Abstract

We analytically investigate the forces due to Surface Plasmon Polariton (SPP) modes between finite and infinitely thick metal slabs separated by an air gap. Using the Drude model and experimentally determined values of the dielectric functions of gold and silver, we study how frequency dispersion and loss in the metals affects the behavior of the SPP modes and the forces generated by them. We calculate the force using the Maxwell Stress Tensor for both the attractive and repulsive modes.

© 2009 Optical Society of America

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2009 (1)

M. Mansuripur and A. R. Zakharian, “Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force,” Phys. Rev. E 79, 10 (2009).
[CrossRef]

2008 (6)

Z. P. Li, M. Kall, and H. Xu, “Optical forces on interacting plasmonic nanoparticles in a focused Gaussian beam,” Phys. Rev. B 77, 085412 (2008).
[CrossRef]

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: Tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100, 186804 (2008).
[CrossRef] [PubMed]

V. Yannopapas, “Optical Forces near a plasmonic nanostructure,” Phys. Rev. B 78,045412 (2008)
[CrossRef]

J. Ng, R. Tang, and C.T. Chan, “Electrodynamic study of plasmonic bonding and antibonding forces in a bisphere,” Phys. Rev. B 77,195407 (2008)
[CrossRef]

E. Cubukcu, N. F. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” IEEE J. Sel. Top. Quantum Electron. 14, 1448–1461 (2008).
[CrossRef]

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46, 157–164 (2008).
[CrossRef]

2007 (7)

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photonics 1, 416–422 (2007).
[CrossRef]

M. Hossein-Zadeh and K. J. Vahala, “Observation of optical spring effect in a microtoroidal optomechanical resonator,” Opt. Lett. 32, 1611–1613 (2007).
[CrossRef] [PubMed]

Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: Analysis of optical properties,” Phys. Rev. B 75, 035411 (2007).
[CrossRef]

F. Liu, Y. Rao, Y. D. Huang, W. Zhang, and J. D. Peng, “Coupling between long range surface plasmon polariton mode and dielectric waveguide mode,” Appl. Phys. Lett. 90, 141101 (2007).
[CrossRef]

P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. 7, 1376–1380 (2007).
[CrossRef] [PubMed]

F. J. G. de Abajo, T. Brixner, and W. Pfeiffer, “Nanoscale force manipulation in the vicinity of a metal nanostructure,” J. Phys. B 40, S249–S258 (2007).
[CrossRef]

R. Quidant, S. Zelenina, and M. Nieto-Vesperinas, “Optical manipulation of plasmonic nanoparticles,” Appl. Phys. A-Materials Science & Processing 89, 233–239 (2007).
[CrossRef]

2006 (9)

G. Volpe, R. Quidant, G. Badenes, and D. Petrov, “Surface plasmon radiation forces,” Phys. Rev. Lett. 96, 238101 (2006).
[CrossRef] [PubMed]

I. Pirozhenko, A. Lambrecht, and V. B. Svetovoy, “Sample dependence of the Casimir force,” New J. Phys. 8, 8238 (2006).
[CrossRef]

S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” Journal of Physics B-Atomic Molecular and Optical Physics 39, S671–S684(2006).
[CrossRef]

H. S. Won, K. C. Kim, S. H. Song, C. H. Oh, P. S. Kim, S. Park, and S. I. Kim, “Vertical coupling of long-range surface plasmon polaritons,” Appl. Phys. Lett. 88, 011110 (2006).
[CrossRef]

H. T. Miyazaki and Y. Kurokawa, “Controlled plasmon resonance in closed metal/insulator/metal nanocavities,” Appl. Phys. Lett. 89, 211126 (2006).
[CrossRef]

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96, 097401 (2006).
[CrossRef] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, M. Centini, C. Sibilia, and J. W. Haus, “Radiation pressure of light pulses and conservation of linear momentum in dispersive media,” Phys. Rev. E 73, 056604 (2006).
[CrossRef]

A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006).
[CrossRef]

2005 (5)

2004 (4)

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philosophical Transactions of the Royal Society a-Mathematical Physical and Engineering Sciences 362, 719–737 (2004).
[CrossRef] [PubMed]

P. Nordlander and E. Prodan, “Plasmon hybridization in nanoparticles near metallic surfaces,” Nano Lett. 4, 2209–2213 (2004).
[CrossRef]

E. Prodan and P. Nordlander, “Plasmon hybridization in spherical nanoparticles,” J. Chem. Phys. 120, 5444–5454 (2004).
[CrossRef] [PubMed]

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. 85, 1466–1468 (2004).
[CrossRef]

2003 (3)

