Abstract

We present a novel approach to study transmission through waveguides in terms of optical streamlines. This theoretical framework combines the computational performance of beam propagation methods with the possibility to monitor the passage of light through the guiding medium by means of these sampler paths. In this way, not only can the optical flow along the waveguide be followed in detail, but also a fair estimate of the transmitted light (intensity) can be accounted for by counting streamline arrivals with starting points statistically distributed according to the input pulse. Furthermore, this approach allows elucidation of the mechanism leading to energy losses, namely, a vortical dynamics, which can be advantageously exploited in optimal waveguide design.

© 2012 Optical Society of America

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  1. J. S. Volker, Y. Ziliang, F. Rupert, W. Yuan, B. Guy, Y. Xiaobo, and Z. Xiang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
    [CrossRef]
  2. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
    [CrossRef]
  3. L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010).
    [CrossRef]
  4. S. Banerjee and A. Sharma, “Propagation characteristic of optical waveguiding structures by direct solution of the Helmholtz equation for total fields,” J. Opt. Soc. Am. A 6, 1884–1894 (1989).
    [CrossRef]
  5. R. Scarmozzino, A. Gopinath, R. Pregla, and S. Herlfert, “Numerical techniques for modeling guided-wave photonic devices,” J. Opt. Soc. Am. A 6, 150–162 (2000).
    [CrossRef]
  6. M. Born and E. Wolf, Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), 7th ed.
  7. J. Campos-Martíınez and R. D. Coalson, “The wide-angle equation and its solution through the short-time iterative Lanczos method,” Appl. Opt. 42, 1732–1742 (2003).
    [CrossRef]
  8. H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge University, 1970).
  9. D. Gloge and D. Marcuse, “Formal quantum theory of light rays,” J. Opt. Soc. Am. 59, 1629–1631 (1969).
    [CrossRef]
  10. W. K. Kahn and S. Yang, “Hamiltonian analysis of beams in an optical slab guide,” J. Opt. Soc. Am. 73, 684–690 (1983).
    [CrossRef]
  11. M. D. Feit, J. J. A. Fleck, and A. Stieger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
    [CrossRef]
  12. M. D. Feit and J. J. A. Fleck, “Solution of the Schrödinger equation by a spectral method II: vibrational energy levels of triatomic molecules,” J. Chem. Phys. 78, 301–308 (1983).
    [CrossRef]
  13. C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
    [CrossRef]
  14. D. K. Pant, R. D. Coalson, M. I. Hernández, and J. Campos-Martínez, “Optimal control theory for the design of optical waveguides,” J. Lightwave Technol. 16, 292–300 (1998).
    [CrossRef]
  15. D. K. Pant, R. D. Coalson, M. I. Hernández, and J. Campos-Martínez, “Optimal control theory for optical waveguide design: application to Y-branch structures,” Appl. Opt. 38, 3917–3923 (1999).
    [CrossRef]
  16. S. Longhi, D. Janner, M. Marano, and P. Laporta, “Quantum-mechanical analogy of beam propagation in waveguides with a bent axis: dynamic-model stabilization and radiation-loss suppression,” Phys. Rev. E 67, 036601 (2003).
    [CrossRef]
  17. M. Davidović, A. S. Sanz, D. Arsenović, M. Božić, and S. Miret-Artés, “Electromagnetic energy flow lines as possible paths of photons,” Phys. Scr. T135, 014009 (2009).
    [CrossRef]
  18. A. S. Sanz, M. Davidović, M. Božić, and S. Miret-Artés, “Understanding interference experiments with polarized light through photon trajectories,” Ann. Phys. 325, 763–784 (2010).
    [CrossRef]
  19. M. Božić, M. Davidović, T. L. Dimitrova, S. Miret-Artés, A. S. Sanz, and A. Weis, “Generalized Arago–Fresnel laws: the eme-flow-line description,” J. Russ. Laser Res. 31, 117–128 (2010).
    [CrossRef]
  20. D. Bohm, “A suggested interpretation of the quantum theory in terms of ‘hidden’ variables. I,” Phys. Rev. 85, 166–179 (1952).
    [CrossRef]
  21. P. R. Holland, The Quantum Theory of Motion (Cambridge University, 1993).
  22. T. L. Dimitrova and A. Weis, “The wave-particle duality of light: a demonstration experiment,” Am. J. Phys. 76, 137–142 (2008).
    [CrossRef]
  23. T. L. Dimitrova and A. Weis, “Lecture demonstrations of interference and quantum erasing with single photons,” Phys. Scr. T135, 014003 (2009).
    [CrossRef]
  24. T. L. Dimitrova and A. Weis, “Single photon quantum erasing: a demonstration experiment,” Eur. J. Phys. 31, 625–637 (2010).
    [CrossRef]
  25. S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
    [CrossRef]
  26. R. D. Coalson, D. K. Pant, A. Ali, and D. W. Langer, “Computing the eigenmodes of lossy field-induced optical waveguides,” J. Lightwave Technol. 12, 1015–1022 (1994).
    [CrossRef]
  27. D.-S. Min, D. W. Langer, D. K. Pant, and R. D. Coalson, “Numerical techniques for modeling guided-wave photonic devices,” Fiber Integr. Opt. 16, 331–342 (1997).
    [CrossRef]
  28. P. K. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
    [CrossRef]
  29. P. K. Tien, “Rules of refractive index and a potential well model of the optical waveguides,” Radio Sci. 16, 437–444 (1981).
    [CrossRef]
  30. E. Madelung, “Quantentheorie in hydrodynamischer Form,” Z. Phys. 40, 322–326 (1926).
    [CrossRef]
  31. I. Bialynicki-Birula, M. Cieplak, and J. Kaminski, Theory of Quanta (Oxford University, 1992).
  32. I. Bialynicki-Birula and Z. Bialynicka-Birula, “Magnetic monopoles in the hydrodynamic formulation of quantum mechanics,” Phys. Rev. D 3, 2410–2412 (1971).
    [CrossRef]
  33. J. O. Hirschfelder, A. C. Christoph, and W. E. Palke, “Quantum mechanical streamlines. I. Square potential barrier,” J. Chem. Phys. 61, 5435–5455 (1974).
    [CrossRef]
  34. J. O. Hirschfelder, C. J. Goebel, and L. W. Bruch, “Quantized vortices around wavefunction nodes. II,” J. Chem. Phys. 61, 5456–5459 (1974).
    [CrossRef]
  35. J. O. Hirschfelder and K. T. Tang, “Quantum mechanical streamlines. III. Idealized reactive atom-diatomic molecule collision,” J. Chem. Phys. 64, 760–785 (1976).
    [CrossRef]
  36. J. O. Hirschfelder and K. T. Tang, “Quantum mechanical streamlines. IV. Collision of two spheres with square potential wells or barriers,” J. Chem. Phys. 65, 470–486 (1976).
    [CrossRef]
  37. R. E. Wyatt, Quantum Dynamics with Trajectories (Springer, 2005).
  38. A. S. Sanz, F. Borondo, and S. Miret-Artés, “Particle diffraction studied using quantum trajectories,” J. Phys. Condens. Matter 14, 6109–6145 (2002).
    [CrossRef]
  39. A. S. Sanz, D. López-Durán, and T. González-Lezana, “Investigating transition state resonances in the time domain by means of Bohmian mechanics: the F + HD reaction,” Chem. Phys., doi:10.1016/j.chemphys.2011.07.017 (to be published).
    [CrossRef]
  40. A. S. Sanz and S. Miret-Artés, “A trajectory-based understanding of quantum interference,” J. Phys. A 41, 435303 (2008).
    [CrossRef]
  41. A. S. Sanz and S. Miret-Artés, “Determining final probabilities directly from the initial state,” http://arxiv.org/abs/1112.3830 .

