Abstract

We show that an attractive potential created by an evanescent wave quantum mechanically reflects an atom with nonzero probability if the atom’s incident de Broglie wavelength is larger than the decay length of the evanescent wave. The amplitude reflection coefficient is calculated by use of an analytical solution of the corresponding Schrödinger equation. We also discuss electromagnetic wave analog of the effect, the partial reflection of light at an exponentially increasing index. We interpret the quantum reflection in terms of a virtual turning point, by means of a complex stationary phase approximation. If the potential remains exponential until it becomes as deep as several hundred recoil energies, the reflection coefficient for a low-energy de Broglie wave is independent of the details of the potential far from the virtual turning point. This means that, if the evanescent wave is strong enough, interactions between the atom and the glass surface at which the evanescent wave is created can be neglected.

© 1996 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  12. C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
    [CrossRef] [PubMed]
  13. W. Seifert, R. Kaiser, A. Aspect, and J. Mlynek, “Reflection of atoms from a dielectric waveguide,” Opt. Commun. 111, 566 (1994).
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  14. M. Kasevich and S. Chu, “Laser cooling below a photon recoil with three-level atoms,” Phys. Rev. Lett. 69, 1741 (1992).
    [CrossRef] [PubMed]
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  17. J. Lawall, F. Bardou, K. Shimizu, M. Leduc, A. Aspect, and C. Cohen-Tannoudji, “Two-dimensional subrecoil laser cooling,” Phys. Rev. Lett. 73, 1915 (1994).
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    [CrossRef]
  21. D. S. Zimmerman and A. J. Berlinsky, “The sticking probability for hydrogen atoms on the surface of liquid 4He,” Can. J. Phys. 61, 508 (1983).
    [CrossRef]
  22. J. J. Berkhout and J. T. M. Walraven, “Scattering of hydrogen atoms from liquid-helium surfaces,” Phys. Rev. B 47, 8886 (1993).
    [CrossRef]
  23. C. Henkel, J.-Y. Courtois, R. Kaiser, C. I. Westbrook, and A. Aspect, “Phase shifts of atomic de Broglie waves at an evanescent wave mirror,” Laser Phys. 4, 1040 (1994).
  24. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 7.
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    [CrossRef] [PubMed]
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  33. T. W. Hijmans, J. T. M. Walraven, and G. V. Shlyapnikov, “Influence of the substrate on the low-temperature limit of the sticking probability of hydrogen atoms on He films,” Phys. Rev. B 45, 2561 (1992).
    [CrossRef]
  34. C. Carraro and M. W. Cole, “Role of long-range forces in H sticking to liquid He,” Phys. Rev. B 45, 12930 (1992).
    [CrossRef]
  35. T. Esslinger, M. Weidemüller, A. Hemmerich, and T. W. Hänsch, “Surface-plasmon mirror for atoms,” Opt. Lett. 18, 450 (1993).
    [CrossRef] [PubMed]
  36. S. Feron, J. Reinhardt, S. Le Boiteaux, O. Gorceix, J. Baudon, M. Ducloy, J. Robert, C. Miniatura, S. NicChormaic, H. Haberland, and V. Lorent, “Reflection of metastable neon atoms by a surface plasmon wave,” Opt. Commun. 102, 83 (1993).
    [CrossRef]
  37. R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evanescent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234 (1994).
    [CrossRef]

1994 (6)

W. Seifert, R. Kaiser, A. Aspect, and J. Mlynek, “Reflection of atoms from a dielectric waveguide,” Opt. Commun. 111, 566 (1994).
[CrossRef]

J. Reichel, O. Morice, G. M. Tino, and C. Salomon, “Subrecoil Raman cooling of cesium atoms,” Europhys. Lett. 28, 477 (1994).
[CrossRef]

F. Bardou, B. Saubamea, J. Lawall, K. Shimizu, O. Emile, C. I. Westbrook, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling with precooled atoms,” C. R. Acad. Sci. Ser.2 318, 877 (1994).

J. Lawall, F. Bardou, K. Shimizu, M. Leduc, A. Aspect, and C. Cohen-Tannoudji, “Two-dimensional subrecoil laser cooling,” Phys. Rev. Lett. 73, 1915 (1994).
[CrossRef] [PubMed]

C. Henkel, J.-Y. Courtois, R. Kaiser, C. I. Westbrook, and A. Aspect, “Phase shifts of atomic de Broglie waves at an evanescent wave mirror,” Laser Phys. 4, 1040 (1994).

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evanescent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234 (1994).
[CrossRef]

1993 (5)

T. Esslinger, M. Weidemüller, A. Hemmerich, and T. W. Hänsch, “Surface-plasmon mirror for atoms,” Opt. Lett. 18, 450 (1993).
[CrossRef] [PubMed]

S. Feron, J. Reinhardt, S. Le Boiteaux, O. Gorceix, J. Baudon, M. Ducloy, J. Robert, C. Miniatura, S. NicChormaic, H. Haberland, and V. Lorent, “Reflection of metastable neon atoms by a surface plasmon wave,” Opt. Commun. 102, 83 (1993).
[CrossRef]

M. Kuś, F. Haake, and D. Delande, “Prebifurcation periodic ghost orbits in semiclassical quantization,” Phys. Rev. Lett. 71, 2167 (1993).
[CrossRef]

J. J. Berkhout and J. T. M. Walraven, “Scattering of hydrogen atoms from liquid-helium surfaces,” Phys. Rev. B 47, 8886 (1993).
[CrossRef]

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[CrossRef] [PubMed]

1992 (3)

M. Kasevich and S. Chu, “Laser cooling below a photon recoil with three-level atoms,” Phys. Rev. Lett. 69, 1741 (1992).
[CrossRef] [PubMed]

T. W. Hijmans, J. T. M. Walraven, and G. V. Shlyapnikov, “Influence of the substrate on the low-temperature limit of the sticking probability of hydrogen atoms on He films,” Phys. Rev. B 45, 2561 (1992).
[CrossRef]

C. Carraro and M. W. Cole, “Role of long-range forces in H sticking to liquid He,” Phys. Rev. B 45, 12930 (1992).
[CrossRef]

1991 (1)

J. M. Doyle, J. C. Sandberg, I. A. Yu, C. L. Cesar, D. Kleppner, and T. J. Greytak, “Hydrogen in the submil-likelvin regime: sticking probability on superfluid 4He,” Phys. Rev. Lett. 67, 603 (1991).
[CrossRef] [PubMed]

1990 (1)

1989 (1)

J. J. Berkhout, O. J. Luiten, I. D. Setija, T. W. Hijmans, T. Mizusaki, and J. T. M. Walraven, “Quantum reflection: focusing of hydrogen atoms with a concave mirror,” Phys. Rev. Lett. 63, 1689 (1989).
[CrossRef] [PubMed]

1987 (2)

V. I. Balykin, V. S. Letokhov, Y. B. Ovchinnikov, and A. I. Sidorov, “Reflection of an atomic beam from a gradient of an optical field,” JETP Lett. 45, 353 (1987).