T. Nikolajsen, K. Leosson, I. Salakhutdinov, and S. I. Bozhevolnyi, “Polymer-based surface-plasmon-polariton stripe waveguides at telecommunication wavelengths,” Appl. Phys. Lett. 82, 668–670 (2003).
[CrossRef]

J. R. Arias-Gonzalez and M. Nieto-Vesperinas, “Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions,” Journal of the Optical Society of America a-Optics Image Science and Vision 20, 1201–1209 (2003).
[CrossRef]

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

2001 (1)

Y. G. Song, B. M. Han, and S. Chang, “Force of surface plasmon-coupled evanescent fields on Mie particles,” Optics Communications 198, 7–19 (2001).
[CrossRef]

1999 (3)

B. M. Han, S. Chang, and S. S. Lee, “Enhancement of the evanescent field pressure on a dielectric film by coupling with surface plasmons,” J. Korean Physical Society 35, 180–185 (1999).

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54, 3–15 (1999).
[CrossRef]

M. I. Antonoyiannakis and J. B. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60, 2363–2374 (1999).
[CrossRef]

1986 (1)

1981 (1)

D. Sarid, “Long-range surface-plasma waves on very thin metal-films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
[CrossRef]

1968 (1)

V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Soviet Physics Uspekhi-Ussr 10, 509–514 (1968).
[CrossRef]

1961 (1)

L. P. Pitaevskii, “Electric forces in a transparent dispersive medium” Soviet Physics Jetp-Ussr 12, 1008–1013 (1961).

1958 (1)

R. L. Garwin, “Solar Sailing – A practical method of propulsion within the solar system,” Jet Propulsion 28, 188–190 (1958).

1903 (1)

E. F. Nichols and G. F. Hull, “The pressure due to radiation (Second paper),” Phys. Rev. 17, 26–50 (1903).

1901 (1)

P. Lebedew, “Testings on the compressive force of light,” Ann. Phys. 6, 433–458 (1901).
[CrossRef]

Antezza, M.

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46, 157–164 (2008).
[CrossRef]

Antonoyiannakis, M. I.

M. I. Antonoyiannakis and J. B. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60, 2363–2374 (1999).
[CrossRef]

Arias-Gonzalez, J. R.

J. R. Arias-Gonzalez and M. Nieto-Vesperinas, “Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions,” Journal of the Optical Society of America a-Optics Image Science and Vision 20, 1201–1209 (2003).
[CrossRef]

Ashkin, A.

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Badenes, G.

G. Volpe, R. Quidant, G. Badenes, and D. Petrov, “Surface plasmon radiation forces,” Phys. Rev. Lett. 96, 238101 (2006).
[CrossRef] [PubMed]

Barnett, S. M.

S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” Journal of Physics B-Atomic Molecular and Optical Physics 39, S671–S684(2006).
[CrossRef]

Barnett, S.M.

R. Loudon, S.M. Barnett, and C. Baxter, “Radiation Pressure and momentum transfer in dielectrics: the photon drag effect,” Phys. Rev. A 71, 063808 (2005).
[CrossRef]

Baxter, C.

R. Loudon, S.M. Barnett, and C. Baxter, “Radiation Pressure and momentum transfer in dielectrics: the photon drag effect,” Phys. Rev. A 71, 063808 (2005).
[CrossRef]

Berini, P.

P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. 7, 1376–1380 (2007).
[CrossRef] [PubMed]

Bjorkholm, J. E.

Bloemer, M. J.

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, M. Centini, C. Sibilia, and J. W. Haus, “Radiation pressure of light pulses and conservation of linear momentum in dispersive media,” Phys. Rev. E 73, 056604 (2006).
[CrossRef]

Boardman, A. D.

A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006).
[CrossRef]

Bozhevolnyi, S. I.

T. Nikolajsen, K. Leosson, I. Salakhutdinov, and S. I. Bozhevolnyi, “Polymer-based surface-plasmon-polariton stripe waveguides at telecommunication wavelengths,” Appl. Phys. Lett. 82, 668–670 (2003).
[CrossRef]

Brixner, T.

F. J. G. de Abajo, T. Brixner, and W. Pfeiffer, “Nanoscale force manipulation in the vicinity of a metal nanostructure,” J. Phys. B 40, S249–S258 (2007).
[CrossRef]

Capasso, F.

Carusotto, I.

F. Riboli, A. Recati, M. Antezza, and I. Carusotto, “Radiation induced force between two planar waveguides,” Eur. Phys. J. D 46, 157–164 (2008).
[CrossRef]

Centini, M.

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, M. Centini, C. Sibilia, and J. W. Haus, “Radiation pressure of light pulses and conservation of linear momentum in dispersive media,” Phys. Rev. E 73, 056604 (2006).
[CrossRef]

Chan, C.T.