2011

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef]

J. S. Volker, Y. Ziliang, F. Rupert, W. Yuan, B. Guy, Y. Xiaobo, and Z. Xiang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

2010

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010).
[CrossRef]

A. S. Sanz, M. Davidović, M. Božić, and S. Miret-Artés, “Understanding interference experiments with polarized light through photon trajectories,” Ann. Phys. 325, 763–784 (2010).
[CrossRef]

M. Božić, M. Davidović, T. L. Dimitrova, S. Miret-Artés, A. S. Sanz, and A. Weis, “Generalized Arago–Fresnel laws: the eme-flow-line description,” J. Russ. Laser Res. 31, 117–128 (2010).
[CrossRef]

T. L. Dimitrova and A. Weis, “Single photon quantum erasing: a demonstration experiment,” Eur. J. Phys. 31, 625–637 (2010).
[CrossRef]

2009

M. Davidović, A. S. Sanz, D. Arsenović, M. Božić, and S. Miret-Artés, “Electromagnetic energy flow lines as possible paths of photons,” Phys. Scr. T135, 014009 (2009).
[CrossRef]

T. L. Dimitrova and A. Weis, “Lecture demonstrations of interference and quantum erasing with single photons,” Phys. Scr. T135, 014003 (2009).
[CrossRef]

2008

T. L. Dimitrova and A. Weis, “The wave-particle duality of light: a demonstration experiment,” Am. J. Phys. 76, 137–142 (2008).
[CrossRef]

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[CrossRef]

A. S. Sanz and S. Miret-Artés, “A trajectory-based understanding of quantum interference,” J. Phys. A 41, 435303 (2008).
[CrossRef]

2003

J. Campos-Martíınez and R. D. Coalson, “The wide-angle equation and its solution through the short-time iterative Lanczos method,” Appl. Opt. 42, 1732–1742 (2003).
[CrossRef]