H. M. Nussenzveig and W. J. Wiscombe, “Diffraction as tunnelling,” Phys. Rev. Lett. 59, 1667 (1987).
[CrossRef] [PubMed]

1983 (1)

D. S. Zimmerman and A. J. Berlinsky, “The sticking probability for hydrogen atoms on the surface of liquid 4He,” Can. J. Phys. 61, 508 (1983).
[CrossRef]

1982 (1)

R. J. Cook and R. K. Hill, “An electromagnetic mirror for neutral atoms,” Opt. Commun. 43, 258 (1982).
[CrossRef]

1971 (1)

M. G. Fuda, “T matrix for the exponential potential,” J. Math. Phys. 12, 1163 (1971).
[CrossRef]

1929 (1)

J. von Neumann and E. Wigner, “Über merkwürdige diskrete Eigenwerte,” Phys. Zeitschr. 30, 465 (1929).

Aminoff, C. G.

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[CrossRef] [PubMed]

Aspect, A.

W. Seifert, R. Kaiser, A. Aspect, and J. Mlynek, “Reflection of atoms from a dielectric waveguide,” Opt. Commun. 111, 566 (1994).
[CrossRef]

F. Bardou, B. Saubamea, J. Lawall, K. Shimizu, O. Emile, C. I. Westbrook, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling with precooled atoms,” C. R. Acad. Sci. Ser.2 318, 877 (1994).

J. Lawall, F. Bardou, K. Shimizu, M. Leduc, A. Aspect, and C. Cohen-Tannoudji, “Two-dimensional subrecoil laser cooling,” Phys. Rev. Lett. 73, 1915 (1994).
[CrossRef] [PubMed]

C. Henkel, J.-Y. Courtois, R. Kaiser, C. I. Westbrook, and A. Aspect, “Phase shifts of atomic de Broglie waves at an evanescent wave mirror,” Laser Phys. 4, 1040 (1994).

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evanescent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234 (1994).
[CrossRef]

Balykin, V. I.

V. I. Balykin, V. S. Letokhov, Y. B. Ovchinnikov, and A. I. Sidorov, “Reflection of an atomic beam from a gradient of an optical field,” JETP Lett. 45, 353 (1987).

Bardou, F.

J. Lawall, F. Bardou, K. Shimizu, M. Leduc, A. Aspect, and C. Cohen-Tannoudji, “Two-dimensional subrecoil laser cooling,” Phys. Rev. Lett. 73, 1915 (1994).
[CrossRef] [PubMed]

F. Bardou, B. Saubamea, J. Lawall, K. Shimizu, O. Emile, C. I. Westbrook, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling with precooled atoms,” C. R. Acad. Sci. Ser.2 318, 877 (1994).

Baudon, J.

S. Feron, J. Reinhardt, S. Le Boiteaux, O. Gorceix, J. Baudon, M. Ducloy, J. Robert, C. Miniatura, S. NicChormaic, H. Haberland, and V. Lorent, “Reflection of metastable neon atoms by a surface plasmon wave,” Opt. Commun. 102, 83 (1993).
[CrossRef]

Berkhout, J. J.

J. J. Berkhout and J. T. M. Walraven, “Scattering of hydrogen atoms from liquid-helium surfaces,” Phys. Rev. B 47, 8886 (1993).
[CrossRef]

J. J. Berkhout, O. J. Luiten, I. D. Setija, T. W. Hijmans, T. Mizusaki, and J. T. M. Walraven, “Quantum reflection: focusing of hydrogen atoms with a concave mirror,” Phys. Rev. Lett. 63, 1689 (1989).
[CrossRef] [PubMed]

J. J. Berkhout and J. T. M. Walraven, in Spin-Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 201.

Berlinsky, A. J.

D. S. Zimmerman and A. J. Berlinsky, “The sticking probability for hydrogen atoms on the surface of liquid 4He,” Can. J. Phys. 61, 508 (1983).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1959).

Bouyer, P.

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[CrossRef] [PubMed]

Carraro, C.

C. Carraro and M. W. Cole, “Role of long-range forces in H sticking to liquid He,” Phys. Rev. B 45, 12930 (1992).
[CrossRef]

Cesar, C. L.

J. M. Doyle, J. C. Sandberg, I. A. Yu, C. L. Cesar, D. Kleppner, and T. J. Greytak, “Hydrogen in the submil-likelvin regime: sticking probability on superfluid 4He,” Phys. Rev. Lett. 67, 603 (1991).
[CrossRef] [PubMed]

Chu, S.

M. Kasevich and S. Chu, “Laser cooling below a photon recoil with three-level atoms,” Phys. Rev. Lett. 69, 1741 (1992).
[CrossRef] [PubMed]

M. A. Kasevich, D. S. Weiss, and S. Chu, “Normal-incidence reflection of slow atoms from an optical evanescent wave,” Opt. Lett. 15, 607 (1990).
[CrossRef] [PubMed]

M. Kasevich, K. Moler, E. Riis, E. Sunderman, D. Weiss, and S. Chu, in Proceedings of the 12th International Conference of Atomic Physics, AIP Proc.233, 47–57 (1991).

Cohen-Tannoudji, C.

F. Bardou, B. Saubamea, J. Lawall, K. Shimizu, O. Emile, C. I. Westbrook, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling with precooled atoms,” C. R. Acad. Sci. Ser.2 318, 877 (1994).

