Abstract

We experimentally map the wavefunction in the vicinity of a radial potential well. We photograph light intensity near the tunneling region as well as measure the spiraling phase structure via interference with a reference wave. This spiraling phase structure is required for conservation of angular momentum. The experimental image reveals the non-intuitive emission of light from a region in space that is empty of material and relatively far from the device.

© 2009 OSA

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2009

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin Lasing with a CaF2 Whispering Gallery Mode Resonator,” Phys. Rev. Lett. 102, (2009).
[CrossRef] [PubMed]

M. Tomes and T. Carmon, “Photonic Micro-Electromechanical Systems Vibrating at X-band (11-GHz) Rates,” Phys. Rev. Lett. 102(11), 4 (2009).
[CrossRef]

2008

T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett. 100, (2008).
[CrossRef] [PubMed]

2007

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Strekalov, and L. Maleki, “Direct observation of stopped light in a whispering-gallery-mode microresonator,” Phys. Rev. A 76, (2007).
[CrossRef]

T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third harmonic generation,” Nat. Phys. 3(6), 470 (2007).
[CrossRef]

C. Sias, A. Zenesini, H. Lignier, S. Wimberger, D. Ciampini, O. Morsch, and E. Arimondo, “Resonantly enhanced tunneling of Bose-Einstein condensates in periodic potentials,” Phys. Rev. Lett. 98(12), 120403 (2007).
[CrossRef] [PubMed]

A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007).
[CrossRef] [PubMed]

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microw. Theory Tech. 55(6), 1209–1218 (2007).
[CrossRef]

2006

H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox,” Phys. Rep.-Rev. Sec. Phys. Lett. 436, 1–69 (2006).

2005

L. Yang, T. Carmon, B. Min, S. M. Spillane, and K. J. Vahala, “Erbium-doped and Raman microlasers on a silicon chip fabricated by the sol-gel process,” Appl.Phys. Lett . 86, (2005).

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[CrossRef] [PubMed]

2004

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett . 93, (2004).
[CrossRef] [PubMed]

2003

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

2002

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415(6872), 621–623 (2002).
[CrossRef] [PubMed]

W. M. Robertson, J. Ash, and J. M. McGaugh, “Breaking the sound barrier: Tunneling of acoustic waves through the forbidden transmission region of a one-dimensional acoustic band gap array,” Am. J. Phys. 70(7), 689–693 (2002).
[CrossRef]

2001

1998

1993

1988

D. W. Pohl, U. C. Fischer, and U. T. Durig, “Scanning near-Field Optical Microscopy (Snom),” J. Microscopy-Oxford 152, 853–861 (1988).
[CrossRef]

1984

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500-a Spatial-Resolution Light-Microscope. 1. Light Is Efficiently Transmitted through Gamma-16 Diameter Apertures,” Ultramicroscopy 13(3), 227–231 (1984).
[CrossRef]

1982

G. Binning, H. Rohrer, C. Gerber, and E. Weibel, “Surface Studies by Scanning Tunneling Microscopy,” Phys. Rev. Lett. 49(1), 57–61 (1982).
[CrossRef]

1972

E. A. Ash and G. Nicholls, “Super-Resolution Aperture Scanning Microscope,” Nature 237, 510 (1972).
[CrossRef] [PubMed]

1958

L. Esaki, “New Phenomenon in Narrow Germanium Para-Normal-Junctions,” Phys. Rev. 109(2), 603–604 (1958).
[CrossRef]

1928

G. Gamow, “Zur Quantentheorie de Atomkernes,” Z. Phys. 51(3-4), 204–212 (1928).
[CrossRef]

Arimondo, E.

C. Sias, A. Zenesini, H. Lignier, S. Wimberger, D. Ciampini, O. Morsch, and E. Arimondo, “Resonantly enhanced tunneling of Bose-Einstein condensates in periodic potentials,” Phys. Rev. Lett. 98(12), 120403 (2007).
[CrossRef] [PubMed]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Ash, E. A.

