On Two-Dimensional Above-Barrier Penetration and Sub-Barrier Tunneling for Non-Relativistic Particles and Photons

ABSTRACT

We study the two-dimensional above-barrier penetration and the sub-barrier tunneling of non-relativistic particles and photons, described in the quasi-monochromatic approximation by simple plane waves. Our scheme represents the motion from the left free-motion zero-potential region to the right zero-potential region through the intermediate region with a one-dimensional rectangular potential barrier along the axis, normal to the both parallel interfaces between all three regions, and with the zero potential along the axis, parallel to the those interfaces. We have firstly obtained the analytical expressions for the infinite series of multiple internal and external reflections and also of multiple transmitted waves of particles and photons, with equal shifts between them along the interfaces for the above-barrier penetration and with various shifts between them in the case of the sub-barrier tunneling. Finally the Hartman and Fletcher effect for any transmitted wave is established.

We study the two-dimensional above-barrier penetration and the sub-barrier tunneling of non-relativistic particles and photons, described in the quasi-monochromatic approximation by simple plane waves. Our scheme represents the motion from the left free-motion zero-potential region to the right zero-potential region through the intermediate region with a one-dimensional rectangular potential barrier along the axis, normal to the both parallel interfaces between all three regions, and with the zero potential along the axis, parallel to the those interfaces. We have firstly obtained the analytical expressions for the infinite series of multiple internal and external reflections and also of multiple transmitted waves of particles and photons, with equal shifts between them along the interfaces for the above-barrier penetration and with various shifts between them in the case of the sub-barrier tunneling. Finally the Hartman and Fletcher effect for any transmitted wave is established.

KEYWORDS

Two-Dimensional (2D) Penetration and Tunneling, Quasi-Monochromatic Approximation, Propagating Plane Waves, Evanescent and Anti-Evanescent Waves

Two-Dimensional (2D) Penetration and Tunneling, Quasi-Monochromatic Approximation, Propagating Plane Waves, Evanescent and Anti-Evanescent Waves

Cite this paper

nullV. Olkhovsky and M. Romaniuk, "On Two-Dimensional Above-Barrier Penetration and Sub-Barrier Tunneling for Non-Relativistic Particles and Photons,"*Journal of Modern Physics*, Vol. 2 No. 10, 2011, pp. 1166-1171. doi: 10.4236/jmp.2011.210145.

nullV. Olkhovsky and M. Romaniuk, "On Two-Dimensional Above-Barrier Penetration and Sub-Barrier Tunneling for Non-Relativistic Particles and Photons,"

References

[1] V. S. Olkhovsky and E. Recami, “Recent Developments in the Time Analysis of Tunnelling Processes,” Physics Reports, Vol. 219, No. 6, 1992, pp. 339-356. doi:10.1016/0370-1573(92)90015-R

[2] R. Landauer and M. Martin, “Barrier Interaction Time in Tunneling,” Reviews of Modern Physics, Vol. 66, No. 1, 1994, pp. 217-228. doi:10.1103/RevModPhys.66.217

[3] V. S. Olkhovsky, E. Recami and J. Jakiel, “Unified Time Analysis of Photon and Particle Tunnelling,” Physics Reports, Vol. 398, No. 3, 2004, pp. 133-178. doi:10.1016/j.physrep.2004.06.001

[4] V. S. Olkhovsky and A. Agresti, “Developments in Time Analysis of Tunnelling Processes,” In: D. Mugnai, A. Ranfagni and L. S. Schulman, Eds., Proc. Adriatico Res. Conf. on Tunneling and Its Implications, Trieste, 30 July- 2 August 1996, World Scientific, Singapore, 1997, pp. 327-355.

[5] V. S. Olkhovsky, “Time Analysis of Tunnelling of Particles and Photons,” Physics of the Alive, Vol. 5, No. 1, 1997, pp. 23-41.

[6] V. S. Olkhovsky, V. Petrillo and A. K. Zaichenko, “Decrease of the Tunneling Time and Violation of the Hartman Effect for Large Barriers,” Physical Review, Vol. A70, No. 1, 2004, pp. 034103-1-4.

[7] J. H. Fermor, “Quantum-Mechanical Tunneling,” American Journal of Physics, Vol. 34, No. 12, 1966, pp. 1168- 1170. doi:10.1119/1.1972543

[8] K. W. McVoy, L. Heller and M. Bolsterli, “Optical Analysis of Potential Well Resonances,” Reviews of Modern Physics, Vol. 39, 1967, pp. 245-258. doi:10.1103/RevModPhys.39.245

[9] A. Anderson, “Multiple Scattering Approach to One- Dimensional Potential Problems,” American Journal of Physics, Vol. 57, No. 3, 1989, pp. 230-235. doi:10.1119/1.16095

[10] S. P. Maydanyuk, V. S. Olkhovsky and A. K. Zaichenko, “Multiple Internal Reflections Method in the Description of Tunnelling Evolution of Nonrelativistic Particles and Photons,” Journal of Physical Studies (Ukraine), Vol. 6, No. 1, 2002, pp. 24-39.

[11] F. Cardone, R. Mignani, S. P. Maidanyuk and V. S. Olkhovsky, “Multiple Internal Reflections during Particle and Photon Tunneling,” Foundations of Physics Letters, Vol. 19, No. 5, 2006, pp. 441-457. doi:10.1007/s10702-006-0903-y

[12] V. S. Olkhovsky, M. V. Romanyuk, “Particle Tunneling and Scattering in a Three-Dimensional Potential with a Hard Core and an External Potential Barrier,” Nuclear Physics and Atomic Energy (Ukraine), Vol. 10, No. 3, 2009, pp. 273-281.