J. Lawall, F. Bardou, K. Shimizu, M. Leduc, A. Aspect, and C. Cohen-Tannoudji, “Two-dimensional subrecoil laser cooling,” Phys. Rev. Lett. 73, 1915 (1994).
[CrossRef] [PubMed]

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[CrossRef] [PubMed]

Cole, M. W.

C. Carraro and M. W. Cole, “Role of long-range forces in H sticking to liquid He,” Phys. Rev. B 45, 12930 (1992).
[CrossRef]

Cook, R. J.

R. J. Cook and R. K. Hill, “An electromagnetic mirror for neutral atoms,” Opt. Commun. 43, 258 (1982).
[CrossRef]

Courtois, J.-Y.

C. Henkel, J.-Y. Courtois, R. Kaiser, C. I. Westbrook, and A. Aspect, “Phase shifts of atomic de Broglie waves at an evanescent wave mirror,” Laser Phys. 4, 1040 (1994).

Dalibard, J.

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[CrossRef] [PubMed]

Delande, D.

M. Kuś, F. Haake, and D. Delande, “Prebifurcation periodic ghost orbits in semiclassical quantization,” Phys. Rev. Lett. 71, 2167 (1993).
[CrossRef]

Delchar, T. A.

D. P. Woodruff and T. A. Delchar, Modern Techniques of Surface Science (Cambridge U. Press, Cambridge, 1986).

Desbiolles, P.

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[CrossRef] [PubMed]

Doyle, J. M.

J. M. Doyle, J. C. Sandberg, I. A. Yu, C. L. Cesar, D. Kleppner, and T. J. Greytak, “Hydrogen in the submil-likelvin regime: sticking probability on superfluid 4He,” Phys. Rev. Lett. 67, 603 (1991).
[CrossRef] [PubMed]

Ducloy, M.

S. Feron, J. Reinhardt, S. Le Boiteaux, O. Gorceix, J. Baudon, M. Ducloy, J. Robert, C. Miniatura, S. NicChormaic, H. Haberland, and V. Lorent, “Reflection of metastable neon atoms by a surface plasmon wave,” Opt. Commun. 102, 83 (1993).
[CrossRef]

Edwards, D. O.

D. O. Edwards and W. F. Saam, in Progress in Low Temperature Physics, D. Brewer, ed. (North-Holland, Amsterdam, 1978), Vol. 7A, p. 285.

Emile, O.

F. Bardou, B. Saubamea, J. Lawall, K. Shimizu, O. Emile, C. I. Westbrook, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling with precooled atoms,” C. R. Acad. Sci. Ser.2 318, 877 (1994).

Esslinger, T.

Feron, S.

S. Feron, J. Reinhardt, S. Le Boiteaux, O. Gorceix, J. Baudon, M. Ducloy, J. Robert, C. Miniatura, S. NicChormaic, H. Haberland, and V. Lorent, “Reflection of metastable neon atoms by a surface plasmon wave,” Opt. Commun. 102, 83 (1993).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. II.

Fock, V. A.

V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, Oxford, 1965).

Fuda, M. G.

M. G. Fuda, “T matrix for the exponential potential,” J. Math. Phys. 12, 1163 (1971).
[CrossRef]

Goldner, L.

K. Helmerson, S. L. Rolston, L. Goldner, and W. D. Phillips, “Optics and interferometry with atoms,” presented at the Ninety-seventh WE-Herdeus-Seminary, Insel Reichenau, Germany, June 10–12, 1992 (WE-Heraes-Stiftung).

Gorceix, O.

S. Feron, J. Reinhardt, S. Le Boiteaux, O. Gorceix, J. Baudon, M. Ducloy, J. Robert, C. Miniatura, S. NicChormaic, H. Haberland, and V. Lorent, “Reflection of metastable neon atoms by a surface plasmon wave,” Opt. Commun. 102, 83 (1993).
[CrossRef]

Greytak, T. J.

J. M. Doyle, J. C. Sandberg, I. A. Yu, C. L. Cesar, D. Kleppner, and T. J. Greytak, “Hydrogen in the submil-likelvin regime: sticking probability on superfluid 4He,” Phys. Rev. Lett. 67, 603 (1991).
[CrossRef] [PubMed]

Haake, F.

M. Kuś, F. Haake, and D. Delande, “Prebifurcation periodic ghost orbits in semiclassical quantization,” Phys. Rev. Lett. 71, 2167 (1993).
[CrossRef]

Haberland, H.

S. Feron, J. Reinhardt, S. Le Boiteaux, O. Gorceix, J. Baudon, M. Ducloy, J. Robert, C. Miniatura, S. NicChormaic, H. Haberland, and V. Lorent, “Reflection of metastable neon atoms by a surface plasmon wave,” Opt. Commun. 102, 83 (1993).
[CrossRef]

Hänsch, T. W.

Helmerson, K.

K. Helmerson, S. L. Rolston, L. Goldner, and W. D. Phillips, “Optics and interferometry with atoms,” presented at the Ninety-seventh WE-Herdeus-Seminary, Insel Reichenau, Germany, June 10–12, 1992 (WE-Heraes-Stiftung).

Hemmerich, A.

Henkel, C.

C. Henkel, J.-Y. Courtois, R. Kaiser, C. I. Westbrook, and A. Aspect, “Phase shifts of atomic de Broglie waves at an evanescent wave mirror,” Laser Phys. 4, 1040 (1994).

Hijmans, T. W.

T. W. Hijmans, J. T. M. Walraven, and G. V. Shlyapnikov, “Influence of the substrate on the low-temperature limit of the sticking probability of hydrogen atoms on He films,” Phys. Rev. B 45, 2561 (1992).
[CrossRef]

J. J. Berkhout, O. J. Luiten, I. D. Setija, T. W. Hijmans, T. Mizusaki, and J. T. M. Walraven, “Quantum reflection: focusing of hydrogen atoms with a concave mirror,” Phys. Rev. Lett. 63, 1689 (1989).
[CrossRef] [PubMed]

Hill, R. K.

R. J. Cook and R. K. Hill, “An electromagnetic mirror for neutral atoms,” Opt. Commun. 43, 258 (1982).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 7.

Kaiser, R.

C. Henkel, J.-Y. Courtois, R. Kaiser, C. I. Westbrook, and A. Aspect, “Phase shifts of atomic de Broglie waves at an evanescent wave mirror,” Laser Phys. 4, 1040 (1994).