E. A. Ash and G. Nicholls, “Super-Resolution Aperture Scanning Microscope,” Nature 237, 510 (1972).
[CrossRef] [PubMed]

Ash, J.

W. M. Robertson, J. Ash, and J. M. McGaugh, “Breaking the sound barrier: Tunneling of acoustic waves through the forbidden transmission region of a one-dimensional acoustic band gap array,” Am. J. Phys. 70(7), 689–693 (2002).
[CrossRef]

Binning, G.

G. Binning, H. Rohrer, C. Gerber, and E. Weibel, “Surface Studies by Scanning Tunneling Microscopy,” Phys. Rev. Lett. 49(1), 57–61 (1982).
[CrossRef]

Cai, M.

Carmon, T.

M. Tomes and T. Carmon, “Photonic Micro-Electromechanical Systems Vibrating at X-band (11-GHz) Rates,” Phys. Rev. Lett. 102(11), 4 (2009).
[CrossRef]

T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett. 100, (2008).
[CrossRef] [PubMed]

T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third harmonic generation,” Nat. Phys. 3(6), 470 (2007).
[CrossRef]

L. Yang, T. Carmon, B. Min, S. M. Spillane, and K. J. Vahala, “Erbium-doped and Raman microlasers on a silicon chip fabricated by the sol-gel process,” Appl.Phys. Lett . 86, (2005).

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[CrossRef] [PubMed]

Ciampini, D.

C. Sias, A. Zenesini, H. Lignier, S. Wimberger, D. Ciampini, O. Morsch, and E. Arimondo, “Resonantly enhanced tunneling of Bose-Einstein condensates in periodic potentials,” Phys. Rev. Lett. 98(12), 120403 (2007).
[CrossRef] [PubMed]

Durig, U. T.

D. W. Pohl, U. C. Fischer, and U. T. Durig, “Scanning near-Field Optical Microscopy (Snom),” J. Microscopy-Oxford 152, 853–861 (1988).
[CrossRef]

Esaki, L.

L. Esaki, “New Phenomenon in Narrow Germanium Para-Normal-Junctions,” Phys. Rev. 109(2), 603–604 (1958).
[CrossRef]

Fischer, U. C.

D. W. Pohl, U. C. Fischer, and U. T. Durig, “Scanning near-Field Optical Microscopy (Snom),” J. Microscopy-Oxford 152, 853–861 (1988).
[CrossRef]

Fisher, P.

A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007).
[CrossRef] [PubMed]

Gamow, G.

G. Gamow, “Zur Quantentheorie de Atomkernes,” Z. Phys. 51(3-4), 204–212 (1928).
[CrossRef]

Gerber, C.

G. Binning, H. Rohrer, C. Gerber, and E. Weibel, “Surface Studies by Scanning Tunneling Microscopy,” Phys. Rev. Lett. 49(1), 57–61 (1982).
[CrossRef]

Grudinin, I. S.

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin Lasing with a CaF2 Whispering Gallery Mode Resonator,” Phys. Rev. Lett. 102, (2009).
[CrossRef] [PubMed]

Harootunian, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500-a Spatial-Resolution Light-Microscope. 1. Light Is Efficiently Transmitted through Gamma-16 Diameter Apertures,” Ultramicroscopy 13(3), 227–231 (1984).
[CrossRef]

Ilchenko, V. S.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Strekalov, and L. Maleki, “Direct observation of stopped light in a whispering-gallery-mode microresonator,” Phys. Rev. A 76, (2007).
[CrossRef]

D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23(4), 247–249 (1998).
[CrossRef]

Isaacson, M.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500-a Spatial-Resolution Light-Microscope. 1. Light Is Efficiently Transmitted through Gamma-16 Diameter Apertures,” Ultramicroscopy 13(3), 227–231 (1984).
[CrossRef]

Joannopoulos, J. D.

A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007).
[CrossRef] [PubMed]

Johnson, B. R.

Kaiser, T.

Karalis, A.

A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007).
[CrossRef] [PubMed]

Kimble, H. J.

