Abstract

We show that light guided in a planar dielectric slab geometry incorporating a biaxial medium has lossless modes with group and phase velocities in opposite directions. Particles in a vacuum gap inserted into the structure experience negative radiation pressure: the particles are pulled by light rather than pushed by it. This effectively one-dimensional dielectric structure represents a new geometry for achieving negative radiation pressure in a wide range of frequencies with minimal loss. Moreover, this geometry provides a straightforward platform for experimentally resolving the Abrahams-Minkowski dilemma.

© 2012 OSA

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    [CrossRef]
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  5. A. Novitsky, C. W. Qiu, and H. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107(20), 203601 (2011).
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  6. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
    [CrossRef] [PubMed]
  7. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
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  12. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002).
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  13. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423(6940), 604–605 (2003).
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2011 (4)

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011).
[CrossRef]

S. Sukhov and A. Dogariu, “Negative nonconservative forces: optical “tractor beams” for arbitrary objects,” Phys. Rev. Lett. 107(20), 203602 (2011).
[CrossRef] [PubMed]

A. Novitsky, C. W. Qiu, and H. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107(20), 203601 (2011).
[CrossRef] [PubMed]

A. Salandrino and D. N. Christodoulides, “Reverse optical forces in negative index dielectric waveguide arrays,” Opt. Lett. 36(16), 3103–3105 (2011).
[CrossRef] [PubMed]

2010 (3)

A. Salandrino and D. N. Christodoulides, “Negative index Clarricoats-Waldron waveguides for terahertz and far infrared applications,” Opt. Express 18(4), 3626–3631 (2010).
[CrossRef] [PubMed]

S. M. Barnett, “Resolution of the abraham-minkowski dilemma,” Phys. Rev. Lett. 104(7), 070401 (2010).
[CrossRef] [PubMed]

S. M. Barnett and R. Loudon, “The enigma of optical momentum in a medium,” Philos. Transact. A Math. Phys. Eng. Sci. 368(1914), 927–939 (2010).
[CrossRef] [PubMed]

2008 (2)

W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” Phys. Rev. Lett. 101(24), 243601 (2008).
[CrossRef] [PubMed]

A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: Recent advances and outlook,” Metamaterials (Amst.) 2(1), 1–17 (2008).
[CrossRef]

2007 (2)

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[CrossRef]

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98(17), 177404 (2007).
[CrossRef]

2006 (2)

2005 (2)

V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71(20), 201101 (2005).
[CrossRef]

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94(17), 170403 (2005).
[CrossRef] [PubMed]

2004 (2)

J. Xu, J. Drelich, and E. M. Nadgorny, “Laser-based patterning of gold nanoparticles into microstructures,” Langmuir 20(4), 1021–1025 (2004).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

2003 (1)

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423(6940), 604–605 (2003).
[CrossRef] [PubMed]

2002 (1)

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002).
[CrossRef]

2001 (1)

2000 (2)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

1998 (1)

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[CrossRef]

1909 (1)

M. Abraham, “Zur elektrodynamik bewegter körper,” Rendiconti del Circolo Matematico di Palermo 28(1), 1–28 (1909).
[CrossRef]

Abraham, M.

M. Abraham, “Zur elektrodynamik bewegter körper,” Rendiconti del Circolo Matematico di Palermo 28(1), 1–28 (1909).
[CrossRef]

Aydin, K.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423(6940), 604–605 (2003).
[CrossRef] [PubMed]

Barnett, S. M.

S. M. Barnett, “Resolution of the abraham-minkowski dilemma,” Phys. Rev. Lett. 104(7), 070401 (2010).
[CrossRef] [PubMed]

S. M. Barnett and R. Loudon, “The enigma of optical momentum in a medium,” Philos. Transact. A Math. Phys. Eng. Sci. 368(1914), 927–939 (2010).
[CrossRef] [PubMed]

Boltasseva, A.

A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: Recent advances and outlook,” Metamaterials (Amst.) 2(1), 1–17 (2008).
[CrossRef]

Boyd, M.

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94(17), 170403 (2005).
[CrossRef] [PubMed]

Campbell, G. K.

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94(17), 170403 (2005).
[CrossRef] [PubMed]

Chan, C. T.

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011).
[CrossRef]

Chen, J.

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011).
[CrossRef]

Christodoulides, D. N.