S. Longhi, D. Janner, M. Marano, and P. Laporta, “Quantum-mechanical analogy of beam propagation in waveguides with a bent axis: dynamic-model stabilization and radiation-loss suppression,” Phys. Rev. E 67, 036601 (2003).
[CrossRef]

2002

A. S. Sanz, F. Borondo, and S. Miret-Artés, “Particle diffraction studied using quantum trajectories,” J. Phys. Condens. Matter 14, 6109–6145 (2002).
[CrossRef]

2000

1999

1998

1997

D.-S. Min, D. W. Langer, D. K. Pant, and R. D. Coalson, “Numerical techniques for modeling guided-wave photonic devices,” Fiber Integr. Opt. 16, 331–342 (1997).
[CrossRef]

1994

R. D. Coalson, D. K. Pant, A. Ali, and D. W. Langer, “Computing the eigenmodes of lossy field-induced optical waveguides,” J. Lightwave Technol. 12, 1015–1022 (1994).
[CrossRef]

1991

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

1989

1983

M. D. Feit and J. J. A. Fleck, “Solution of the Schrödinger equation by a spectral method II: vibrational energy levels of triatomic molecules,” J. Chem. Phys. 78, 301–308 (1983).
[CrossRef]

W. K. Kahn and S. Yang, “Hamiltonian analysis of beams in an optical slab guide,” J. Opt. Soc. Am. 73, 684–690 (1983).
[CrossRef]

1982

M. D. Feit, J. J. A. Fleck, and A. Stieger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
[CrossRef]

1981

P. K. Tien, “Rules of refractive index and a potential well model of the optical waveguides,” Radio Sci. 16, 437–444 (1981).
[CrossRef]

1977

P. K. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[CrossRef]

1976

J. O. Hirschfelder and K. T. Tang, “Quantum mechanical streamlines. III. Idealized reactive atom-diatomic molecule collision,” J. Chem. Phys. 64, 760–785 (1976).
[CrossRef]

J. O. Hirschfelder and K. T. Tang, “Quantum mechanical streamlines. IV. Collision of two spheres with square potential wells or barriers,” J. Chem. Phys. 65, 470–486 (1976).
[CrossRef]

1974

J. O. Hirschfelder, A. C. Christoph, and W. E. Palke, “Quantum mechanical streamlines. I. Square potential barrier,” J. Chem. Phys. 61, 5435–5455 (1974).
[CrossRef]

J. O. Hirschfelder, C. J. Goebel, and L. W. Bruch, “Quantized vortices around wavefunction nodes. II,” J. Chem. Phys. 61, 5456–5459 (1974).
[CrossRef]

1971

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Magnetic monopoles in the hydrodynamic formulation of quantum mechanics,” Phys. Rev. D 3, 2410–2412 (1971).
[CrossRef]

1969

1952

D. Bohm, “A suggested interpretation of the quantum theory in terms of ‘hidden’ variables. I,” Phys. Rev. 85, 166–179 (1952).
[CrossRef]

1926

E. Madelung, “Quantentheorie in hydrodynamischer Form,” Z. Phys. 40, 322–326 (1926).
[CrossRef]

Ali, A.

R. D. Coalson, D. K. Pant, A. Ali, and D. W. Langer, “Computing the eigenmodes of lossy field-induced optical waveguides,” J. Lightwave Technol. 12, 1015–1022 (1994).
[CrossRef]

Arsenovic, D.

M. Davidović, A. S. Sanz, D. Arsenović, M. Božić, and S. Miret-Artés, “Electromagnetic energy flow lines as possible paths of photons,” Phys. Scr. T135, 014009 (2009).
[CrossRef]

Banerjee, S.

Bialynicka-Birula, Z.

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Magnetic monopoles in the hydrodynamic formulation of quantum mechanics,” Phys. Rev. D 3, 2410–2412 (1971).
[CrossRef]

Bialynicki-Birula, I.

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Magnetic monopoles in the hydrodynamic formulation of quantum mechanics,” Phys. Rev. D 3, 2410–2412 (1971).
[CrossRef]

I. Bialynicki-Birula, M. Cieplak, and J. Kaminski, Theory of Quanta (Oxford University, 1992).

Bisseling, R. H.

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

Bohm, D.

D. Bohm, “A suggested interpretation of the quantum theory in terms of ‘hidden’ variables. I,” Phys. Rev. 85, 166–179 (1952).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), 7th ed.

Borondo, F.

A. S. Sanz, F. Borondo, and S. Miret-Artés, “Particle diffraction studied using quantum trajectories,” J. Phys. Condens. Matter 14, 6109–6145 (2002).
[CrossRef]

Božic, M.