[13] R. Y. Chiao, P. G. Kwiat and A. M. Steinberg, “Analogies between Electron and Photon Tunneling: A Proposed Experiment to Measure Photon Tunneling Times,” Physica B, Vol. 175, 1991, pp. 257-262. doi:10.1016/0921-4526(91)90724-S

[14] A. M. Steinberg and R. Y. Chiao, “Tunneling Delay Times in One and Two Dimensions,” Physical Review, Vol. A, No. 49, 1994, pp.3283-3285.

[15] C. K. Carniglia and L. Mandel, “Phase-Shift Measurement of Evanescent Electromagnetic Waves,” Journal of the Optical Society of America, Vol. 61, No. 8, 1971, pp. 1035-1043. doi:10.1364/JOSA.61.001035

[16] S. Zhu, A. W. Yu, D. Hawley and R. Roy, “Frustrated total Internal Reflection: A Demonstration and Review,” American Journal of Physics, Vol. 54, No. 7, 1986, pp. 601-606. doi:10.1119/1.14514

[1] V. S. Olkhovsky and E. Recami, “Recent Developments in the Time Analysis of Tunnelling Processes,” Physics Reports, Vol. 219, No. 6, 1992, pp. 339-356. doi:10.1016/0370-1573(92)90015-R

[2] R. Landauer and M. Martin, “Barrier Interaction Time in Tunneling,” Reviews of Modern Physics, Vol. 66, No. 1, 1994, pp. 217-228. doi:10.1103/RevModPhys.66.217

[3] V. S. Olkhovsky, E. Recami and J. Jakiel, “Unified Time Analysis of Photon and Particle Tunnelling,” Physics Reports, Vol. 398, No. 3, 2004, pp. 133-178. doi:10.1016/j.physrep.2004.06.001

[4] V. S. Olkhovsky and A. Agresti, “Developments in Time Analysis of Tunnelling Processes,” In: D. Mugnai, A. Ranfagni and L. S. Schulman, Eds., Proc. Adriatico Res. Conf. on Tunneling and Its Implications, Trieste, 30 July- 2 August 1996, World Scientific, Singapore, 1997, pp. 327-355.

[5] V. S. Olkhovsky, “Time Analysis of Tunnelling of Particles and Photons,” Physics of the Alive, Vol. 5, No. 1, 1997, pp. 23-41.

[6] V. S. Olkhovsky, V. Petrillo and A. K. Zaichenko, “Decrease of the Tunneling Time and Violation of the Hartman Effect for Large Barriers,” Physical Review, Vol. A70, No. 1, 2004, pp. 034103-1-4.

[7] J. H. Fermor, “Quantum-Mechanical Tunneling,” American Journal of Physics, Vol. 34, No. 12, 1966, pp. 1168- 1170. doi:10.1119/1.1972543

[8] K. W. McVoy, L. Heller and M. Bolsterli, “Optical Analysis of Potential Well Resonances,” Reviews of Modern Physics, Vol. 39, 1967, pp. 245-258. doi:10.1103/RevModPhys.39.245

[9] A. Anderson, “Multiple Scattering Approach to One- Dimensional Potential Problems,” American Journal of Physics, Vol. 57, No. 3, 1989, pp. 230-235. doi:10.1119/1.16095

[10] S. P. Maydanyuk, V. S. Olkhovsky and A. K. Zaichenko, “Multiple Internal Reflections Method in the Description of Tunnelling Evolution of Nonrelativistic Particles and Photons,” Journal of Physical Studies (Ukraine), Vol. 6, No. 1, 2002, pp. 24-39.

[11] F. Cardone, R. Mignani, S. P. Maidanyuk and V. S. Olkhovsky, “Multiple Internal Reflections during Particle and Photon Tunneling,” Foundations of Physics Letters, Vol. 19, No. 5, 2006, pp. 441-457. doi:10.1007/s10702-006-0903-y

[12] V. S. Olkhovsky, M. V. Romanyuk, “Particle Tunneling and Scattering in a Three-Dimensional Potential with a Hard Core and an External Potential Barrier,” Nuclear Physics and Atomic Energy (Ukraine), Vol. 10, No. 3, 2009, pp. 273-281.

[13] R. Y. Chiao, P. G. Kwiat and A. M. Steinberg, “Analogies between Electron and Photon Tunneling: A Proposed Experiment to Measure Photon Tunneling Times,” Physica B, Vol. 175, 1991, pp. 257-262. doi:10.1016/0921-4526(91)90724-S

[14] A. M. Steinberg and R. Y. Chiao, “Tunneling Delay Times in One and Two Dimensions,” Physical Review, Vol. A, No. 49, 1994, pp.3283-3285.

[15] C. K. Carniglia and L. Mandel, “Phase-Shift Measurement of Evanescent Electromagnetic Waves,” Journal of the Optical Society of America, Vol. 61, No. 8, 1971, pp. 1035-1043. doi:10.1364/JOSA.61.001035

[16] S. Zhu, A. W. Yu, D. Hawley and R. Roy, “Frustrated total Internal Reflection: A Demonstration and Review,” American Journal of Physics, Vol. 54, No. 7, 1986, pp. 601-606. doi:10.1119/1.14514