Abstract

By considering the transverse spin angular momentum (SAM) that results from the rotation of the electric-field component of a surface mode as a longitudinal SAM of an elliptically polarized light propagating through a homogeneous medium, an alternate route to deriving the formula of the Abraham SAM carried by the surface mode can be achieved. The findings prove in an explicit manner that it is the Abraham SAM that is directly related to the rotation of the electric field.

© 2014 Optical Society of America

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References

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  1. K.-Y. Kim, J. Kim, I.-M. Lee, and B. Lee, Phys. Rev. A 85, 023840 (2012).
    [CrossRef]
  2. K.-Y. Kim and B. Lee, “Angular momentum of surface modes in metamaterial waveguides,” in Proceedings of the 2nd Japan–Korea Metamaterials Forum, June26–28, 2012, pp. 40–41.
  3. K. Y. Bliokh and F. Nori, Phys. Rev. A 85, 061801(R) (2012).
    [CrossRef]
  4. K.-Y. Kim, I.-M. Lee, J. Kim, J. Jung, and B. Lee, Phys. Rev. A 86, 063805 (2012).
    [CrossRef]
  5. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, New J. Phys. 15, 033026 (2013).
    [CrossRef]
  6. K.-Y. Kim and B. Lee, “Abraham and Minkowski spin angular momenta of surface modes in singly negative metamaterial layers,” in Proceedings of the 3rd Korea–Japan Metamaterials Forum, June26–28, 2013, pp. 34–35.
  7. A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, Phys. Rev. A 88, 033831 (2013).
    [CrossRef]
  8. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” arXiv:1308.0547 (2013).
  9. A. Canaguier-Durand and C. Genet, “Transverse spinning of a sphere in a plasmonic field,” arXiv:1311.7481 (2013).
  10. R. Peierls, More Surprises in Theoretical Physics (Princeton University, 1991).
  11. R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. A 79, 023813 (2009).
    [CrossRef]
  12. S. M. Barnett, Phys. Rev. Lett. 104, 070401 (2010).
    [CrossRef]
  13. B. A. Kemp, J. Appl. Phys. 109, 111101 (2011).
    [CrossRef]
  14. We note that the divergence-free term is the electric displacement D that is given by D=ε0E+P, where P denotes the electric polarization. Since we have ρv=−∇·P, the polarization charge density can be written as ρv=ε0∇·E.
  15. K.-Y. Kim and J. Kim, Opt. Lett. 36, 4065 (2011).
    [CrossRef]
  16. M. Padgett, S. M. Barnett, and R. Loudon, J. Mod. Opt. 50, 1555 (2003).
  17. K.-Y. Kim, IEEE Photon. J. 4, 2333 (2012).
    [CrossRef]
  18. S. T. Ali, Phys. Rev. D 7, 1668 (1973).
    [CrossRef]
  19. A. A. Stahlhofen and G. Nimtz, Europhys. Lett. 76, 189 (2006).
    [CrossRef]

2013 (2)

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, New J. Phys. 15, 033026 (2013).
[CrossRef]

A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, Phys. Rev. A 88, 033831 (2013).
[CrossRef]

2012 (4)

K.-Y. Kim, J. Kim, I.-M. Lee, and B. Lee, Phys. Rev. A 85, 023840 (2012).
[CrossRef]

K. Y. Bliokh and F. Nori, Phys. Rev. A 85, 061801(R) (2012).
[CrossRef]

K.-Y. Kim, I.-M. Lee, J. Kim, J. Jung, and B. Lee, Phys. Rev. A 86, 063805 (2012).
[CrossRef]

K.-Y. Kim, IEEE Photon. J. 4, 2333 (2012).
[CrossRef]

2011 (2)

B. A. Kemp, J. Appl. Phys. 109, 111101 (2011).
[CrossRef]

K.-Y. Kim and J. Kim, Opt. Lett. 36, 4065 (2011).
[CrossRef]

2010 (1)

S. M. Barnett, Phys. Rev. Lett. 104, 070401 (2010).
[CrossRef]

2009 (1)

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. A 79, 023813 (2009).
[CrossRef]

2006 (1)

A. A. Stahlhofen and G. Nimtz, Europhys. Lett. 76, 189 (2006).
[CrossRef]

2003 (1)

M. Padgett, S. M. Barnett, and R. Loudon, J. Mod. Opt. 50, 1555 (2003).

1973 (1)

S. T. Ali, Phys. Rev. D 7, 1668 (1973).
[CrossRef]

Ali, S. T.

S. T. Ali, Phys. Rev. D 7, 1668 (1973).
[CrossRef]

Barnett, S. M.

S. M. Barnett, Phys. Rev. Lett. 104, 070401 (2010).
[CrossRef]

M. Padgett, S. M. Barnett, and R. Loudon, J. Mod. Opt. 50, 1555 (2003).

Bekshaev, A. Y.