A. S. Sanz, M. Davidović, M. Božić, and S. Miret-Artés, “Understanding interference experiments with polarized light through photon trajectories,” Ann. Phys. 325, 763–784 (2010).
[CrossRef]

M. Božić, M. Davidović, T. L. Dimitrova, S. Miret-Artés, A. S. Sanz, and A. Weis, “Generalized Arago–Fresnel laws: the eme-flow-line description,” J. Russ. Laser Res. 31, 117–128 (2010).
[CrossRef]

M. Davidović, A. S. Sanz, D. Arsenović, M. Božić, and S. Miret-Artés, “Electromagnetic energy flow lines as possible paths of photons,” Phys. Scr. T135, 014009 (2009).
[CrossRef]

Braverman, B.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef]

Bruch, L. W.

J. O. Hirschfelder, C. J. Goebel, and L. W. Bruch, “Quantized vortices around wavefunction nodes. II,” J. Chem. Phys. 61, 5456–5459 (1974).
[CrossRef]

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge University, 1970).

Campos-Martíinez, J.

Campos-Martínez, J.

Cerjan, C.

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

Christoph, A. C.

J. O. Hirschfelder, A. C. Christoph, and W. E. Palke, “Quantum mechanical streamlines. I. Square potential barrier,” J. Chem. Phys. 61, 5435–5455 (1974).
[CrossRef]

Cieplak, M.

I. Bialynicki-Birula, M. Cieplak, and J. Kaminski, Theory of Quanta (Oxford University, 1992).

Coalson, R. D.

Crespi, A.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010).
[CrossRef]

Cryan, M. J.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[CrossRef]

Davidovic, M.

M. Božić, M. Davidović, T. L. Dimitrova, S. Miret-Artés, A. S. Sanz, and A. Weis, “Generalized Arago–Fresnel laws: the eme-flow-line description,” J. Russ. Laser Res. 31, 117–128 (2010).
[CrossRef]

A. S. Sanz, M. Davidović, M. Božić, and S. Miret-Artés, “Understanding interference experiments with polarized light through photon trajectories,” Ann. Phys. 325, 763–784 (2010).
[CrossRef]

M. Davidović, A. S. Sanz, D. Arsenović, M. Božić, and S. Miret-Artés, “Electromagnetic energy flow lines as possible paths of photons,” Phys. Scr. T135, 014009 (2009).
[CrossRef]

Dimitrova, T. L.

T. L. Dimitrova and A. Weis, “Single photon quantum erasing: a demonstration experiment,” Eur. J. Phys. 31, 625–637 (2010).
[CrossRef]

M. Božić, M. Davidović, T. L. Dimitrova, S. Miret-Artés, A. S. Sanz, and A. Weis, “Generalized Arago–Fresnel laws: the eme-flow-line description,” J. Russ. Laser Res. 31, 117–128 (2010).
[CrossRef]

T. L. Dimitrova and A. Weis, “Lecture demonstrations of interference and quantum erasing with single photons,” Phys. Scr. T135, 014003 (2009).
[CrossRef]

T. L. Dimitrova and A. Weis, “The wave-particle duality of light: a demonstration experiment,” Am. J. Phys. 76, 137–142 (2008).
[CrossRef]

Feit, M. D.

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

M. D. Feit and J. J. A. Fleck, “Solution of the Schrödinger equation by a spectral method II: vibrational energy levels of triatomic molecules,” J. Chem. Phys. 78, 301–308 (1983).
[CrossRef]

M. D. Feit, J. J. A. Fleck, and A. Stieger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
[CrossRef]

Fleck, J. J. A.

M. D. Feit and J. J. A. Fleck, “Solution of the Schrödinger equation by a spectral method II: vibrational energy levels of triatomic molecules,” J. Chem. Phys. 78, 301–308 (1983).
[CrossRef]

M. D. Feit, J. J. A. Fleck, and A. Stieger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
[CrossRef]

Friesner, R.

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

Gloge, D.

Goebel, C. J.

J. O. Hirschfelder, C. J. Goebel, and L. W. Bruch, “Quantized vortices around wavefunction nodes. II,” J. Chem. Phys. 61, 5456–5459 (1974).
[CrossRef]

González-Lezana, T.

A. S. Sanz, D. López-Durán, and T. González-Lezana, “Investigating transition state resonances in the time domain by means of Bohmian mechanics: the F + HD reaction,” Chem. Phys., doi:10.1016/j.chemphys.2011.07.017 (to be published).
[CrossRef]

Gopinath, A.

Guldberg, A.

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

Guy, B.

J. S. Volker, Y. Ziliang, F. Rupert, W. Yuan, B. Guy, Y. Xiaobo, and Z. Xiang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

Hammerich, A.

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

Herlfert, S.

Hernández, M. I.

Hirschfelder, J. O.