W. Seifert, R. Kaiser, A. Aspect, and J. Mlynek, “Reflection of atoms from a dielectric waveguide,” Opt. Commun. 111, 566 (1994).
[CrossRef]

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evanescent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234 (1994).
[CrossRef]

Kasevich, M.

M. Kasevich and S. Chu, “Laser cooling below a photon recoil with three-level atoms,” Phys. Rev. Lett. 69, 1741 (1992).
[CrossRef] [PubMed]

M. Kasevich, K. Moler, E. Riis, E. Sunderman, D. Weiss, and S. Chu, in Proceedings of the 12th International Conference of Atomic Physics, AIP Proc.233, 47–57 (1991).

Kasevich, M. A.

Kleppner, D.

J. M. Doyle, J. C. Sandberg, I. A. Yu, C. L. Cesar, D. Kleppner, and T. J. Greytak, “Hydrogen in the submil-likelvin regime: sticking probability on superfluid 4He,” Phys. Rev. Lett. 67, 603 (1991).
[CrossRef] [PubMed]

Kus, M.

M. Kuś, F. Haake, and D. Delande, “Prebifurcation periodic ghost orbits in semiclassical quantization,” Phys. Rev. Lett. 71, 2167 (1993).
[CrossRef]

Lawall, J.

J. Lawall, F. Bardou, K. Shimizu, M. Leduc, A. Aspect, and C. Cohen-Tannoudji, “Two-dimensional subrecoil laser cooling,” Phys. Rev. Lett. 73, 1915 (1994).
[CrossRef] [PubMed]

F. Bardou, B. Saubamea, J. Lawall, K. Shimizu, O. Emile, C. I. Westbrook, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling with precooled atoms,” C. R. Acad. Sci. Ser.2 318, 877 (1994).

Le Boiteaux, S.

S. Feron, J. Reinhardt, S. Le Boiteaux, O. Gorceix, J. Baudon, M. Ducloy, J. Robert, C. Miniatura, S. NicChormaic, H. Haberland, and V. Lorent, “Reflection of metastable neon atoms by a surface plasmon wave,” Opt. Commun. 102, 83 (1993).
[CrossRef]

Leduc, M.

J. Lawall, F. Bardou, K. Shimizu, M. Leduc, A. Aspect, and C. Cohen-Tannoudji, “Two-dimensional subrecoil laser cooling,” Phys. Rev. Lett. 73, 1915 (1994).
[CrossRef] [PubMed]

Leipold, D.

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evanescent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234 (1994).
[CrossRef]

Letokhov, V. S.

V. I. Balykin, V. S. Letokhov, Y. B. Ovchinnikov, and A. I. Sidorov, “Reflection of an atomic beam from a gradient of an optical field,” JETP Lett. 45, 353 (1987).

Lévy, Y.

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evanescent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234 (1994).
[CrossRef]

Lorent, V.

S. Feron, J. Reinhardt, S. Le Boiteaux, O. Gorceix, J. Baudon, M. Ducloy, J. Robert, C. Miniatura, S. NicChormaic, H. Haberland, and V. Lorent, “Reflection of metastable neon atoms by a surface plasmon wave,” Opt. Commun. 102, 83 (1993).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Luiten, O. J.

J. J. Berkhout, O. J. Luiten, I. D. Setija, T. W. Hijmans, T. Mizusaki, and J. T. M. Walraven, “Quantum reflection: focusing of hydrogen atoms with a concave mirror,” Phys. Rev. Lett. 63, 1689 (1989).
[CrossRef] [PubMed]

Messiah, A.

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1961), Vol. I.

Miniatura, C.

S. Feron, J. Reinhardt, S. Le Boiteaux, O. Gorceix, J. Baudon, M. Ducloy, J. Robert, C. Miniatura, S. NicChormaic, H. Haberland, and V. Lorent, “Reflection of metastable neon atoms by a surface plasmon wave,” Opt. Commun. 102, 83 (1993).
[CrossRef]

Mizusaki, T.

J. J. Berkhout, O. J. Luiten, I. D. Setija, T. W. Hijmans, T. Mizusaki, and J. T. M. Walraven, “Quantum reflection: focusing of hydrogen atoms with a concave mirror,” Phys. Rev. Lett. 63, 1689 (1989).
[CrossRef] [PubMed]

Mlynek, J.

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evanescent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234 (1994).
[CrossRef]

W. Seifert, R. Kaiser, A. Aspect, and J. Mlynek, “Reflection of atoms from a dielectric waveguide,” Opt. Commun. 111, 566 (1994).
[CrossRef]

Moler, K.

M. Kasevich, K. Moler, E. Riis, E. Sunderman, D. Weiss, and S. Chu, in Proceedings of the 12th International Conference of Atomic Physics, AIP Proc.233, 47–57 (1991).

Morice, O.

J. Reichel, O. Morice, G. M. Tino, and C. Salomon, “Subrecoil Raman cooling of cesium atoms,” Europhys. Lett. 28, 477 (1994).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. II.

NicChormaic, S.

S. Feron, J. Reinhardt, S. Le Boiteaux, O. Gorceix, J. Baudon, M. Ducloy, J. Robert, C. Miniatura, S. NicChormaic, H. Haberland, and V. Lorent, “Reflection of metastable neon atoms by a surface plasmon wave,” Opt. Commun. 102, 83 (1993).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig and W. J. Wiscombe, “Diffraction as tunnelling,” Phys. Rev. Lett. 59, 1667 (1987).
[CrossRef] [PubMed]

Ovchinnikov, Y. B.

V. I. Balykin, V. S. Letokhov, Y. B. Ovchinnikov, and A. I. Sidorov, “Reflection of an atomic beam from a gradient of an optical field,” JETP Lett. 45, 353 (1987).

Phillips, W. D.

K. Helmerson, S. L. Rolston, L. Goldner, and W. D. Phillips, “Optics and interferometry with atoms,” presented at the Ninety-seventh WE-Herdeus-Seminary, Insel Reichenau, Germany, June 10–12, 1992 (WE-Heraes-Stiftung).

Reichel, J.