Cubukcu, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423(6940), 604–605 (2003).
[CrossRef] [PubMed]

Dogariu, A.

S. Sukhov and A. Dogariu, “Negative nonconservative forces: optical “tractor beams” for arbitrary objects,” Phys. Rev. Lett. 107(20), 203602 (2011).
[CrossRef] [PubMed]

Drelich, J.

J. Xu, J. Drelich, and E. M. Nadgorny, “Laser-based patterning of gold nanoparticles into microstructures,” Langmuir 20(4), 1021–1025 (2004).
[CrossRef] [PubMed]

El-Ganainy, R.

Feng, R.

W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” Phys. Rev. Lett. 101(24), 243601 (2008).
[CrossRef] [PubMed]

Foteinopoulou, S.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423(6940), 604–605 (2003).
[CrossRef] [PubMed]

Hazel, J.

Hodgkinson, I.

Joannopoulos, J. D.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001).
[CrossRef] [PubMed]

Johnson, S. G.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001).
[CrossRef] [PubMed]

Ketterle, W.

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94(17), 170403 (2005).
[CrossRef] [PubMed]

Leanhardt, A. E.

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94(17), 170403 (2005).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt, “Optics: momentum in an uncertain light,” Nature 444(7121), 823–824 (2006).
[CrossRef] [PubMed]

Lin, Z.

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011).
[CrossRef]

Loudon, R.

S. M. Barnett and R. Loudon, “The enigma of optical momentum in a medium,” Philos. Transact. A Math. Phys. Eng. Sci. 368(1914), 927–939 (2010).
[CrossRef] [PubMed]

Luo, C.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002).
[CrossRef]

Mokhov, S.

Mun, J.

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94(17), 170403 (2005).
[CrossRef] [PubMed]

Nadgorny, E. M.

J. Xu, J. Drelich, and E. M. Nadgorny, “Laser-based patterning of gold nanoparticles into microstructures,” Langmuir 20(4), 1021–1025 (2004).
[CrossRef] [PubMed]

Narimanov, E. E.

V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71(20), 201101 (2005).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Ng, J.

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011).
[CrossRef]

Novitsky, A.

A. Novitsky, C. W. Qiu, and H. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107(20), 203601 (2011).
[CrossRef] [PubMed]

Ozbay, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423(6940), 604–605 (2003).
[CrossRef] [PubMed]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002).
[CrossRef]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Podolskiy, V. A.

V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71(20), 201101 (2005).
[CrossRef]

Pritchard, D. E.

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94(17), 170403 (2005).
[CrossRef] [PubMed]

Qiu, C. W.

A. Novitsky, C. W. Qiu, and H. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107(20), 203601 (2011).
[CrossRef] [PubMed]

Salandrino, A.

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Shalaev, V. M.

A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: Recent advances and outlook,” Metamaterials (Amst.) 2(1), 1–17 (2008).
[CrossRef]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[CrossRef]

She, W.

W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” Phys. Rev. Lett. 101(24), 243601 (2008).
[CrossRef] [PubMed]

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Soukoulis, C. M.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423(6940), 604–605 (2003).
[CrossRef] [PubMed]

Stockman, M. I.

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98(17), 177404 (2007).
[CrossRef]

Streed, E. W.

G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Photon recoil momentum in dispersive media,” Phys. Rev. Lett. 94(17), 170403 (2005).
[CrossRef] [PubMed]

Sukhov, S.

S. Sukhov and A. Dogariu, “Negative nonconservative forces: optical “tractor beams” for arbitrary objects,” Phys. Rev. Lett. 107(20), 203602 (2011).
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Wang, H.

A. Novitsky, C. W. Qiu, and H. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107(20), 203601 (2011).
[CrossRef] [PubMed]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

Wu, Q. H.

Xu, J.

J. Xu, J. Drelich, and E. M. Nadgorny, “Laser-based patterning of gold nanoparticles into microstructures,” Langmuir 20(4), 1021–1025 (2004).
[CrossRef] [PubMed]

Yu, J.