J. O. Hirschfelder and K. T. Tang, “Quantum mechanical streamlines. IV. Collision of two spheres with square potential wells or barriers,” J. Chem. Phys. 65, 470–486 (1976).
[CrossRef]

J. O. Hirschfelder and K. T. Tang, “Quantum mechanical streamlines. III. Idealized reactive atom-diatomic molecule collision,” J. Chem. Phys. 64, 760–785 (1976).
[CrossRef]

J. O. Hirschfelder, A. C. Christoph, and W. E. Palke, “Quantum mechanical streamlines. I. Square potential barrier,” J. Chem. Phys. 61, 5435–5455 (1974).
[CrossRef]

J. O. Hirschfelder, C. J. Goebel, and L. W. Bruch, “Quantized vortices around wavefunction nodes. II,” J. Chem. Phys. 61, 5456–5459 (1974).
[CrossRef]

Holland, P. R.

P. R. Holland, The Quantum Theory of Motion (Cambridge University, 1993).

Janner, D.

S. Longhi, D. Janner, M. Marano, and P. Laporta, “Quantum-mechanical analogy of beam propagation in waveguides with a bent axis: dynamic-model stabilization and radiation-loss suppression,” Phys. Rev. E 67, 036601 (2003).
[CrossRef]

Jolicard, G.

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

Kahn, W. K.

Kaminski, J.

I. Bialynicki-Birula, M. Cieplak, and J. Kaminski, Theory of Quanta (Oxford University, 1992).

Karrlein, W.

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

Kocsis, S.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef]

Kosloff, R.

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

Langer, D. W.

D.-S. Min, D. W. Langer, D. K. Pant, and R. D. Coalson, “Numerical techniques for modeling guided-wave photonic devices,” Fiber Integr. Opt. 16, 331–342 (1997).
[CrossRef]

R. D. Coalson, D. K. Pant, A. Ali, and D. W. Langer, “Computing the eigenmodes of lossy field-induced optical waveguides,” J. Lightwave Technol. 12, 1015–1022 (1994).
[CrossRef]

Laporta, P.

S. Longhi, D. Janner, M. Marano, and P. Laporta, “Quantum-mechanical analogy of beam propagation in waveguides with a bent axis: dynamic-model stabilization and radiation-loss suppression,” Phys. Rev. E 67, 036601 (2003).
[CrossRef]

Leforestier, C.

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

Lipkin, N.

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

Longhi, S.

S. Longhi, D. Janner, M. Marano, and P. Laporta, “Quantum-mechanical analogy of beam propagation in waveguides with a bent axis: dynamic-model stabilization and radiation-loss suppression,” Phys. Rev. E 67, 036601 (2003).
[CrossRef]

López-Durán, D.

A. S. Sanz, D. López-Durán, and T. González-Lezana, “Investigating transition state resonances in the time domain by means of Bohmian mechanics: the F + HD reaction,” Chem. Phys., doi:10.1016/j.chemphys.2011.07.017 (to be published).
[CrossRef]

Madelung, E.

E. Madelung, “Quantentheorie in hydrodynamischer Form,” Z. Phys. 40, 322–326 (1926).
[CrossRef]

Marano, M.

S. Longhi, D. Janner, M. Marano, and P. Laporta, “Quantum-mechanical analogy of beam propagation in waveguides with a bent axis: dynamic-model stabilization and radiation-loss suppression,” Phys. Rev. E 67, 036601 (2003).
[CrossRef]

Marcuse, D.

Mataloni, P.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010).
[CrossRef]

Meyer, H.-D.

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

Min, D.-S.

D.-S. Min, D. W. Langer, D. K. Pant, and R. D. Coalson, “Numerical techniques for modeling guided-wave photonic devices,” Fiber Integr. Opt. 16, 331–342 (1997).
[CrossRef]

Miret-Artés, S.

M. Božić, M. Davidović, T. L. Dimitrova, S. Miret-Artés, A. S. Sanz, and A. Weis, “Generalized Arago–Fresnel laws: the eme-flow-line description,” J. Russ. Laser Res. 31, 117–128 (2010).
[CrossRef]

A. S. Sanz, M. Davidović, M. Božić, and S. Miret-Artés, “Understanding interference experiments with polarized light through photon trajectories,” Ann. Phys. 325, 763–784 (2010).
[CrossRef]

M. Davidović, A. S. Sanz, D. Arsenović, M. Božić, and S. Miret-Artés, “Electromagnetic energy flow lines as possible paths of photons,” Phys. Scr. T135, 014009 (2009).
[CrossRef]

A. S. Sanz and S. Miret-Artés, “A trajectory-based understanding of quantum interference,” J. Phys. A 41, 435303 (2008).
[CrossRef]

A. S. Sanz, F. Borondo, and S. Miret-Artés, “Particle diffraction studied using quantum trajectories,” J. Phys. Condens. Matter 14, 6109–6145 (2002).
[CrossRef]

Mirin, R. P.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef]

O’Brien, J. L.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[CrossRef]

Osellame, R.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010).
[CrossRef]

Palke, W. E.

J. O. Hirschfelder, A. C. Christoph, and W. E. Palke, “Quantum mechanical streamlines. I. Square potential barrier,” J. Chem. Phys. 61, 5435–5455 (1974).
[CrossRef]

Pant, D. K.