J. Reichel, O. Morice, G. M. Tino, and C. Salomon, “Subrecoil Raman cooling of cesium atoms,” Europhys. Lett. 28, 477 (1994).
[CrossRef]

Reinhardt, J.

S. Feron, J. Reinhardt, S. Le Boiteaux, O. Gorceix, J. Baudon, M. Ducloy, J. Robert, C. Miniatura, S. NicChormaic, H. Haberland, and V. Lorent, “Reflection of metastable neon atoms by a surface plasmon wave,” Opt. Commun. 102, 83 (1993).
[CrossRef]

Riis, E.

M. Kasevich, K. Moler, E. Riis, E. Sunderman, D. Weiss, and S. Chu, in Proceedings of the 12th International Conference of Atomic Physics, AIP Proc.233, 47–57 (1991).

Robert, J.

S. Feron, J. Reinhardt, S. Le Boiteaux, O. Gorceix, J. Baudon, M. Ducloy, J. Robert, C. Miniatura, S. NicChormaic, H. Haberland, and V. Lorent, “Reflection of metastable neon atoms by a surface plasmon wave,” Opt. Commun. 102, 83 (1993).
[CrossRef]

Rolston, S. L.

K. Helmerson, S. L. Rolston, L. Goldner, and W. D. Phillips, “Optics and interferometry with atoms,” presented at the Ninety-seventh WE-Herdeus-Seminary, Insel Reichenau, Germany, June 10–12, 1992 (WE-Heraes-Stiftung).

Saam, W. F.

D. O. Edwards and W. F. Saam, in Progress in Low Temperature Physics, D. Brewer, ed. (North-Holland, Amsterdam, 1978), Vol. 7A, p. 285.

Salomon, C.

J. Reichel, O. Morice, G. M. Tino, and C. Salomon, “Subrecoil Raman cooling of cesium atoms,” Europhys. Lett. 28, 477 (1994).
[CrossRef]

Sandberg, J. C.

J. M. Doyle, J. C. Sandberg, I. A. Yu, C. L. Cesar, D. Kleppner, and T. J. Greytak, “Hydrogen in the submil-likelvin regime: sticking probability on superfluid 4He,” Phys. Rev. Lett. 67, 603 (1991).
[CrossRef] [PubMed]

Saubamea, B.

F. Bardou, B. Saubamea, J. Lawall, K. Shimizu, O. Emile, C. I. Westbrook, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling with precooled atoms,” C. R. Acad. Sci. Ser.2 318, 877 (1994).

Seifert, W.

W. Seifert, R. Kaiser, A. Aspect, and J. Mlynek, “Reflection of atoms from a dielectric waveguide,” Opt. Commun. 111, 566 (1994).
[CrossRef]

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evanescent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234 (1994).
[CrossRef]

Setija, I. D.

J. J. Berkhout, O. J. Luiten, I. D. Setija, T. W. Hijmans, T. Mizusaki, and J. T. M. Walraven, “Quantum reflection: focusing of hydrogen atoms with a concave mirror,” Phys. Rev. Lett. 63, 1689 (1989).
[CrossRef] [PubMed]

Shimizu, K.

F. Bardou, B. Saubamea, J. Lawall, K. Shimizu, O. Emile, C. I. Westbrook, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling with precooled atoms,” C. R. Acad. Sci. Ser.2 318, 877 (1994).

J. Lawall, F. Bardou, K. Shimizu, M. Leduc, A. Aspect, and C. Cohen-Tannoudji, “Two-dimensional subrecoil laser cooling,” Phys. Rev. Lett. 73, 1915 (1994).
[CrossRef] [PubMed]

Shlyapnikov, G. V.

T. W. Hijmans, J. T. M. Walraven, and G. V. Shlyapnikov, “Influence of the substrate on the low-temperature limit of the sticking probability of hydrogen atoms on He films,” Phys. Rev. B 45, 2561 (1992).
[CrossRef]

Sidorov, A. I.

V. I. Balykin, V. S. Letokhov, Y. B. Ovchinnikov, and A. I. Sidorov, “Reflection of an atomic beam from a gradient of an optical field,” JETP Lett. 45, 353 (1987).

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Steane, A. M.

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[CrossRef] [PubMed]

Sunderman, E.

M. Kasevich, K. Moler, E. Riis, E. Sunderman, D. Weiss, and S. Chu, in Proceedings of the 12th International Conference of Atomic Physics, AIP Proc.233, 47–57 (1991).

Tino, G. M.

J. Reichel, O. Morice, G. M. Tino, and C. Salomon, “Subrecoil Raman cooling of cesium atoms,” Europhys. Lett. 28, 477 (1994).
[CrossRef]

Vansteenkiste, N.

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evanescent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234 (1994).
[CrossRef]

von Neumann, J.

J. von Neumann and E. Wigner, “Über merkwürdige diskrete Eigenwerte,” Phys. Zeitschr. 30, 465 (1929).

Walraven, J. T. M.

J. J. Berkhout and J. T. M. Walraven, “Scattering of hydrogen atoms from liquid-helium surfaces,” Phys. Rev. B 47, 8886 (1993).
[CrossRef]

T. W. Hijmans, J. T. M. Walraven, and G. V. Shlyapnikov, “Influence of the substrate on the low-temperature limit of the sticking probability of hydrogen atoms on He films,” Phys. Rev. B 45, 2561 (1992).
[CrossRef]

J. J. Berkhout, O. J. Luiten, I. D. Setija, T. W. Hijmans, T. Mizusaki, and J. T. M. Walraven, “Quantum reflection: focusing of hydrogen atoms with a concave mirror,” Phys. Rev. Lett. 63, 1689 (1989).
[CrossRef] [PubMed]

J. J. Berkhout and J. T. M. Walraven, in Spin-Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 201.

Watson,

Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1962).

Weidemüller, M.

Weiss, D.

M. Kasevich, K. Moler, E. Riis, E. Sunderman, D. Weiss, and S. Chu, in Proceedings of the 12th International Conference of Atomic Physics, AIP Proc.233, 47–57 (1991).

Weiss, D. S.

Westbrook, C. I.

F. Bardou, B. Saubamea, J. Lawall, K. Shimizu, O. Emile, C. I. Westbrook, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling with precooled atoms,” C. R. Acad. Sci. Ser.2 318, 877 (1994).

