Abstract

Spin and orbital angular momenta (AM) of light are well studied for free-space electromagnetic fields, even nonparaxial. One of the important applications of these concepts is the information transfer using AM modes, often via optical fibers and other guiding systems. However, the self-consistent description of the spin and orbital AM of light in optical media (including dispersive and metallic cases) was provided only recently [Bliokh et al., Phys. Rev. Lett. 119, 073901 (2017) [CrossRef]  ]. Here we present the first accurate calculations, both analytical and numerical, of the spin and orbital AM, as well as the helicity and other properties, for the full-vector eigenmodes of cylindrical dielectric and metallic (nanowire) waveguides. We find remarkable fundamental relations, such as the quantization of the canonical total AM of cylindrical guided modes in the general nonparaxial case. This quantization, as well as the noninteger values of the spin and orbital AM, are determined by the generalized geometric and dynamical phases in the mode fields. Moreover, we show that the spin AM of metallic-wire modes is determined, in the geometrical-optics approximation, by the transverse spin of surface plasmon polaritons propagating along helical trajectories on the wire surface. Our work provides a solid platform for future studies and applications of the AM and helicity properties of guided optical and plasmonic waves.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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2018 (1)

F. Alpeggiani, K. Bliokh, F. Nori, and L. Kuipers, “Electromagnetic helicity in complex media,” Phys. Rev. Lett. 120, 243605 (2018).
[Crossref]

2017 (6)

M. Partanen, T. Häyrynen, J. Oksanen, and J. Tulkki, “Photon mass drag and the momentum of light in a medium,” Phys. Rev. A 95, 063850 (2017).
[Crossref]

D. Garoli, P. Zilio, F. De Angelis, and Y. Gorodetski, “Helicity locking of chiral light emitted from a plasmonic nanotaper,” Nanoscale 9, 6965–6969 (2017).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Optical momentum, spin, and angular momentum in dispersive media,” Phys. Rev. Lett. 119, 073901 (2017).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Optical momentum and angular momentum in complex media: from the Abraham-Minkowski debate to unusual properties of surface plasmon-polaritons,” New J. Phys. 19, 123014 (2017).
[Crossref]

S. A. H. Gangaraj, M. G. Silveirinha, and G. W. Hanson, “Berry phase, Berry connection, and Chern number for a continuum bianisotropic material from a classical electromagnetics perspective,” IEEE J. Multiscale Multiphys. Comput. Tech. 2, 3–17 (2017).
[Crossref]

F. L. Kien, T. Busch, V. G. Truong, and S. N. Chormaic, “Higher-order modes of vacuum-clad ultrathin optical fibers,” Phys. Rev. A 96, 023835 (2017).
[Crossref]

2016 (2)

K. van Kruining and J. B. Götte, “The conditions for the preservation of duality symmetry in a linear medium,” J. Opt. 18, 085601 (2016).
[Crossref]

E. Leader, “The photon angular momentum controversy: resolution of a conflict between laser optics and particle physics,” Phys. Lett. B 756, 303–308 (2016).
[Crossref]

2015 (5)

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
[Crossref]

A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9, 789–795 (2015).
[Crossref]

K. Y. Bliokh, F. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

P. Gregg, P. Kristensen, and S. Ramachandran, “Conservation of orbital angular momentum in air-core optical fibers,” Optica 2, 267–270 (2015).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

2014 (2)

S. Golowich, “Asymptotic theory of strong spin-orbit coupling in optical fiber,” Opt. Lett. 39, 92–95 (2014).
[Crossref]

K. Y. Bliokh, J. Dressel, and F. Nori, “Conservation of the spin and orbital angular momenta in electromagnetism,” New J. Phys. 16, 093037 (2014).
[Crossref]

2013 (5)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic Maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[Crossref]

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

S. M. Barnett and M. Berry, “Superweak momentum transfer near optical vortices,” J. Opt. 15, 125701 (2013).
[Crossref]

2012 (5)

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity, optical spin and related quantities in electromagnetic theory,” New J. Phys. 14, 053050 (2012).
[Crossref]

K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85, 061801 (2012).
[Crossref]

T. G. Philbin and O. Allanson, “Optical angular momentum in dispersive media,” Phys. Rev. A 86, 055802 (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, 488–496 (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, 033001 (2012).
[Crossref]

2011 (4)

B. Kemp, “Resolution of the Abraham-Minkowski debate: implications for the electromagnetic wave theory of light in matter,” J. Appl. Phys. 109, 111101 (2011).
[Crossref]

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Canonical separation of angular momentum of light into its orbital and spin parts,” J. Opt. 13, 064014 (2011).
[Crossref]

T. G. Philbin, “Electromagnetic energy momentum in dispersive media,” Phys. Rev. A 83, 013823 (2011).
[Crossref]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
[Crossref]