D. K. Pant, R. D. Coalson, M. I. Hernández, and J. Campos-Martínez, “Optimal control theory for optical waveguide design: application to Y-branch structures,” Appl. Opt. 38, 3917–3923 (1999).
[CrossRef]

D. K. Pant, R. D. Coalson, M. I. Hernández, and J. Campos-Martínez, “Optimal control theory for the design of optical waveguides,” J. Lightwave Technol. 16, 292–300 (1998).
[CrossRef]

D.-S. Min, D. W. Langer, D. K. Pant, and R. D. Coalson, “Numerical techniques for modeling guided-wave photonic devices,” Fiber Integr. Opt. 16, 331–342 (1997).
[CrossRef]

R. D. Coalson, D. K. Pant, A. Ali, and D. W. Langer, “Computing the eigenmodes of lossy field-induced optical waveguides,” J. Lightwave Technol. 12, 1015–1022 (1994).
[CrossRef]

Politi, A.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[CrossRef]

Pregla, R.

Ramponi, R.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010).
[CrossRef]

Rarity, J. G.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[CrossRef]

Ravets, S.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef]

Roncero, O.

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

Rupert, F.

J. S. Volker, Y. Ziliang, F. Rupert, W. Yuan, B. Guy, Y. Xiaobo, and Z. Xiang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

Sansoni, L.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010).
[CrossRef]

Sanz, A. S.

A. S. Sanz, M. Davidović, M. Božić, and S. Miret-Artés, “Understanding interference experiments with polarized light through photon trajectories,” Ann. Phys. 325, 763–784 (2010).
[CrossRef]

M. Božić, M. Davidović, T. L. Dimitrova, S. Miret-Artés, A. S. Sanz, and A. Weis, “Generalized Arago–Fresnel laws: the eme-flow-line description,” J. Russ. Laser Res. 31, 117–128 (2010).
[CrossRef]

M. Davidović, A. S. Sanz, D. Arsenović, M. Božić, and S. Miret-Artés, “Electromagnetic energy flow lines as possible paths of photons,” Phys. Scr. T135, 014009 (2009).
[CrossRef]

A. S. Sanz and S. Miret-Artés, “A trajectory-based understanding of quantum interference,” J. Phys. A 41, 435303 (2008).
[CrossRef]

A. S. Sanz, F. Borondo, and S. Miret-Artés, “Particle diffraction studied using quantum trajectories,” J. Phys. Condens. Matter 14, 6109–6145 (2002).
[CrossRef]

A. S. Sanz, D. López-Durán, and T. González-Lezana, “Investigating transition state resonances in the time domain by means of Bohmian mechanics: the F + HD reaction,” Chem. Phys., doi:10.1016/j.chemphys.2011.07.017 (to be published).
[CrossRef]

Scarmozzino, R.

Sciarrino, F.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010).
[CrossRef]

Shalm, L. K.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef]

Sharma, A.

Steinberg, A. M.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef]

Stevens, M. J.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef]

Stieger, A.

M. D. Feit, J. J. A. Fleck, and A. Stieger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
[CrossRef]

Tang, K. T.

J. O. Hirschfelder and K. T. Tang, “Quantum mechanical streamlines. IV. Collision of two spheres with square potential wells or barriers,” J. Chem. Phys. 65, 470–486 (1976).
[CrossRef]

J. O. Hirschfelder and K. T. Tang, “Quantum mechanical streamlines. III. Idealized reactive atom-diatomic molecule collision,” J. Chem. Phys. 64, 760–785 (1976).
[CrossRef]

Tien, P. K.

P. K. Tien, “Rules of refractive index and a potential well model of the optical waveguides,” Radio Sci. 16, 437–444 (1981).
[CrossRef]

P. K. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[CrossRef]

Vallone, G.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010).
[CrossRef]

Volker, J. S.

J. S. Volker, Y. Ziliang, F. Rupert, W. Yuan, B. Guy, Y. Xiaobo, and Z. Xiang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

Weis, A.

M. Božić, M. Davidović, T. L. Dimitrova, S. Miret-Artés, A. S. Sanz, and A. Weis, “Generalized Arago–Fresnel laws: the eme-flow-line description,” J. Russ. Laser Res. 31, 117–128 (2010).
[CrossRef]

T. L. Dimitrova and A. Weis, “Single photon quantum erasing: a demonstration experiment,” Eur. J. Phys. 31, 625–637 (2010).
[CrossRef]

T. L. Dimitrova and A. Weis, “Lecture demonstrations of interference and quantum erasing with single photons,” Phys. Scr. T135, 014003 (2009).
[CrossRef]

T. L. Dimitrova and A. Weis, “The wave-particle duality of light: a demonstration experiment,” Am. J. Phys. 76, 137–142 (2008).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), 7th ed.

Wyatt, R. E.

R. E. Wyatt, Quantum Dynamics with Trajectories (Springer, 2005).

Xiang, Z.