C. Henkel, J.-Y. Courtois, R. Kaiser, C. I. Westbrook, and A. Aspect, “Phase shifts of atomic de Broglie waves at an evanescent wave mirror,” Laser Phys. 4, 1040 (1994).

Wigner, E.

J. von Neumann and E. Wigner, “Über merkwürdige diskrete Eigenwerte,” Phys. Zeitschr. 30, 465 (1929).

Wiscombe, W. J.

H. M. Nussenzveig and W. J. Wiscombe, “Diffraction as tunnelling,” Phys. Rev. Lett. 59, 1667 (1987).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1959).

Woodruff, D. P.

D. P. Woodruff and T. A. Delchar, Modern Techniques of Surface Science (Cambridge U. Press, Cambridge, 1986).

Yu, I. A.

J. M. Doyle, J. C. Sandberg, I. A. Yu, C. L. Cesar, D. Kleppner, and T. J. Greytak, “Hydrogen in the submil-likelvin regime: sticking probability on superfluid 4He,” Phys. Rev. Lett. 67, 603 (1991).
[CrossRef] [PubMed]

Zimmerman, D. S.

D. S. Zimmerman and A. J. Berlinsky, “The sticking probability for hydrogen atoms on the surface of liquid 4He,” Can. J. Phys. 61, 508 (1983).
[CrossRef]

C. R. Acad. Sci. Ser.2 (1)

F. Bardou, B. Saubamea, J. Lawall, K. Shimizu, O. Emile, C. I. Westbrook, A. Aspect, and C. Cohen-Tannoudji, “Subrecoil laser cooling with precooled atoms,” C. R. Acad. Sci. Ser.2 318, 877 (1994).

Can. J. Phys. (1)

D. S. Zimmerman and A. J. Berlinsky, “The sticking probability for hydrogen atoms on the surface of liquid 4He,” Can. J. Phys. 61, 508 (1983).
[CrossRef]

Europhys. Lett. (1)

J. Reichel, O. Morice, G. M. Tino, and C. Salomon, “Subrecoil Raman cooling of cesium atoms,” Europhys. Lett. 28, 477 (1994).
[CrossRef]

J. Math. Phys. (1)

M. G. Fuda, “T matrix for the exponential potential,” J. Math. Phys. 12, 1163 (1971).
[CrossRef]

JETP Lett. (1)

V. I. Balykin, V. S. Letokhov, Y. B. Ovchinnikov, and A. I. Sidorov, “Reflection of an atomic beam from a gradient of an optical field,” JETP Lett. 45, 353 (1987).

Laser Phys. (1)

C. Henkel, J.-Y. Courtois, R. Kaiser, C. I. Westbrook, and A. Aspect, “Phase shifts of atomic de Broglie waves at an evanescent wave mirror,” Laser Phys. 4, 1040 (1994).

Opt. Commun. (4)

W. Seifert, R. Kaiser, A. Aspect, and J. Mlynek, “Reflection of atoms from a dielectric waveguide,” Opt. Commun. 111, 566 (1994).
[CrossRef]

R. J. Cook and R. K. Hill, “An electromagnetic mirror for neutral atoms,” Opt. Commun. 43, 258 (1982).
[CrossRef]

S. Feron, J. Reinhardt, S. Le Boiteaux, O. Gorceix, J. Baudon, M. Ducloy, J. Robert, C. Miniatura, S. NicChormaic, H. Haberland, and V. Lorent, “Reflection of metastable neon atoms by a surface plasmon wave,” Opt. Commun. 102, 83 (1993).
[CrossRef]

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evanescent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234 (1994).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. B (3)

J. J. Berkhout and J. T. M. Walraven, “Scattering of hydrogen atoms from liquid-helium surfaces,” Phys. Rev. B 47, 8886 (1993).
[CrossRef]

T. W. Hijmans, J. T. M. Walraven, and G. V. Shlyapnikov, “Influence of the substrate on the low-temperature limit of the sticking probability of hydrogen atoms on He films,” Phys. Rev. B 45, 2561 (1992).
[CrossRef]

C. Carraro and M. W. Cole, “Role of long-range forces in H sticking to liquid He,” Phys. Rev. B 45, 12930 (1992).
[CrossRef]

Phys. Rev. Lett. (7)

H. M. Nussenzveig and W. J. Wiscombe, “Diffraction as tunnelling,” Phys. Rev. Lett. 59, 1667 (1987).
[CrossRef] [PubMed]

M. Kuś, F. Haake, and D. Delande, “Prebifurcation periodic ghost orbits in semiclassical quantization,” Phys. Rev. Lett. 71, 2167 (1993).
[CrossRef]

M. Kasevich and S. Chu, “Laser cooling below a photon recoil with three-level atoms,” Phys. Rev. Lett. 69, 1741 (1992).
[CrossRef] [PubMed]

J. J. Berkhout, O. J. Luiten, I. D. Setija, T. W. Hijmans, T. Mizusaki, and J. T. M. Walraven, “Quantum reflection: focusing of hydrogen atoms with a concave mirror,” Phys. Rev. Lett. 63, 1689 (1989).
[CrossRef] [PubMed]

C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard, and C. Cohen-Tannoudji, “Cesium atoms bouncing in a stable gravitational cavity,” Phys. Rev. Lett. 71, 3083 (1993).
[CrossRef] [PubMed]

J. M. Doyle, J. C. Sandberg, I. A. Yu, C. L. Cesar, D. Kleppner, and T. J. Greytak, “Hydrogen in the submil-likelvin regime: sticking probability on superfluid 4He,” Phys. Rev. Lett. 67, 603 (1991).
[CrossRef] [PubMed]

J. Lawall, F. Bardou, K. Shimizu, M. Leduc, A. Aspect, and C. Cohen-Tannoudji, “Two-dimensional subrecoil laser cooling,” Phys. Rev. Lett. 73, 1915 (1994).
[CrossRef] [PubMed]

Phys. Zeitschr. (1)

J. von Neumann and E. Wigner, “Über merkwürdige diskrete Eigenwerte,” Phys. Zeitschr. 30, 465 (1929).

Other (14)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 9th ed. (Dover, New York, 1972).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. II.

D. O. Edwards and W. F. Saam, in Progress in Low Temperature Physics, D. Brewer, ed. (North-Holland, Amsterdam, 1978), Vol. 7A, p. 285.