2010 (5)

P. W. Milonni and R. W. Boyd, “Momentum of light in a dielectric medium,” Adv. Opt. Photon. 2, 519–553 (2010).
[Crossref]

S. M. Barnett and R. Loudon, “The enigma of optical momentum in a medium,” Philos. Trans. R. Soc. A 368, 927–939 (2010).
[Crossref]

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

S. M. Barnett, “Rotation of electromagnetic fields and the nature of optical angular momentum,” J. Mod. Opt. 57, 1339–1343 (2010).
[Crossref]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[Crossref]

2009 (2)

P. B. Catrysse and S. Fan, “Understanding the dispersion of coaxial plasmonic structures through a connection with the planar metal-insulator-metal geometry,” Appl. Phys. Lett. 94, 231111 (2009).
[Crossref]

M. V. Berry, “Optical currents,” J. Opt. A 11, 094001 (2009).
[Crossref]

2008 (2)

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2, 748–753 (2008).
[Crossref]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101, 030404 (2008).
[Crossref]

2007 (1)

R. N. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197–1216 (2007).
[Crossref]

2006 (1)

F. Le Kien, V. Balykin, and K. Hakuta, “Angular momentum of light in an optical nanofiber,” Phys. Rev. A 73, 053823 (2006).
[Crossref]

2004 (2)

K. Y. Bliokh and Y. P. Bliokh, “Modified geometrical optics of a smoothly inhomogeneous isotropic medium: the anisotropy, berry phase, and the optical magnus effect,” Phys. Rev. E 70, 026605 (2004).
[Crossref]

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, 5448–5456 (2004).
[Crossref]

2003 (2)

V. Garcés-Chávez, D. McGloin, M. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[Crossref]

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

2002 (1)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

2001 (2)

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

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).
[Crossref]

1999 (1)

L. Allen, M. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

1998 (1)

A. Alexeyev, T. Fadeyeva, A. Volyar, and M. Soskin, “Optical vortices and the flow of their angular momentum in a multimode fiber,” Semicond. Phys. Quantum Electron. Optoelectron. 1, 82–89 (1998).

1997 (1)

R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281, 1–64 (1997).
[Crossref]

1996 (3)

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K. Y. Bliokh, M. A. Alonso, and M. R. Dennis are preparing a manuscript to be called “Space-variant geometric phases and topological indices in 3D polarized fields.”

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K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Optical momentum and angular momentum in complex media: from the Abraham-Minkowski debate to unusual properties of surface plasmon-polaritons,” New J. Phys. 19, 123014 (2017).
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K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Optical momentum, spin, and angular momentum in dispersive media,” Phys. Rev. Lett. 119, 073901 (2017).
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K. Y. Bliokh, J. Dressel, and F. Nori, “Conservation of the spin and orbital angular momenta in electromagnetism,” New J. Phys. 16, 093037 (2014).
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K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
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K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
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K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
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K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101, 030404 (2008).
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K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2, 748–753 (2008).
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K. Y. Bliokh, M. A. Alonso, and M. R. Dennis are preparing a manuscript to be called “Space-variant geometric phases and topological indices in 3D polarized fields.”

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N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
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F. L. Kien, T. Busch, V. G. Truong, and S. N. Chormaic, “Higher-order modes of vacuum-clad ultrathin optical fibers,” Phys. Rev. A 96, 023835 (2017).
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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, 5448–5456 (2004).
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J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
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M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).
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K. Y. Bliokh, M. A. Alonso, and M. R. Dennis are preparing a manuscript to be called “Space-variant geometric phases and topological indices in 3D polarized fields.”

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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, 488–496 (2012).
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H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
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Figures (4)