Kippenberg, T. J.

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[CrossRef] [PubMed]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett . 93, (2004).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415(6872), 621–623 (2002).
[CrossRef] [PubMed]

Kurs, A.

A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007).
[CrossRef] [PubMed]

Lange, S.

Lewis, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500-a Spatial-Resolution Light-Microscope. 1. Light Is Efficiently Transmitted through Gamma-16 Diameter Apertures,” Ultramicroscopy 13(3), 227–231 (1984).
[CrossRef]

Lignier, H.

C. Sias, A. Zenesini, H. Lignier, S. Wimberger, D. Ciampini, O. Morsch, and E. Arimondo, “Resonantly enhanced tunneling of Bose-Einstein condensates in periodic potentials,” Phys. Rev. Lett. 98(12), 120403 (2007).
[CrossRef] [PubMed]

Mabuchi, H.

Maleki, L.

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin Lasing with a CaF2 Whispering Gallery Mode Resonator,” Phys. Rev. Lett. 102, (2009).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Strekalov, and L. Maleki, “Direct observation of stopped light in a whispering-gallery-mode microresonator,” Phys. Rev. A 76, (2007).
[CrossRef]

Matsko, A. B.

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin Lasing with a CaF2 Whispering Gallery Mode Resonator,” Phys. Rev. Lett. 102, (2009).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Strekalov, and L. Maleki, “Direct observation of stopped light in a whispering-gallery-mode microresonator,” Phys. Rev. A 76, (2007).
[CrossRef]

McGaugh, J. M.

W. M. Robertson, J. Ash, and J. M. McGaugh, “Breaking the sound barrier: Tunneling of acoustic waves through the forbidden transmission region of a one-dimensional acoustic band gap array,” Am. J. Phys. 70(7), 689–693 (2002).
[CrossRef]

Min, B.

L. Yang, T. Carmon, B. Min, S. M. Spillane, and K. J. Vahala, “Erbium-doped and Raman microlasers on a silicon chip fabricated by the sol-gel process,” Appl.Phys. Lett . 86, (2005).

Moffatt, R.

A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007).
[CrossRef] [PubMed]

Morsch, O.

C. Sias, A. Zenesini, H. Lignier, S. Wimberger, D. Ciampini, O. Morsch, and E. Arimondo, “Resonantly enhanced tunneling of Bose-Einstein condensates in periodic potentials,” Phys. Rev. Lett. 98(12), 120403 (2007).
[CrossRef] [PubMed]

Muray, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500-a Spatial-Resolution Light-Microscope. 1. Light Is Efficiently Transmitted through Gamma-16 Diameter Apertures,” Ultramicroscopy 13(3), 227–231 (1984).
[CrossRef]

Nicholls, G.

E. A. Ash and G. Nicholls, “Super-Resolution Aperture Scanning Microscope,” Nature 237, 510 (1972).
[CrossRef] [PubMed]

Oxborrow, M.

T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett. 100, (2008).
[CrossRef] [PubMed]

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microw. Theory Tech. 55(6), 1209–1218 (2007).
[CrossRef]

Pohl, D. W.

D. W. Pohl, U. C. Fischer, and U. T. Durig, “Scanning near-Field Optical Microscopy (Snom),” J. Microscopy-Oxford 152, 853–861 (1988).
[CrossRef]

Robertson, W. M.

W. M. Robertson, J. Ash, and J. M. McGaugh, “Breaking the sound barrier: Tunneling of acoustic waves through the forbidden transmission region of a one-dimensional acoustic band gap array,” Am. J. Phys. 70(7), 689–693 (2002).
[CrossRef]

Rohrer, H.

G. Binning, H. Rohrer, C. Gerber, and E. Weibel, “Surface Studies by Scanning Tunneling Microscopy,” Phys. Rev. Lett. 49(1), 57–61 (1982).
[CrossRef]

Rokhsari, H.

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[CrossRef] [PubMed]

Roll, G.