J. J. Berkhout and J. T. M. Walraven, in Spin-Polarized Quantum Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p. 201.

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1961), Vol. I.

D. P. Woodruff and T. A. Delchar, Modern Techniques of Surface Science (Cambridge U. Press, Cambridge, 1986).

M. Kasevich, K. Moler, E. Riis, E. Sunderman, D. Weiss, and S. Chu, in Proceedings of the 12th International Conference of Atomic Physics, AIP Proc.233, 47–57 (1991).

K. Helmerson, S. L. Rolston, L. Goldner, and W. D. Phillips, “Optics and interferometry with atoms,” presented at the Ninety-seventh WE-Herdeus-Seminary, Insel Reichenau, Germany, June 10–12, 1992 (WE-Heraes-Stiftung).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 7.

V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, Oxford, 1965).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1959).

I. S. Gradshteyn and I. M. Ryzhik, eds., Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1962).

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Figures (6)

Fig. 1
Fig. 1

Classical trajectory [Eq. (2.2), thick curve] of a particle being attracted into the region of large negative potential (region to the right). It actually reaches infinity (z = +∞) at time t = t. Note that time t may be interpreted as proportional to the x coordinate because the atom is moving freely in the direction parallel to the plane z = 0. Thin curve: asymptotic trajectory [expression (2.3)] for the free-field region z → −∞. The horizon ζhor(vz) [Eq. (2.4)] is defined as the position of the asymptotic trajectory at time t = t. Position and time coordinates are shown relative to ζhor(vz) and t, respectively.

Fig. 2
Fig. 2

(a) Wave function (2.13) for a particle in an attractive exponential potential. Short-dashed (long-dashed) curves: real (imaginary) part of ψkz (z). Thick curves: envelope ±|ψkz (z)|, showing oscillations that are due to the interference between the incident and the reflected waves. The incident wave vector is kz = (π/4)κ (incident de Broglie wavelength λdB = 8κ−1). The amplitude reflection coefficient has magnitude |r| ≈ 0.08. The z coordinate is shown relative to the classical horizon ζhor(vz) [Eq. (2.4)]. (b) Effective step potential [Eq. (2.19), dashed curve], giving the same amplitude reflection coefficient as the exponential potential (thick curve), calculated for the incident energy of the wave function in (a). The step position equals ζeff (vz) ≈ ζhor(vz) − 0.14κ−1, and the potential decrease Veff ≈ 0.25ћ2κ2/2M. The potential energy is given in units of the recoil energy, Erecћ2κ2/2M.

Fig. 3
Fig. 3

Thick curve: intensity reflection coefficient R [Eq. (2.16)] for quantum reflection at an attractive exponential potential as a function of the incident velocity vz. The reflection coefficient decays exponentially, with a characteristic velocity scale given by ћκ/2πM (cf. the inset on a logarithmic scale). The incident normal velocity vz is given in units of the recoil velocity, ћκ/M. Dashed curve: intensity reflection coefficient for the low-energy limit of the effective step potential [expressions (2.20)]. It coincides with the reflection coefficient for the exponential potential for vzћκ/2πM, in the deep quantum regime.

Fig. 4
Fig. 4

Geometrical rays for the exponentially increasing index. The index increases toward the right and equals n = 2 for z = 0. Thick, solid curve: ray incident from the low-index region [Eq. (3.4)] at an angle θi = 60°. Thick, dashed curve: ray trapped in the (high-index) region z ≥ 0 [Eq. (3.5)] with complex angle of incidence θi = π/2 + iα such that sinh α = 1. (This ray corresponds to the mirage effect, i.e., total internal reflection of a beam incident upon a decreasing index of refraction above the critical angle.) Thin, dashed curves: asymptotic rays for the regions z → ±∞. Positions are given in units of 1/κ.

Fig. 5
Fig. 5

Semiclassical interpretation of quantum reflection. Lower curve: trajectory [Eq. (2.2)] of an incident atom, reaching infinity (z = +∞) at time t = t. Upper (dashed) curve: trajectory of an atom at the same energy, which starts from z = +∞ at t = t and escapes into the region z → −∞ of vanishing potential. Position and time coordinates are shown relative to the classical horizon ζhor(vz) and the time t, respectively.

Fig. 6
Fig. 6

Intensity reflection coefficient R for a truncated potential as a function of the value of the potential at z = 0, given in terms of the parameter km/κ [see Eq. (5.1)]. The velocities corresponding to the curves are given in recoil units, PMvz/ћκ. Thick, solid curves: exact result [Eqs. (B7)]; long-dashed curves: asymptotic expansion (5.3); short-dashed lines: reflection coefficient R for the unlimited potential.

Equations (71)