Fig. 1.
Fig. 1. Schematic pictures of the eigenmodes of (a) a dielectric fiber and (b) a metallic wire. The geometrical-optics skew rays with their polarizations (transverse circular in dielectrics and in-plane elliptical for surface plasmon polaritons [6,36,49,50]) are shown by cyan and magenta, respectively. These helical rays and their corresponding polarizations illustrate the origin of the orbital (L) and spin (S) AM of the cylindrical guided modes.
Fig. 2.
Fig. 2. Numerically calculated eigenmodes (a) of a multimode dielectric fiber with parameters r0=200  nm, ϵ1=2.1, and ϵ2=1 and (b) of a metallic wire with parameters r0=150  nm, ϵ1=1ωp2/ω2, ωp=1.3262×1016  s16.63c/r0, ϵ2=1. The frequency ω was varied in these calculations [note that varying the radius r0 would result in different curves in the panels (b)]. The upper panels depict the normalized propagation constants β, which characterize the canonical momentum (9) of the modes (exceeding k0 per photon). The lower panels show the subluminal group velocities (9) of the modes. The small greyscale panels show typical transverse energy distributions W(x,y) in different modes. The dielectric fiber modes are marked by the total-AM quantum number =m+σ, as well as by the three (orbital, spin, and radial) quantum numbers (m,σ,n) in Eq. (8). The metallic-wire modes are marked by the single total-AM quantum number . The dotted curves in (b) correspond to the surface-plasmon geometrical-optics model [Eqs. (21) and (22)].
Fig. 3.
Fig. 3. Numerically calculated canonical spin, orbital, and total AM [Eqs. (1), (10), (11), and (17)] as well as the helicity [Eqs. (4) and (17)] and the Abraham–Poynting total AM [Eq. (3)] of the modes of a dielectric fiber are shown in Fig. 2(a). Here, plotted are the normalized integral values (in units of per photon), defined as ¯=ω/W. One can see the quantization of the canonical total AM J¯z=L¯z+S¯z=, the noninteger Poynting–Abraham AM J¯z, and the differing spin AM and helicity S¯zS¯. In the large-radius (paraxial) limit k0r01, the canonical spin and orbital AM tend to the quantized values L¯zm and S¯zS¯σ.
Fig. 4.
Fig. 4. Same as in Fig. 3 but for the metallic-wire modes shown in Fig. 2(b). The parameters are the same as in Fig. 2(b), and the frequency ω was varied in these plots (varying the radius r0 would result in different curves). The main difference in the behavior of the depicted quantities as compared to Fig. 3 is that in the large-radius (paraxial) limit k0r01, the canonical spin and orbital AM tend to the values L¯z and S¯zS¯0, whereas, surprisingly, the Poynting–Abraham total AM also vanishes: J¯z0. The red dotted curves correspond to the geometrical-optics model for the spin AM [Eq. (23)], based on the transverse spin of surface plasmon polaritons.

Equations (26)

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W=14(ϵ˜|E|2+μ˜|H|2),P=14ωIm[ϵ˜E*·()E+μ˜H*·()H],S=14ωIm(ϵ˜E*×E+μ˜H*×H),L=r×P,J=L+S.
P=12cRe(E*×H),
J=r×P.
S=12ω|n˜|Im(H*·E)=14ω|ϵ˜μϵ+μ˜ϵμ|Im(H*·E),
ϵ={ϵ1,for  r<r0ϵ2,for  r>r0andμ=1.
E±=iϵκ(±βA+ikB)J1(ρ)ei(1)ϕ+iβz,H±=iμκ(±βBikA)J1(ρ)ei(1)ϕ+iβz,Ez=2ϵAJ(ρ)eiϕ+iβz,Hz=2μBJ(ρ)eiϕ+iβz.
Jα(ρ)Hα(1)(ρ),(A,B)(C,D),
Lz=Sz=S=Jz=Jz=0for  =0.
(m,σ,n)=(orbital,spin,radial),
PzW=PzW=βω,υg=c2PzW=ωβ,
ωLzW=(1)W++(+1)W+WzW,wSzW=W+WW.
ωJzW=ωJzW=ωLzW+ωSzW=.
ωJzW.
ϕD=12CArg(Ψ)·dr.
ϕ=Ckloc·drImCψ*·()ψψ*·ψ·dr.
ϕG=ϕϕD.
ωLzW=ϕ̶,ωSzW=ϕ̶G,ωJzW=ϕ̶D=,
W=14[bξG+(aξ++ζ)F],Pz=14ϵμc[bξ+F+aξG],Sz=14ω[aξF+bξ+G],S=14ω[bξF+(aξ++ζ)G].
ξ±(ρ)=|J1(ρ)|2±|J+1(ρ)|2,ζ(ρ)=2|J(ρ)|2,a=k2+β2|κ|2,b=2kβ|κ|2,F=|A|2+|B|2,G=2Im(AB*).
ωSzWωSWσ,ωLzWm,for  k0r01.
ωSzWωSWωJzW0,ωLzW,for  k0r01.
β(ω)kp2(ω)2r02,kp(ω)=ϵ1(ω)1+ϵ1(ω)ωc.
vgc(1+ϵ1)21+ϵ12ϵ11+ϵ12k02r02.
ωSzW1ϵ1(2+ϵ1)1+ϵ12k0r0.
M^=(ϵ2J0ϵ1H(1)0ϵ2βκ12r0Jiϵ2k1κ1Jϵ1βκ22r0H(1)iϵ1k2κ2H(1)0J0H(1)ik1κ1Jβκ12r0Jik2κ2H(1)βκ22r0Hl(1)).
J1J0ϵ2ϵ1κ1κ2H1(1)H0(1)=0(TM),J1J0κ1κ2H1(1)H0(1)=0(TE),

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