Savchenkov, A. A.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Strekalov, and L. Maleki, “Direct observation of stopped light in a whispering-gallery-mode microresonator,” Phys. Rev. A 76, (2007).
[CrossRef]

Schwefel, H. G. L.

T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett. 100, (2008).
[CrossRef] [PubMed]

Schweiger, G.

Sias, C.

C. Sias, A. Zenesini, H. Lignier, S. Wimberger, D. Ciampini, O. Morsch, and E. Arimondo, “Resonantly enhanced tunneling of Bose-Einstein condensates in periodic potentials,” Phys. Rev. Lett. 98(12), 120403 (2007).
[CrossRef] [PubMed]

Soljacic, M.

A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007).
[CrossRef] [PubMed]

Spillane, S. M.

L. Yang, T. Carmon, B. Min, S. M. Spillane, and K. J. Vahala, “Erbium-doped and Raman microlasers on a silicon chip fabricated by the sol-gel process,” Appl.Phys. Lett . 86, (2005).

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett . 93, (2004).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415(6872), 621–623 (2002).
[CrossRef] [PubMed]

Stone, A. D.

T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett. 100, (2008).
[CrossRef] [PubMed]

Streed, E. W.

Strekalov, D.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Strekalov, and L. Maleki, “Direct observation of stopped light in a whispering-gallery-mode microresonator,” Phys. Rev. A 76, (2007).
[CrossRef]

Tomes, M.

M. Tomes and T. Carmon, “Photonic Micro-Electromechanical Systems Vibrating at X-band (11-GHz) Rates,” Phys. Rev. Lett. 102(11), 4 (2009).
[CrossRef]

Vahala, K.

Vahala, K. J.

T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett. 100, (2008).
[CrossRef] [PubMed]

T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third harmonic generation,” Nat. Phys. 3(6), 470 (2007).
[CrossRef]

L. Yang, T. Carmon, B. Min, S. M. Spillane, and K. J. Vahala, “Erbium-doped and Raman microlasers on a silicon chip fabricated by the sol-gel process,” Appl.Phys. Lett . 86, (2005).

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“A movie is presented in the file “tomes dynamics of the electric field in the tunneling region near a dielectric sphere.avi.”

Supplementary Material (1)

» Media 1: AVI (6436 KB)     

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Figures (4)

Fig. 1
Fig. 1

(a) Picture of light escaping the cavity. Light on the left marks the bound circulating mode via forward scattering. Light on the right is the unbound radiation after passing through the evanescent region without being seen. There are no objects or surfaces at the place where light emerges on the right. (b) Graph of spherical-resonator potential vs. radius in the equatorial plane. Colors indicate whether the region is bound, unbound, or evanescent. The radius of the sphere is a. (c) Calculation of waveform outside of the cavity alongside experimental data. The dotted lines are a guide for the eye.

Fig. 2
Fig. 2

Top: Analytical [3], and numerical solution to the field outside of the cavity in a transverse plane. Numerical solution calculated via finite element method as described in [18]. Bottom: The same field along the dashed line in the upper panels. A movie provided online shows that the field starts propagating radially only when crossing the outer turning point to the unbound region. (Media 1)

Fig. 3
Fig. 3

(a) Side-view image of cavity showing the internal mode scattering at the dielectric boundary (left spot) and also emitting at the edge of the tunnel barrier (right spot). Light is coupled to the cavity via a tapered fiber at the far edge of the cavity, hidden by the resonator and out of focus due to distance. We verified that no light is scattered from the taper alone by careful examination without a nearby resonator. (b) Calculation of the expected intensity distribution as would be imaged by the microscope (no surface scattering is assumed) (c) Intensity along the line shown in (a). (d) Intensity along the line shown in (b).

Fig. 4
Fig. 4

Plot of tunneling distance vs. cavity radius with experimental results as points and calculation as a line.

Equations (2)

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Ε[ψ(kr)/kr][ei(mϕωt)]
d2ψ(r)dr2+V(r)ψ(r)=Eψ(r),

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