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V ( z ) = ½ M v m 2 exp ( 2 κ z ) .
z ( t ) = κ 1 log { v m v z sinh [ κ v z ( t t ) ] } , t < t ,
z ( t ) ζ hor + v z ( t t ) , t t τ ,
ζ hor ( v z ) = κ 1 log ( 2 v z / v m ) .
d 2 ψ d z 2 k m 2 exp ( 2 κ z ) ψ = k z 2 ψ ,
ψ ± ( z ) = C ± exp ( ± i k z z ) , z 1 / κ ,
ψ ± ( z ) C ± exp ( κ z / 2 ) exp [ ± i k m κ exp ( κ z ) ] , z + 1 / κ ,
u ( z ) k m κ exp ( κ z ) ,
z z l , k m k m exp ( κ l ) .
u 2 d 2 ψ d u 2 + u d ψ d u + ( u 2 + P 2 ) ψ = 0 ,
P k z κ = M v z ћ κ
ψ ( u ) = c 1 J iP ( u ) + c 2 J iP ( u ) .
ψ k z ( z ) = [ M π sinh ( π P ) ћ κ exp ( 2 π P ) ] 1 / 2 H iP ( 1 ) [ u ( z ) ] ,
ψ k z ( z ) = C [ exp ( i k z z ) r exp ( i k z z ) ] , z ,
r = exp ( π P ) exp [ 2 iP log ( 2 κ k m ) ] Γ ( 1 + iP ) Γ ( 1 iP ) ,
R reflected current incident current = | r | 2 = exp ( 2 π P ) = exp ( 2 π v z ћ κ / M ) .
R 1 { λ dB ( 2 π ) 2 / κ or v z ћ κ / 2 π M .
T 2 π v z ћ κ / M ,
V eff ( z ) = { 0 for z < ζ eff V eff for z > ζ eff .
V eff ћ 2 2 M ( 2 κ π ) 2 , ζ eff ζ hor ( ћ κ / M ) + γ κ 1 , k z κ ,
Δ φ arg r = 2 k z ζ eff = 2 k z κ log ( 2 κ k m ) + 2 arg Γ ( 1 + i k z κ ) ,
τ ћ Δ φ E inc = 1 v z Δ φ k z
τ 2 ζ eff v z , v z ћ κ / M .
[ 2 + k 0 2 n 2 ( r ) ] E ( r ) = 0 ,
n 2 ( r ) = 1 + m 2 exp ( 2 κ z ) ,
V n ( r ) = ( ћ k 0 ) 2 2 M [ n 2 ( r ) 1 ]
n ( r ) sin θ ( r ) = const . = sin θ i ,
z ( x ) = κ 1 log { m cos θ i sinh [ κ tan θ i ( x x ) ] } ,
z ( x ) = κ 1 log { m sinh α sin [ κ ( x x ) tanh α ] } .
R = exp ( 2 π k 0 cos θ i κ ) ,
cos θ i κ λ opt ( 2 π ) 2 .
κ / k z ( 2 π ) 2 .
ψ k z ( z ) = C d ζ exp [ i k z ζ + i u ( z ) cosh κ ζ ] ,
ϕ ( ζ ; z ) = k z ζ + u ( z ) cosh κ ζ
ζ ¯ inc ( z ) = k 1 arcsinh [ k z κ u ( z ) ] ,
ζ ¯ out ( z ) = ζ ¯ inc + i π κ 1 ,
ζ ¯ inc ( z ) = v z ( t t )
ψ k z ( z ) C ( 2 π / κ 2 ) 1 / 2 [ u 2 ( z ) + ( k z / κ ) 2 ] 1 / 4 × ( exp { i ( k z / κ ) arcsinh [ k z κ u ( z ) ] + i [ u 2 ( z ) + ( k z / κ ) 2 ] 1 / 2 + i π 4 } + exp { π k z κ + i ( k z / κ ) arcsinh [ k z κ u ( z ) ] i [ u 2 ( z ) + ( k z / κ ) 2 ] 1 / 2 i π 4 } ) .
ψ k z ( z ) C ( 2 π k z κ ) 1 / 2 [ exp ( i k z z ) exp ( i φ inc + i π / 4 ) + exp ( i k z z ) exp ( i φ out i π / 4 ) ] , z ,
φ inc = k z [ 1 κ ζ hor ( v z ) ] ,
φ out = k z [ 1 κ ζ hor ( v z ) ] + i π k z κ .
R = exp ( 2 π k z / κ ) ,
ϕ ( ζ ¯ inc ; z ) = 1 ћ z 0 z p inc ( z ) d z ,
ϕ ( ζ ¯ out ; z ) = 1 ћ z 0 z p out ( z ) d z + i π k z κ .
z out ( t ) = k 1 log { v m v z sinh [ κ v z ( t t ) ] } , t > t ,
z 0 ( v z ) ζ hor ( v z ) 1.10 κ 1 ,
ψ k z ( z ) C [ 2 π ћ M κ v ( z ) ] 1 / 2 { exp [ i ћ z 0 z p inc ( z ) d z + i π 4 ] + exp [ i ћ z 0 z p out ( z ) d z i π 4 π k z κ ] } , z z 0 ,
ψ WKB ( z ) = N p ( z ) cos [ 1 ћ z turn z p ( z )d z + i π 4 ] , z < z turn ,
Δ φ ( st.ph. ) = Re φ out φ inc ( π / 2 ) ,
1 2 2 ϕ ζ 2 | ζ ¯ ( ζ ζ ¯ ) 2 π .
V ( z ) = { ( ћ k m ) 2 2 M exp ( 2 κ z ) for z 0 ( ћ k m ) 2 2 M for z 0 ,
R exp ( 2 π k z / κ ) ,
R R [ 1 sinh ( π k z κ ) cos ( 2 k m / κ ) k m / κ ] , k m κ , k z 2 / κ .
u ( z ) = k m κ exp ( κ z ) ,
J ± iP ( u ) = ( 2 π u ) 1 / 2 [ cos χ + 4 P 2 + 1 8 u sin χ + O ( 1 u 2 ) ] , u ,
χ = u π 4 i π P 2 .
H iP ( 1 ) = 1 sinh ( π P ) [ exp ( π P ) J iP J iP ] .
H iP ( 1 ) ( u ) = ( 2 π u ) 1 / 2 exp ( i χ ) [ 1 + 4 P 2 + 1 8 iu + O ( 1 u 2 ) ] , u ,
J iP ( u ) = 1 Γ ( 1 + iP ) ( u 2 ) iP + O ( u 2 ) , u 0 .
j inc = ћ k z M 1 | Γ ( 1 + iP ) | 2 = ћ κ M π sinh π P ,
| Γ ( 1 + iP ) | = ( π P sinh π P ) 1 / 2
ψ ( z ) = { j k z ( z ) rj k z ( z ) for z 0 t exp ( ik t z ) for z 0 .
j ± k z ( z ) exp [ ± iP log ( 2 κ / k m ) ] Γ ( 1 ± iP ) J ± iP [ u ( z ) ] ,
k t = k z 2 + k m 2 .
r = exp [ 2 iP log ( u m / 2 ) ] × Γ ( 1 + iP ) Γ ( 1 iP ) k m J iP ( u m ) ik t J iP ( u m ) k m J iP ( u m ) ik t J iP ( u m ) ,
t = exp [ iP log ( u m / 2 ) ] Γ ( 1 iP ) 2 ik z k m J iP ( u m ) ik t J iP ( u m ) ,
u m k m / κ .
J iP ( u ) J iP ( u ) J iP ( u ) J iP ( u ) = 2 sinh ( π P ) i π u .
R = | r | 2 , T = k t k z | t | 2 .
r k t k z k t + k z , t 2 k z k t + k z ,
k t k z 2 + k m 2 .

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