J. Ng, R. Tang, and C.T. Chan, “Electrodynamic study of plasmonic bonding and antibonding forces in a bisphere,” Phys. Rev. B 77,195407 (2008)
[CrossRef]

Chang, S.

Y. G. Song, B. M. Han, and S. Chang, “Force of surface plasmon-coupled evanescent fields on Mie particles,” Optics Communications 198, 7–19 (2001).
[CrossRef]

B. M. Han, S. Chang, and S. S. Lee, “Enhancement of the evanescent field pressure on a dielectric film by coupling with surface plasmons,” J. Korean Physical Society 35, 180–185 (1999).

Charbonneau, R.

P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. 7, 1376–1380 (2007).
[CrossRef] [PubMed]

Chaumet, P. C.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philosophical Transactions of the Royal Society a-Mathematical Physical and Engineering Sciences 362, 719–737 (2004).
[CrossRef] [PubMed]

Chu, S.

Crozier, K. B.

E. Cubukcu, N. F. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” IEEE J. Sel. Top. Quantum Electron. 14, 1448–1461 (2008).
[CrossRef]

Cubukcu, E.

E. Cubukcu, N. F. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” IEEE J. Sel. Top. Quantum Electron. 14, 1448–1461 (2008).
[CrossRef]

D’Aguanno, G.

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, M. Centini, C. Sibilia, and J. W. Haus, “Radiation pressure of light pulses and conservation of linear momentum in dispersive media,” Phys. Rev. E 73, 056604 (2006).
[CrossRef]

de Abajo, F. J. G.

F. J. G. de Abajo, T. Brixner, and W. Pfeiffer, “Nanoscale force manipulation in the vicinity of a metal nanostructure,” J. Phys. B 40, S249–S258 (2007).
[CrossRef]

Diehl, L.

E. Cubukcu, N. F. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” IEEE J. Sel. Top. Quantum Electron. 14, 1448–1461 (2008).
[CrossRef]

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Dziedzic, J. M.

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).

Eichenfield, M.

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photonics 1, 416–422 (2007).
[CrossRef]

Garwin, R. L.

R. L. Garwin, “Solar Sailing – A practical method of propulsion within the solar system,” Jet Propulsion 28, 188–190 (1958).

Gauglitz, G.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54, 3–15 (1999).
[CrossRef]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).

Ginzburg, V. L.

V. L. Ginzburg, Applications of Electrodynamics in Theoretical Physics and Astrophysics (Gordon and Breach Science Publishers, New York, 1989).

Girard, C.

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: Tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100, 186804 (2008).
[CrossRef] [PubMed]

Grady, N. K.

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Figures (8)

Fig. 1.
Fig. 1.

The Metal-Insulator-Metal (MIM, (a)) and Insulator-Metal-Insulator-Metal-Insulator (IMIMI, (b)) geometries. ε 1 is the electrical permittivity of the metal and ε2 is the permittivity of the dielectric. The roman numerals in the IMIMI geometry correspond to the regions defined in Eq. (3). In both geometries, the origin is placed at the center of the dielectric gap of width 2w, and SPP propagation is in the -z-direction in the calculations.

Fig. 2.
Fig. 2.

Ez field shapes and naming conventions for the modes supported by the IMIMI and MIM geometries. (a) shows two isolated IMI stripe waveguides each supporting a Long-Range Surface Plasmon Polariton (LRSPP) mode. When these waveguides are brought in proximity to one another, LRSPP 1 and LRSPP 2 will couple symmetrically (b) and anti-symmetrically (c). The symmetric Short Range Surface Plasmon Polariton (SRSPP) modes supported by the IMI waveguide (d) will also couple symmetrically (e) and antisymmetrically (f). The MIM geometry supports only two modes, known here as S 0 (g) and A 0 (h).

Fig. 3.
Fig. 3.

Drude Plasmon dispersion for the MIM ((a) and (c)) and IMIMI ((b) and (d)) geometries for gap widths, 2w, of 30 nm (a) and (b) and 100 nm (c) and (d), respectively, modeled with the plasma frequency and damping coefficient for gold: ωp =1.37×1016 s-1 (νp =ωp /2π) and γ=3.68×1013 s-1. The values for silver do not differ from these values enough to produce plots that are distinguishable from those shown here. The thicknesses of the metal slabs in the IMIMI geometry are held constant at 20 nm.

Fig. 4.
Fig. 4.

SPP Dispersion for the MIM A 0 (red lines (a), (c)) and IMIMI A s (red lines, (b), (d)), and S s (blue lines, (b), (d)) modes for gap widths of 30 nm (a) and (b) and 100 nm (c) and (d), respectively, modeled with the dielectric data for gold, taken from Ref. [44]. Grey dots represent the modes calculated with the Drude model. The thicknesses of the metal slabs are held constant at 20 nm.

