Abstract

Resonators fold the path of light by reflections leading to a phase balance and thus constructive addition of propagating waves. However, amplitude decrease of these waves due to incomplete reflection or material absorption leads to a finite quality factor of all resonances. Here we report on our discovery that evanescent waves can lead to a perfect phase and amplitude balance causing an ideal Fabry-Perot resonance condition in spite of material absorption and non-ideal reflectivities. This counterintuitive resonance occurs if and only if the metallic Fabry-Perot plates are in relative motion to each other separated by a critical distance. We show that the energy needed to approach the resonance arises from the conversion of the mechanical energy of motion to electromagnetic energy. The phenomenon is similar to lasing where the losses in the cavity resonance are exactly compensated by optical gain media instead of mechanical motion. Nonlinearities and non-localities in material response will inevitably curtail any singularities however we show the giant enhancement in non-equilibrium phenomena due to such resonances in moving media.

© 2014 Optical Society of America

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References

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  1. M. Wolf and E. Born, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1980).
  2. J. A. Kong, Electromagnetic Wave Theory (Wiley, 1990), Vol. 2.
  3. L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).
  4. H. Raether, Surface Plasmons on Smooth Surfaces (Springer, 1988).
  5. F. Intravaia and A. Lambrecht, “Surface plasmon modes and the casimir energy,” Phys. Rev. Lett. 94(11), 110404 (2005).
    [Crossref] [PubMed]
  6. J. B. Pendry, “Shearing the vacuum - quantum friction,” J. Phys. Condens. Matter 9(47), 10301–10320 (1997).
    [Crossref]
  7. A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
    [Crossref] [PubMed]
  8. R. Zhao, A. Manjavacas, F. J. García de Abajo, and J. B. Pendry, “Rotational quantum friction,” Phys. Rev. Lett. 109(12), 123604 (2012).
    [Crossref] [PubMed]
  9. A. I. Volokitin and B. N. J. Persson, “Quantum friction,” Phys. Rev. Lett. 106(9), 094502 (2011).
    [Crossref] [PubMed]
  10. C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337(6098), 1072–1074 (2012).
    [Crossref] [PubMed]
  11. A. Moreau, C. Ciracì, and D. R. Smith, “Impact of nonlocal response on metallodielectric multilayers and optical patch antennas,” Phys. Rev. B 87(4), 045401 (2013).
    [Crossref]
  12. M. F. Maghrebi, R. Golestanian, and M. Kardar, “Scattering approach to the dynamical Casimir effect,” Phys. Rev. D Part. Fields Gravit. Cosmol. 87(2), 025016 (2013).
    [Crossref]
  13. S.-A. Biehs, E. Rousseau, and J.-J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105(23), 234301 (2010).
    [Crossref] [PubMed]
  14. A. I. Volokitin and B. N. J. Persson, “Theory of the interaction forces and the radiative heat transfer between moving bodies,” Phys. Rev. B 78(15), 155437 (2008).
    [Crossref]
  15. J. B. Pendry, “Radiative exchange of heat between nanostructures,” J. Phys. Condens. Matter 11(35), 6621–6633 (1999).
    [Crossref]
  16. S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9(8), 2909–2913 (2009).
    [Crossref] [PubMed]
  17. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
  18. M. Kardar and R. Golestanian, “The “friction” of vacuum, and other fluctuation-induced forces,” Rev. Mod. Phys. 71(4), 1233–1245 (1999).
    [Crossref]

2013 (2)

A. Moreau, C. Ciracì, and D. R. Smith, “Impact of nonlocal response on metallodielectric multilayers and optical patch antennas,” Phys. Rev. B 87(4), 045401 (2013).
[Crossref]

M. F. Maghrebi, R. Golestanian, and M. Kardar, “Scattering approach to the dynamical Casimir effect,” Phys. Rev. D Part. Fields Gravit. Cosmol. 87(2), 025016 (2013).
[Crossref]

2012 (2)

R. Zhao, A. Manjavacas, F. J. García de Abajo, and J. B. Pendry, “Rotational quantum friction,” Phys. Rev. Lett. 109(12), 123604 (2012).
[Crossref] [PubMed]

C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337(6098), 1072–1074 (2012).
[Crossref] [PubMed]

2011 (1)

A. I. Volokitin and B. N. J. Persson, “Quantum friction,” Phys. Rev. Lett. 106(9), 094502 (2011).
[Crossref] [PubMed]

2010 (1)

S.-A. Biehs, E. Rousseau, and J.-J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105(23), 234301 (2010).
[Crossref] [PubMed]

2009 (1)

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9(8), 2909–2913 (2009).
[Crossref] [PubMed]

2008 (1)