J. S. Volker, Y. Ziliang, F. Rupert, W. Yuan, B. Guy, Y. Xiaobo, and Z. Xiang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

Xiaobo, Y.

J. S. Volker, Y. Ziliang, F. Rupert, W. Yuan, B. Guy, Y. Xiaobo, and Z. Xiang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

Yang, S.

Yu, S.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[CrossRef]

Yuan, W.

J. S. Volker, Y. Ziliang, F. Rupert, W. Yuan, B. Guy, Y. Xiaobo, and Z. Xiang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

Ziliang, Y.

J. S. Volker, Y. Ziliang, F. Rupert, W. Yuan, B. Guy, Y. Xiaobo, and Z. Xiang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

Am. J. Phys.

T. L. Dimitrova and A. Weis, “The wave-particle duality of light: a demonstration experiment,” Am. J. Phys. 76, 137–142 (2008).
[CrossRef]

Ann. Phys.

A. S. Sanz, M. Davidović, M. Božić, and S. Miret-Artés, “Understanding interference experiments with polarized light through photon trajectories,” Ann. Phys. 325, 763–784 (2010).
[CrossRef]

Appl. Opt.

Chem. Phys.

A. S. Sanz, D. López-Durán, and T. González-Lezana, “Investigating transition state resonances in the time domain by means of Bohmian mechanics: the F + HD reaction,” Chem. Phys., doi:10.1016/j.chemphys.2011.07.017 (to be published).
[CrossRef]

Eur. J. Phys.

T. L. Dimitrova and A. Weis, “Single photon quantum erasing: a demonstration experiment,” Eur. J. Phys. 31, 625–637 (2010).
[CrossRef]

Fiber Integr. Opt.

D.-S. Min, D. W. Langer, D. K. Pant, and R. D. Coalson, “Numerical techniques for modeling guided-wave photonic devices,” Fiber Integr. Opt. 16, 331–342 (1997).
[CrossRef]

J. Chem. Phys.

J. O. Hirschfelder, A. C. Christoph, and W. E. Palke, “Quantum mechanical streamlines. I. Square potential barrier,” J. Chem. Phys. 61, 5435–5455 (1974).
[CrossRef]

J. O. Hirschfelder, C. J. Goebel, and L. W. Bruch, “Quantized vortices around wavefunction nodes. II,” J. Chem. Phys. 61, 5456–5459 (1974).
[CrossRef]

J. O. Hirschfelder and K. T. Tang, “Quantum mechanical streamlines. III. Idealized reactive atom-diatomic molecule collision,” J. Chem. Phys. 64, 760–785 (1976).
[CrossRef]

J. O. Hirschfelder and K. T. Tang, “Quantum mechanical streamlines. IV. Collision of two spheres with square potential wells or barriers,” J. Chem. Phys. 65, 470–486 (1976).
[CrossRef]

M. D. Feit and J. J. A. Fleck, “Solution of the Schrödinger equation by a spectral method II: vibrational energy levels of triatomic molecules,” J. Chem. Phys. 78, 301–308 (1983).
[CrossRef]

J. Comp. Phys.

C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comp. Phys. 94, 59–80 (1991).
[CrossRef]

J. Comput. Phys.

M. D. Feit, J. J. A. Fleck, and A. Stieger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
[CrossRef]

J. Lightwave Technol.

R. D. Coalson, D. K. Pant, A. Ali, and D. W. Langer, “Computing the eigenmodes of lossy field-induced optical waveguides,” J. Lightwave Technol. 12, 1015–1022 (1994).
[CrossRef]

D. K. Pant, R. D. Coalson, M. I. Hernández, and J. Campos-Martínez, “Optimal control theory for the design of optical waveguides,” J. Lightwave Technol. 16, 292–300 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

A. S. Sanz and S. Miret-Artés, “A trajectory-based understanding of quantum interference,” J. Phys. A 41, 435303 (2008).
[CrossRef]

J. Phys. Condens. Matter

A. S. Sanz, F. Borondo, and S. Miret-Artés, “Particle diffraction studied using quantum trajectories,” J. Phys. Condens. Matter 14, 6109–6145 (2002).
[CrossRef]

J. Russ. Laser Res.

M. Božić, M. Davidović, T. L. Dimitrova, S. Miret-Artés, A. S. Sanz, and A. Weis, “Generalized Arago–Fresnel laws: the eme-flow-line description,” J. Russ. Laser Res. 31, 117–128 (2010).
[CrossRef]

Nat. Commun.

J. S. Volker, Y. Ziliang, F. Rupert, W. Yuan, B. Guy, Y. Xiaobo, and Z. Xiang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[CrossRef]

Phys. Rev.