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, New J. Phys. 15, 033026 (2013).
[CrossRef]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” arXiv:1308.0547 (2013).

Bliokh, K. Y.

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, New J. Phys. 15, 033026 (2013).
[CrossRef]

K. Y. Bliokh and F. Nori, Phys. Rev. A 85, 061801(R) (2012).
[CrossRef]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” arXiv:1308.0547 (2013).

Canaguier-Durand, A.

A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, Phys. Rev. A 88, 033831 (2013).
[CrossRef]

A. Canaguier-Durand and C. Genet, “Transverse spinning of a sphere in a plasmonic field,” arXiv:1311.7481 (2013).

Cuche, A.

A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, Phys. Rev. A 88, 033831 (2013).
[CrossRef]

Ebbesen, T. W.

A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, Phys. Rev. A 88, 033831 (2013).
[CrossRef]

Genet, C.

A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, Phys. Rev. A 88, 033831 (2013).
[CrossRef]

A. Canaguier-Durand and C. Genet, “Transverse spinning of a sphere in a plasmonic field,” arXiv:1311.7481 (2013).

Heckenberg, N. R.

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. A 79, 023813 (2009).
[CrossRef]

Jung, J.

K.-Y. Kim, I.-M. Lee, J. Kim, J. Jung, and B. Lee, Phys. Rev. A 86, 063805 (2012).
[CrossRef]

Kemp, B. A.

B. A. Kemp, J. Appl. Phys. 109, 111101 (2011).
[CrossRef]

Kim, J.

K.-Y. Kim, I.-M. Lee, J. Kim, J. Jung, and B. Lee, Phys. Rev. A 86, 063805 (2012).
[CrossRef]

K.-Y. Kim, J. Kim, I.-M. Lee, and B. Lee, Phys. Rev. A 85, 023840 (2012).
[CrossRef]

K.-Y. Kim and J. Kim, Opt. Lett. 36, 4065 (2011).
[CrossRef]

Kim, K.-Y.

K.-Y. Kim, IEEE Photon. J. 4, 2333 (2012).
[CrossRef]

K.-Y. Kim, J. Kim, I.-M. Lee, and B. Lee, Phys. Rev. A 85, 023840 (2012).
[CrossRef]

K.-Y. Kim, I.-M. Lee, J. Kim, J. Jung, and B. Lee, Phys. Rev. A 86, 063805 (2012).
[CrossRef]

K.-Y. Kim and J. Kim, Opt. Lett. 36, 4065 (2011).
[CrossRef]

K.-Y. Kim and B. Lee, “Angular momentum of surface modes in metamaterial waveguides,” in Proceedings of the 2nd Japan–Korea Metamaterials Forum, June26–28, 2012, pp. 40–41.

K.-Y. Kim and B. Lee, “Abraham and Minkowski spin angular momenta of surface modes in singly negative metamaterial layers,” in Proceedings of the 3rd Korea–Japan Metamaterials Forum, June26–28, 2013, pp. 34–35.

Lee, B.

K.-Y. Kim, J. Kim, I.-M. Lee, and B. Lee, Phys. Rev. A 85, 023840 (2012).
[CrossRef]

K.-Y. Kim, I.-M. Lee, J. Kim, J. Jung, and B. Lee, Phys. Rev. A 86, 063805 (2012).
[CrossRef]

K.-Y. Kim and B. Lee, “Angular momentum of surface modes in metamaterial waveguides,” in Proceedings of the 2nd Japan–Korea Metamaterials Forum, June26–28, 2012, pp. 40–41.

K.-Y. Kim and B. Lee, “Abraham and Minkowski spin angular momenta of surface modes in singly negative metamaterial layers,” in Proceedings of the 3rd Korea–Japan Metamaterials Forum, June26–28, 2013, pp. 34–35.

Lee, I.-M.

K.-Y. Kim, J. Kim, I.-M. Lee, and B. Lee, Phys. Rev. A 85, 023840 (2012).
[CrossRef]

K.-Y. Kim, I.-M. Lee, J. Kim, J. Jung, and B. Lee, Phys. Rev. A 86, 063805 (2012).
[CrossRef]

Loudon, R.

M. Padgett, S. M. Barnett, and R. Loudon, J. Mod. Opt. 50, 1555 (2003).

Nieminen, T. A.

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. A 79, 023813 (2009).
[CrossRef]

Nimtz, G.

A. A. Stahlhofen and G. Nimtz, Europhys. Lett. 76, 189 (2006).
[CrossRef]

Nori, F.

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, New J. Phys. 15, 033026 (2013).
[CrossRef]

K. Y. Bliokh and F. Nori, Phys. Rev. A 85, 061801(R) (2012).
[CrossRef]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” arXiv:1308.0547 (2013).

