3525–3550 (2011).
[CrossRef]

Phys. Plasmas (1)

H.-I. Lee and J. Mok, “On the cubic zero-order solution of electromagnetic waves. II. isolated particles with lossy plasmas,” Phys. Plasmas 17, 072109 (2010).
[CrossRef]

Phys. Rev. E (1)

S. L. Prosvirnin and N. I. Zheludev, “Polarization effects in the diffraction of light by a planar chiral structure,” Phys. Rev. E 71, 037603 (2005).
[CrossRef]

Phys. Rev. Lett. (5)

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101, 043903 (2008).
[CrossRef]

G. Ferrari and G. Cuoghi, “Schrödinger equation for a particle on a curved surface in an electric and magnetic field,” Phys. Rev. Lett. 100, 230403 (2008).
[CrossRef]

N. R. Cooper and J. Dalibard, “Reaching fractional quantum Hall states with optical flux lattices,” Phys. Rev. Lett. 110, 185301 (2013).
[CrossRef]

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, “Optical bistability in a nonlinear optical coupler with a negative index channel,” Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

Phys. Rev. X (1)

M. Selvanayagam and G. V. Eleftheriades, “Experimental demonstration of active electromagnetic cloaking,” Phys. Rev. X 3, 041011 (2013).
[CrossRef]

Rep. Prog. Phys. (1)

J. M. Seoane and M. A. F. Sanjuán, “New developments in classical chaotic scattering,” Rep. Prog. Phys. 76, 016001 (2013).
[CrossRef]

Science (2)

N. M. Litchinitser, “Structured light meets structured matter,” Science 337, 1054–1055 (2012).
[CrossRef]

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339, 1405–1407 (2013).
[CrossRef]

SIAM Rev. (1)

Q. Du, M. D. Gunzburger, and J. S. Peterson, “Analysis and approximation of the Ginzburg–Landau model of superconductivity,” SIAM Rev. 34, 54–81 (1992).
[CrossRef]

Other (3)

S. Boyd and L. Vandenberghe, Convex Optimzation (Cambridge University, 2004).

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 9th ed. (Wiley, 2010).

M. Abramowitz and N. C. Stegun, Handbook of Mathematical Functions (Dover, 1970).

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Figures (5)

Fig. 1.
Fig. 1.

Trajectories on the rectangular (x,y) plane, which simulate the PVFs with the parameters a=1 and α=β=1. (a) Backward trajectories emanating from a finite-thickness annulus of gain medium (in yellow color), obtained from the exact solutions θ(t)=θ0+aτ and Eq. (12). (b) Forward trajectories running toward the unit circle, obtained from the exact solutions θ(t)=θ0at and Eq. (9).

Fig. 2.
Fig. 2.

Vortex motions on the rectangular (x,y) plane for the opposite-signs couplings with α=β=1 obtained from numerical solutions to Eq. (1). The initial positions are located (a) in the exterior of the unit circle with r0=2 and (b) in the interior of the unit circle with r0=0.3. Angular positions at the initial time are given by θ0=(1/2)mπ with m=0,1,,7.

Fig. 3.
Fig. 3.

Time versus angle t(θ) in dashed curves from Eq. (4), and angle versus time θ(t) in solid curves from Eq. (5): (a) for β=1, (b) for β=4. Besides, a=1 and α=1 for both panels.

Fig. 4.
Fig. 4.

Trajectories on the rectangular (x,y) plane obtained from numerical integrations with α=1 and β=4<0 (opposite-signed couplings). The additional parameter a is written on each panel.

Fig. 5.
Fig. 5.

Trajectories obtained by solving the cubic Schrödinger Eq. (17) with α=β˜=1 on the basis of θ˜(τ)=θ˜0at˜. Note that K=0 here. The ellipticity is varied such that (a) a=1, (b) a=(1/4), and (c) a=4.

Equations (22)

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

dxdt=a2αyx(x2a2+y2b21),
dydt=b2βxy(x2a2+y2b21).
drdt=12a(α+β)rsin(2θ)+r(1r2),
dθdt=12a[(α+β)cos(2θ)+βα].
at=1αβarctan[ααβtan(θ)],αβ<0.
tan(θ)=αβαtan(atαβ),αβ<0.
x,yexp[(1±iaαβ)t],αβ<0.
dxdt=x+yx(x2+y2),
dydt=x+yy(x2+y2).
drdt=r(1r2).
r=[1+(1r021)e2at]1/2.
drdθ=(α+β)rsin(2θ)+2a1r(1r2)(α+β)cos(2θ)+βα.
r(0)r0={rex>1,r(τ)rexrin<1,r(τ)rin.
r=[1+(1r021)e2aτ]1/2.
τ<τlim=12ln(rex2rex21).
t=i×t˜,β=β˜,tanθ=itanθ˜.
drdt˜=12a(αβ˜)rtan(2θ˜)i×r(1|r|2),
dθ˜dt˜=12a[(α+β˜)cos(2θ˜)+β˜α].
d|r|2dt˜=a(αβ˜)|r|2tan(2θ˜)+K|r|2(1|r|2).
|r|2|r0|2=|cos(2θ˜)||cos(2θ˜0)|.
a2cos2θ˜+sin2θ˜=aa2cos2(θ˜±12π)+sin2(θ˜±12π).
V(x,y;a,b,g)=(x2a2+gxy+y2b2)1.

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