A. I. Volokitin and B. N. J. Persson, “Theory of the interaction forces and the radiative heat transfer between moving bodies,” Phys. Rev. B 78(15), 155437 (2008).
[Crossref]

2007 (1)

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

2005 (1)

F. Intravaia and A. Lambrecht, “Surface plasmon modes and the casimir energy,” Phys. Rev. Lett. 94(11), 110404 (2005).
[Crossref] [PubMed]

1999 (2)

J. B. Pendry, “Radiative exchange of heat between nanostructures,” J. Phys. Condens. Matter 11(35), 6621–6633 (1999).
[Crossref]

M. Kardar and R. Golestanian, “The “friction” of vacuum, and other fluctuation-induced forces,” Rev. Mod. Phys. 71(4), 1233–1245 (1999).
[Crossref]

1997 (1)

J. B. Pendry, “Shearing the vacuum - quantum friction,” J. Phys. Condens. Matter 9(47), 10301–10320 (1997).
[Crossref]

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Biehs, S.-A.

S.-A. Biehs, E. Rousseau, and J.-J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105(23), 234301 (2010).
[Crossref] [PubMed]

Chen, G.

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9(8), 2909–2913 (2009).
[Crossref] [PubMed]

Chilkoti, A.

C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337(6098), 1072–1074 (2012).
[Crossref] [PubMed]

Ciracì, C.

A. Moreau, C. Ciracì, and D. R. Smith, “Impact of nonlocal response on metallodielectric multilayers and optical patch antennas,” Phys. Rev. B 87(4), 045401 (2013).
[Crossref]

C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337(6098), 1072–1074 (2012).
[Crossref] [PubMed]

Fernández-Domínguez, A. I.

C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337(6098), 1072–1074 (2012).
[Crossref] [PubMed]

Franz, K. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

García de Abajo, F. J.

R. Zhao, A. Manjavacas, F. J. García de Abajo, and J. B. Pendry, “Rotational quantum friction,” Phys. Rev. Lett. 109(12), 123604 (2012).
[Crossref] [PubMed]

Gmachl, C.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Golestanian, R.

M. F. Maghrebi, R. Golestanian, and M. Kardar, “Scattering approach to the dynamical Casimir effect,” Phys. Rev. D Part. Fields Gravit. Cosmol. 87(2), 025016 (2013).
[Crossref]

M. Kardar and R. Golestanian, “The “friction” of vacuum, and other fluctuation-induced forces,” Rev. Mod. Phys. 71(4), 1233–1245 (1999).
[Crossref]

Greffet, J.-J.

S.-A. Biehs, E. Rousseau, and J.-J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105(23), 234301 (2010).
[Crossref] [PubMed]

Hill, R. T.

C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337(6098), 1072–1074 (2012).
[Crossref] [PubMed]

Hoffman, A. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Howard, S. S.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Intravaia, F.

F. Intravaia and A. Lambrecht, “Surface plasmon modes and the casimir energy,” Phys. Rev. Lett. 94(11), 110404 (2005).
[Crossref] [PubMed]

Kardar, M.

M. F. Maghrebi, R. Golestanian, and M. Kardar, “Scattering approach to the dynamical Casimir effect,” Phys. Rev. D Part. Fields Gravit. Cosmol. 87(2), 025016 (2013).
[Crossref]

M. Kardar and R. Golestanian, “The “friction” of vacuum, and other fluctuation-induced forces,” Rev. Mod. Phys. 71(4), 1233–1245 (1999).
[Crossref]

Lambrecht, A.

F. Intravaia and A. Lambrecht, “Surface plasmon modes and the casimir energy,” Phys. Rev. Lett. 94(11), 110404 (2005).
[Crossref] [PubMed]

Maghrebi, M. F.

M. F. Maghrebi, R. Golestanian, and M. Kardar, “Scattering approach to the dynamical Casimir effect,” Phys. Rev. D Part. Fields Gravit. Cosmol. 87(2), 025016 (2013).
[Crossref]

Maier, S. A.

C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337(6098), 1072–1074 (2012).
[Crossref] [PubMed]

Manjavacas, A.

R. Zhao, A. Manjavacas, F. J. García de Abajo, and J. B. Pendry, “Rotational quantum friction,” Phys. Rev. Lett. 109(12), 123604 (2012).
[Crossref] [PubMed]

Mock, J. J.

C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337(6098), 1072–1074 (2012).
[Crossref] [PubMed]

Moreau, A.

A. Moreau, C. Ciracì, and D. R. Smith, “Impact of nonlocal response on metallodielectric multilayers and optical patch antennas,” Phys. Rev. B 87(4), 045401 (2013).
[Crossref]

Narayanaswamy, A.