Padgett, M.

M. Padgett, S. M. Barnett, and R. Loudon, J. Mod. Opt. 50, 1555 (2003).

Peierls, R.

R. Peierls, More Surprises in Theoretical Physics (Princeton University, 1991).

Pfeifer, R. N. C.

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. A 79, 023813 (2009).
[CrossRef]

Rubinsztein-Dunlop, H.

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. A 79, 023813 (2009).
[CrossRef]

Stahlhofen, A. A.

A. A. Stahlhofen and G. Nimtz, Europhys. Lett. 76, 189 (2006).
[CrossRef]

Europhys. Lett. (1)

A. A. Stahlhofen and G. Nimtz, Europhys. Lett. 76, 189 (2006).
[CrossRef]

IEEE Photon. J. (1)

K.-Y. Kim, IEEE Photon. J. 4, 2333 (2012).
[CrossRef]

J. Appl. Phys. (1)

B. A. Kemp, J. Appl. Phys. 109, 111101 (2011).
[CrossRef]

J. Mod. Opt. (1)

M. Padgett, S. M. Barnett, and R. Loudon, J. Mod. Opt. 50, 1555 (2003).

New J. Phys. (1)

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, New J. Phys. 15, 033026 (2013).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (5)

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. A 79, 023813 (2009).
[CrossRef]

K.-Y. Kim, J. Kim, I.-M. Lee, and B. Lee, Phys. Rev. A 85, 023840 (2012).
[CrossRef]

K. Y. Bliokh and F. Nori, Phys. Rev. A 85, 061801(R) (2012).
[CrossRef]

K.-Y. Kim, I.-M. Lee, J. Kim, J. Jung, and B. Lee, Phys. Rev. A 86, 063805 (2012).
[CrossRef]

A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, Phys. Rev. A 88, 033831 (2013).
[CrossRef]

Phys. Rev. D (1)

S. T. Ali, Phys. Rev. D 7, 1668 (1973).
[CrossRef]

Phys. Rev. Lett. (1)

S. M. Barnett, Phys. Rev. Lett. 104, 070401 (2010).
[CrossRef]

Other (6)

K.-Y. Kim and B. Lee, “Abraham and Minkowski spin angular momenta of surface modes in singly negative metamaterial layers,” in Proceedings of the 3rd Korea–Japan Metamaterials Forum, June26–28, 2013, pp. 34–35.

We note that the divergence-free term is the electric displacement D that is given by D=ε0E+P, where P denotes the electric polarization. Since we have ρv=−∇·P, the polarization charge density can be written as ρv=ε0∇·E.

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” arXiv:1308.0547 (2013).

A. Canaguier-Durand and C. Genet, “Transverse spinning of a sphere in a plasmonic field,” arXiv:1311.7481 (2013).

R. Peierls, More Surprises in Theoretical Physics (Princeton University, 1991).

K.-Y. Kim and B. Lee, “Angular momentum of surface modes in metamaterial waveguides,” in Proceedings of the 2nd Japan–Korea Metamaterials Forum, June26–28, 2012, pp. 40–41.

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Figures (2)

Fig. 1.
Fig. 1.

(a) Schematic showing the electric-field component of a TM surface mode and polarization charges. (b) Change in the peak positions of the polarization charge density and (c) resultant directions of the electric-field rotation in the case of a forward surface mode. (d) and (e) Same as (b) and (c) with a backward mode.

Fig. 2.
Fig. 2.

(a) Polarization property of a TM surface mode. We can say that it is in a hybrid polarization state between linear and elliptical ones. (b) We assumed in deriving Eq. (4) that the transverse SAM via the rotation of the electric-field component of a surface mode is equal to the longitudinal SAM carried by an elliptically polarized light propagating through a homogeneous medium of refractive index n=(εrμr)1/2. (c) This longitudinal SAM is a weighted sum of the SAMs carried by left- (σ+) and right-handed (σ) circularly polarized waves.

Equations (9)

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E=η0ϕεr[neffx^+jκy^]ej(neffkyωt),
J=η0ϕ2εr[neff+κ2(1j)+neffκ2(1j)].
ψ|S^|ψ=η02ϕ22εr2[(neff+κ)2(neffκ)2]ngnz^.
srot=wω×ψ|S^|ψ,=μ0neffϕ2ωεr2κμrz^,
sA=(ε0/μr)R(E)×R(Arot)=μ0neffϕ2ωεr2κμrz^,
srot(M)=ngnsrot=μ0neffϕ2ωκεrngnz^=ngnsM,
SA=sAdx=μ0neff2ωk(1μr1εr121μr2εr22)z^.
SM=μ0neff2ωk(1εr11εr2)z^,
srot=wω×η02ϕ22εr2(4neffκ)jng|εr||μr|z^.

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