D. Bohm, “A suggested interpretation of the quantum theory in terms of ‘hidden’ variables. I,” Phys. Rev. 85, 166–179 (1952).
[CrossRef]

Phys. Rev. D

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Magnetic monopoles in the hydrodynamic formulation of quantum mechanics,” Phys. Rev. D 3, 2410–2412 (1971).
[CrossRef]

Phys. Rev. E

S. Longhi, D. Janner, M. Marano, and P. Laporta, “Quantum-mechanical analogy of beam propagation in waveguides with a bent axis: dynamic-model stabilization and radiation-loss suppression,” Phys. Rev. E 67, 036601 (2003).
[CrossRef]

Phys. Rev. Lett.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010).
[CrossRef]

Phys. Scr.

M. Davidović, A. S. Sanz, D. Arsenović, M. Božić, and S. Miret-Artés, “Electromagnetic energy flow lines as possible paths of photons,” Phys. Scr. T135, 014009 (2009).
[CrossRef]

T. L. Dimitrova and A. Weis, “Lecture demonstrations of interference and quantum erasing with single photons,” Phys. Scr. T135, 014003 (2009).
[CrossRef]

Radio Sci.

P. K. Tien, “Rules of refractive index and a potential well model of the optical waveguides,” Radio Sci. 16, 437–444 (1981).
[CrossRef]

Rev. Mod. Phys.

P. K. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[CrossRef]

Science

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
[CrossRef]

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[CrossRef]

Z. Phys.

E. Madelung, “Quantentheorie in hydrodynamischer Form,” Z. Phys. 40, 322–326 (1926).
[CrossRef]

Other

I. Bialynicki-Birula, M. Cieplak, and J. Kaminski, Theory of Quanta (Oxford University, 1992).

P. R. Holland, The Quantum Theory of Motion (Cambridge University, 1993).

A. S. Sanz and S. Miret-Artés, “Determining final probabilities directly from the initial state,” http://arxiv.org/abs/1112.3830 .

R. E. Wyatt, Quantum Dynamics with Trajectories (Springer, 2005).

M. Born and E. Wolf, Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), 7th ed.

H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge University, 1970).

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Figures (4)

Fig. 1.
Fig. 1.

Geometrical layout of the Y-junction design considered here, where the significant (geometrical) parameters are also displayed (see Table 1).

Fig. 2.
Fig. 2.

(a) Initial and (b) final intensity transmitted through the waveguide. In each panel, black squares denote the optical streamline calculation (histogram), while the red solid line accounts for the value obtained from the standard optical calculation (paraxial equation). The parameters used in these calculations are given in Table 1.

Fig. 3.
Fig. 3.

Three-dimensional representation showing the evolution of the initial pulse ρ ( x , 0 ) as it propagates (along the z coordinate) throughout the waveguide (below, the corresponding contour plot is also displayed). In particular, ρ ( x , z ) has been obtained as a histogramlike distribution of optical streamlines at each z value, which corresponds fairly well with the representation one would find by standard wave propagation methods (see text for details). The parameters used in these calculations are given in Table 1.

Fig. 4.
Fig. 4.

(a) Optical streamlines and (b) contour-plot of the evolution of the histogram illustrating the flow of light throughout the waveguide. In both figures, the straight black lines mark the waveguide boundaries (see Fig. 1). In part (a), blue and red lines are used to emphasize the fact that optical streamlines also satisfy the noncrossing rule that characterizes both quantum [40] and optical fluxes [18]. In part (b), the maximum value of the contours is set up to 0.1 in order to appreciate the outgoing light flow. White labels correspond to different vortices that appear during the evolution of the flow (see text for details).

Tables (1)

Tables Icon

Table 1. Values of the Different Parameters Considered in the Design of the Waveguide Used in the Calculations Shown Here (See Text for Details)

Equations (14)

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

2 Φ + n 2 k 2 Φ = 0 ,
Φ ( r ) = ϕ ( r ) e i k z z .
2 i k z ϕ z + 2 ϕ z 2 = 2 ϕ + ( k z 2 k 2 n 2 ) ϕ .
2 i β 0 ϕ z = 2 ϕ + ( β 0 2 k 2 n 2 ) ϕ .
i ϕ z = [ 1 2 k n 0 2 + V ( r ) ] ϕ ,
V ( r ) = k 2 n 0 [ n 0 2 n 2 ( r ) ]
ϕ ( R , z ) = ρ 1 / 2 ( R , z ) e i S ( R , z ) ,
ρ z + · J = 0 ,
S z + ( S ) 2 2 μ + V eff = 0 .
v = S μ = 2 μ i ( ϕ * ϕ ϕ ϕ * ϕ * ϕ ) .
Q 1 2 μ 2 ρ 1 / 2 ρ 1 / 2 = 1 4 μ [ 1 2 ( ρ ρ ) 2 2 ρ ρ ] ,
V = k 2 n 0 ( n 0 2 n 1 2 ) ,
V = k 2 n 0 ( n 0 2 n 2 2 ) .
ϕ ( x , z ) = cos ( ω z ) φ 1 ( x ) + sin ( ω z ) φ 2 ( x ) .

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