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9(8), 2909–2913 (2009).
[Crossref] [PubMed]

Narimanov, E. E.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Pendry, J. B.

R. Zhao, A. Manjavacas, F. J. García de Abajo, and J. B. Pendry, “Rotational quantum friction,” Phys. Rev. Lett. 109(12), 123604 (2012).
[Crossref] [PubMed]

C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337(6098), 1072–1074 (2012).
[Crossref] [PubMed]

J. B. Pendry, “Radiative exchange of heat between nanostructures,” J. Phys. Condens. Matter 11(35), 6621–6633 (1999).
[Crossref]

J. B. Pendry, “Shearing the vacuum - quantum friction,” J. Phys. Condens. Matter 9(47), 10301–10320 (1997).
[Crossref]

Persson, B. N. J.

A. I. Volokitin and B. N. J. Persson, “Quantum friction,” Phys. Rev. Lett. 106(9), 094502 (2011).
[Crossref] [PubMed]

A. I. Volokitin and B. N. J. Persson, “Theory of the interaction forces and the radiative heat transfer between moving bodies,” Phys. Rev. B 78(15), 155437 (2008).
[Crossref]

Podolskiy, V. A.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Rousseau, E.

S.-A. Biehs, E. Rousseau, and J.-J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105(23), 234301 (2010).
[Crossref] [PubMed]

Shen, S.

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9(8), 2909–2913 (2009).
[Crossref] [PubMed]

Sivco, D. L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Smith, D. R.

A. Moreau, C. Ciracì, and D. R. Smith, “Impact of nonlocal response on metallodielectric multilayers and optical patch antennas,” Phys. Rev. B 87(4), 045401 (2013).
[Crossref]

C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337(6098), 1072–1074 (2012).
[Crossref] [PubMed]

Urzhumov, Y.

C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337(6098), 1072–1074 (2012).
[Crossref] [PubMed]

Volokitin, A. I.

A. I. Volokitin and B. N. J. Persson, “Quantum friction,” Phys. Rev. Lett. 106(9), 094502 (2011).
[Crossref] [PubMed]

A. I. Volokitin and B. N. J. Persson, “Theory of the interaction forces and the radiative heat transfer between moving bodies,” Phys. Rev. B 78(15), 155437 (2008).
[Crossref]

Wasserman, D.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Zhao, R.

R. Zhao, A. Manjavacas, F. J. García de Abajo, and J. B. Pendry, “Rotational quantum friction,” Phys. Rev. Lett. 109(12), 123604 (2012).
[Crossref] [PubMed]

J. Phys. Condens. Matter (2)

J. B. Pendry, “Shearing the vacuum - quantum friction,” J. Phys. Condens. Matter 9(47), 10301–10320 (1997).
[Crossref]

J. B. Pendry, “Radiative exchange of heat between nanostructures,” J. Phys. Condens. Matter 11(35), 6621–6633 (1999).
[Crossref]

Nano Lett. (1)

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9(8), 2909–2913 (2009).
[Crossref] [PubMed]

Nat. Mater. (1)

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Phys. Rev. B (2)

A. Moreau, C. Ciracì, and D. R. Smith, “Impact of nonlocal response on metallodielectric multilayers and optical patch antennas,” Phys. Rev. B 87(4), 045401 (2013).
[Crossref]

A. I. Volokitin and B. N. J. Persson, “Theory of the interaction forces and the radiative heat transfer between moving bodies,” Phys. Rev. B 78(15), 155437 (2008).
[Crossref]

Phys. Rev. D Part. Fields Gravit. Cosmol. (1)

M. F. Maghrebi, R. Golestanian, and M. Kardar, “Scattering approach to the dynamical Casimir effect,” Phys. Rev. D Part. Fields Gravit. Cosmol. 87(2), 025016 (2013).
[Crossref]

Phys. Rev. Lett. (4)

S.-A. Biehs, E. Rousseau, and J.-J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105(23), 234301 (2010).
[Crossref] [PubMed]

F. Intravaia and A. Lambrecht, “Surface plasmon modes and the casimir energy,” Phys. Rev. Lett. 94(11), 110404 (2005).
[Crossref] [PubMed]

R. Zhao, A. Manjavacas, F. J. García de Abajo, and J. B. Pendry, “Rotational quantum friction,” Phys. Rev. Lett. 109(12), 123604 (2012).
[Crossref] [PubMed]

A. I. Volokitin and B. N. J. Persson, “Quantum friction,” Phys. Rev. Lett. 106(9), 094502 (2011).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

M. Kardar and R. Golestanian, “The “friction” of vacuum, and other fluctuation-induced forces,” Rev. Mod. Phys. 71(4), 1233–1245 (1999).
[Crossref]