W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” Phys. Rev. Lett. 101(24), 243601 (2008).
[CrossRef] [PubMed]

Appl. Opt. (1)

Langmuir (1)

J. Xu, J. Drelich, and E. M. Nadgorny, “Laser-based patterning of gold nanoparticles into microstructures,” Langmuir 20(4), 1021–1025 (2004).
[CrossRef] [PubMed]

Metamaterials (Amst.) (1)

A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: Recent advances and outlook,” Metamaterials (Amst.) 2(1), 1–17 (2008).
[CrossRef]

Nat. Photonics (2)

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011).
[CrossRef]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[CrossRef]

Nature (2)

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423(6940), 604–605 (2003).
[CrossRef] [PubMed]

U. Leonhardt, “Optics: momentum in an uncertain light,” Nature 444(7121), 823–824 (2006).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (1)

Philos. Transact. A Math. Phys. Eng. Sci. (1)

S. M. Barnett and R. Loudon, “The enigma of optical momentum in a medium,” Philos. Transact. A Math. Phys. Eng. Sci. 368(1914), 927–939 (2010).
[CrossRef] [PubMed]

Phys. Rev. B (2)

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002).
[CrossRef]

V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71(20), 201101 (2005).
[CrossRef]

Phys. Rev. Lett. (8)

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98(17), 177404 (2007).
[CrossRef]

S. Sukhov and A. Dogariu, “Negative nonconservative forces: optical “tractor beams” for arbitrary objects,” Phys. Rev. Lett. 107(20), 203602 (2011).
[CrossRef] [PubMed]

A. Novitsky, C. W. Qiu, and H. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107(20), 203601 (2011).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

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Rendiconti del Circolo Matematico di Palermo (1)

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Other (5)

It should be noted that particles can be pulled backward even when radiation pressure is positive e.g., when the momentum of scattered waves is larger than the incoming momentum flux.

H. Minkowski, “Die grundgleichungen für die elektromagnetischen vorgänge in bewegten körpern,” Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111 (1908).

H. Lezec and K. J. Chau, “Negative radiation-pressure response of a left-handed plasmonic metamaterial,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JWE1.

M. Ben-Artzi and J. Nemirovsky, “Resolvent estimates for Schrodinger-type and Maxwell equations with applications,” in Spectral and Scattering Theory, A.G. Ramm ed. (Plenum Press, 1998), pp. 19–31.

See Fig. 6.3–8 in B.E.A. Saleh and M.C. Teich, John Wiley & Sons, Inc. 215 (1991).

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Figures (4)

Fig. 1
Fig. 1

(a) Sketch of the three-dimensional isofrequency surface of a homogeneous biaxial medium, where the axes are the principal axes of the index ellipsoid, and ε i ω/c= ε i 2π/ λ 0 mark the intersections of the ellipsoid with these axes. Here, k = (kx,ky,kz) designates the wavevector of a plane wave propagating in this medium. (b) Rotated version of (a), showing the electric field E and magnetic field H of the plane wave at a particular point on the isofrequency surface; note that the vectors k, E, D (electric displacement), and S (Poynting vector) lie in the same plane. (c) Two-dimensional intersection of the isofrequency surface with the yz-plane. For a given k z there are four possible plane waves in the yz-plane. Two of them (k1 and k4) have group velocities with negative z-components.

Fig. 2
Fig. 2

(a) Cross-section of a block of biaxial medium with a slab-hole within it, for the insertion of small particles. As shown, and discussed in the text, NRP cannot be achieved in this geometry. (b) Cross-section of a waveguide in which negative modes and NRP can be realized. The ray diagram shows that while light flows in the positive z-direction (the direction of the group velocity), radiation pressure in the particle gap is in the negative z-direction (the direction of the phase velocity). (c) Full three-dimensional depiction of the waveguide geometry. The sphere and the corresponding arrow indicate the direction in which a small particle is forced within the gap.

Fig. 3
Fig. 3

Dispersion diagram for the slab-waveguide arrangement depicted in Fig. 2 and described in the text. In the frequency range shown, roughly half of the modes have negative slope, corresponding to negative group velocities: these are the negative modes that act to produce NRP. These dispersion curves are for the configuration with a “particle gap,” as depicted in Fig. 2(b). The presence of the gap does not qualitatively effect the presence of the negative modes. Negative modes are indicated with arrows.

Fig. 4
Fig. 4

(a) Time-averaged Poynting flux (in the z-direction) of a negative mode with an air gap in the center. (b) Same as (a) but for a positive mode (i.e. positive phase and group velocity). In the gap, particles experience a force opposite that of the direction of energy transport of the wave. In both plots the positive direction is defined as the direction of propagation of the group velocity.

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