Fig. 5.
Fig. 5.

SPP Dispersion for the MIM A 0 (red lines, (a), (c)) and IMIMI A s (red lines, (b), (d)) and S s (blue lines, (b), (d)) modes for gap widths of 30 nm (a) and (b) and 100 nm (c) and (d), respectively, modeled with the dielectric data for silver, taken from Ref. [44]. Grey dots represents the modes calculated using the Drude model. The thicknesses of the metal slabs in the IMIMI geometry are held constant at 20 nm.

Fig. 6.
Fig. 6.

SPP Wavevectors for the MIM A 0 (a) and IMIMI A s (b) and S s (c) modes for as the gap width is varied, modeled with the dielectric data for gold (green lines) silver (blue lines), taken from Ref. [44], and the Drude model (red lines). The thickness of the metal slabs in the IMIMI geometry is 20 nm.

Fig. 7.
Fig. 7.

(a) and (b): The force from the SPP modes in the IMIMI geometry, calculated using three models for the metal: tabulated data for gold (green lines) and silver (blue lines), and the Drude Model (red lines) at an operating wavelength of λ 0=600nm. Plotted in (a) is the magnitude of the attractive A s mode force, while the repulsive S s mode force is plotted in (b). (c) and (d): The A s and S s mode forces between silver slabs at λ 0=450nm (cyan lines), λ 0=600nm (blue lines), λ 0=1000nm (magenta lines). The MIM A 0 mode behaves like the IMIMI A s mode, and so is not plotted here.

Fig. 8.
Fig. 8.

IMIMI energy density crossections at λ 0=450nm for geometries using Drude metals. The plots show the energy density of the modes for gap widths between 10 and 400 nm. In (a), the crossections for the A s mode. In (b), the crossections for the S s mode. Note that the colormaps in the two panels are not of the same scale.

Equations (25)

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k02 n2 = kz2 +k2 .
k02 n2 = kz2 ky2.
Hx (y,z,t)={𝓐exp(ky2y)i.y>d+w𝓑exp(ky1y)+𝓒exp(ky1y)ii.w<y<d+w𝓓exp(ky2y)+𝓕exp(ky2y)iii.w<y<w𝓖exp(ky1y)+𝓗exp(ky1y)iv.(d+w)<y<w𝓙exp(ky2y)v.y<(d+w)
Ey (y,z,t)=k2ωεHx(y,z,t)
Ez (y,z,t)=1iωεyHx(y,z,t)
ky2ε1ky1ε2tanh(ky2w)=[ky1ε1sinh(ky1d)+ky2ε2cosh(ky1d)ky1ε1cosh(ky1d)+ky2ε2sinh(ky1d)].
ε1(ω)ε0 =1 ωp2ω2+γ2+iωp2γω(γ2+ω2).
𝓐=2 𝒟 ky1ε1cosh(ky2w)ky1ε1cosh(ky1d)+ky2ε2sinh(ky2d) exp (ky2[w+d])
𝓑=𝒟 cosh(ky2w)(ky1ε1+ky2ε2)ky1ε1cosh(ky1d)+ky2ε2sinh(ky1d) exp (ky1[w+d])
𝓒=𝓓 cosh(ky2w)(ky1ε1ky2ε2)ky1ε1cosh(ky1d)+ky2ε2sinh(ky1d) exp (ky1[w+d]).
pz=Re {S·ẑdxdy}
𝒫=pzw=Re {kzωε}0Hx2dy,
𝓓2 =ω 𝒫 ×
{βε2[𝓐¯2exp(2ky2[w+d])2ky2+sinh(2ky2t)ky2+sin(2ky2t)ky2]
+βε1+αε1ε12×
[(𝓑¯2exp(ky1[2w+d])ky1+𝓒¯2exp(ky1[2w+d])ky1)sinh(ky1d)
+2Re{𝓑¯𝓒¯*exp[iky1(2w+d)]}ky1sin(ky1d)]}1
AT(r,t)·n(r)da=ddtV(E×H)c2d3r+V[(ρP·)E+(J+Pt)×B]d3r,
T=[ε0EE+μ0HH12(ε0E·E+μ0H·H)I]
ddtV1c2(E×H)d3r=dGfielddt,
F=dGmechdt=V(P·)E+(Pt)×Bd3r,
Fy=μ02(1neff2)𝓓2,
Fy=μ02(cky2ω2)𝓓2,
F=dUdwkz,
u(r)=14ε(1+ωεdεdω)[E(r,t)·E*(r,t)]+14μ0[H(r,t)·H*(r,t).]

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