Science (1)

C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337(6098), 1072–1074 (2012).
[Crossref] [PubMed]

Other (5)

M. Wolf and E. Born, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1980).

J. A. Kong, Electromagnetic Wave Theory (Wiley, 1990), Vol. 2.

L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

H. Raether, Surface Plasmons on Smooth Surfaces (Springer, 1988).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

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Figures (5)

Fig. 1
Fig. 1 Singular Fabry-Perot (FP) resonance of evanescent waves can be achieved by setting the FP plates in relative motion. Plate 1 is stationary while plate 2 is moving at a constant velocity V along the x direction. The reflection coefficients and the distance for the moving case can lead to a perfect balance of both phase and amplitude which cannot occur for stationary plates.
Fig. 2
Fig. 2 Phase (a) and amplitude (b) of reflection coefficients at the SPR frequency from stationary and moving plates. The velocity of the moving plate is 104m/s. The x-axis is the lateral wavevector (kx/k0) normalized by 1/β. For a stationary plate (green curves), the reflection coefficient is almost a constant at these huge wavevectors. However, the frequencies perceived by the moving plates are negative, ranging from ω´ = −0.5ωSPR to ω´ = −1.5ωSPR, so the phases of the reflection coefficients are negative. Furthermore, the phase at the phase balance wavevector is exactly opposite to that of the stationary plate (see blue arrow). In (b), we can clearly see that the amplitude of r can be larger than unity which is essential for amplitude balance. The amplitude for the moving plate has a unique dispersion which peaks at the phase balance wavevector where ω´ = −ωSPR.
Fig. 3
Fig. 3 Contribution to exchanged photon number resolved by frequency and lateral wavevector kx (normalized to free space wavevector) at (a) V = 0.5V0 and (b) V→V0. The right panels are 3-dimensional plots. In both (a) and (b), we see two bright curves, both of which are due to surface plasmon resonances. In (a), at a velocity away from V0, the singular resonance condition is far from being satisfied. However, the bright curves remain due to the SPR at the two interfaces. In (b), the large peak in the middle is due to the singular resonance that arises since the amplitude balance condition is satisfied when V→V0 and phase balance condition is satisfied at kx = 2/β. This leads to giant photon exchange between moving plates at phase and amplitude balance (PAB) condition.
Fig. 4
Fig. 4 (a) Heat transfer on the moving plates (a) resolved by the wavevector kx at V→V0 and V = 0.5V0. A major contribution to the heat arises from modes at the perfect phase balance wavevector. Note, at 0.5V0, the amplitude of friction is significantly smaller. (b) The distance velocity of heat transfer at velocities near V0. It is very clear that heat transfer goes up rapidly near the critical velocity V0. In the inset, the x axis is (V0−V)/V0 and in log scale. We clearly see a linear increasing behavior as V approaches V0. This is consistent with the theoretical scaling law which predicts a giant heat transfer.
Fig. 5
Fig. 5 (a) Amplitude of reflection coefficients at various velocities and non-locality parameters. The red and blue regions denote that amplitudes are larger than unity, where amplitude balance is possible, while gray region denotes amplitudes less than unity which cannot satisfy amplitude balance. We emphasize that at a higher velocity comparable to the Fermi velocity of the electrons, the amplitude can be larger than one with non-locality taken into account.

Equations (15)

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

r 1 (ω) r 2 (ω) e 2i k z d =1,
r 2 (ω)= r 1 (ω).
r 2 (ω)= r 1 (ω).
k x PB =2 ω V = 2 k 0 β ( ω =ω),
r 1 (ω) r 2 mov (ω) e 2i k z d = r 1 (ω) r 1 * (ω) e 2| k z |d = | r 1 (ω) | 2 e 2| k z |d =1.
arg( r 1 (ω))+arg( r 1 (ω))+arg( e 2| k z |d )=0,
| r p ( ω SPR )|>1.
k x PB =2 ω SPR V .
| r p ( ω SPR ) | 2 e 4 ω SPR d/V =1,
V 0 = 2 ω SPR d ln| r p ( ω SPR ) |
r 2 e 2| k z |d =1.  ( impossible in stationary plates )
| r | 2 e 2| k z |d =1.  ( possible in moving plates )
S z = 1 2 Re (E× H ) z | k z | ω 2Im(r).
N(ω, k x , k y )= 2Im[ r 1 (ω) ] | e i k z d | 2 2Im[ r 2 mov (ω) ](n(ω', T 2 )n(ω, T 1 )) | 1 r 1 (ω) r 2 mov (ω) e 2i k z d | 2 .
H= dω 2π d k x 2π d k y 2π h(ω, k x , k y ) .

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