Abstract

Partial reflection of linearly polarized Laguerre–Gaussian beams incident at a dielectric interface are studied beyond the paraxial regime. Based on the angular spectrum method and Taylor series expansion, we derive exact analytical expressions for the reflected electric field. This result holds in both the paraxial and nonparaxial regimes. The result is then extended to beams of arbitrary polarization and used to analytically calculate the transverse and longitudinal shifts of the beams’ center of gravity. Finally, several numerical examples are performed to verify the analytical formulas we derived near the Brewster angle.

© 2013 Optical Society of America

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  1. F. Goos and H. Hanchen, “Ein neuer und fundamentaler Versuch zur total reflexion,” Ann. Phys. 436, 333–346 (1947).
    [CrossRef]
  2. F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465–467 (1955).
  3. C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
    [CrossRef]
  4. K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75, 066609 (2007).
    [CrossRef]
  5. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hanchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008).
    [CrossRef]
  6. M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3, 337–340 (2009).
    [CrossRef]
  7. K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
    [CrossRef]
  8. O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
    [CrossRef]
  9. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef]
  10. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, E. Wolf, ed., Vol. 39 (Elsevier, 1999), pp. 291–372.
  11. V. G. Fedoseyev, “Spin-independent transverse shift of the centre of gravity of a reflected and of a refracted light beam,” Opt. Commun. 193, 9–18 (2001).
    [CrossRef]
  12. R. Dasgupta and P. K. Gupta, “Experimental observation of spin-independent transverse shift of the centre of gravity of a reflected Laguerre-Gaussian light beam,” Opt. Commun. 257, 91–96 (2006).
    [CrossRef]
  13. H. Okuda and H. Sasada, “Huge transverse deformation in nonspecular reflection of a light beam possessing orbital angular momentum near critical incidence,” Opt. Express 14, 8393–8402 (2006).
    [CrossRef]
  14. H. Okuda and H. Sasada, “Significant deformations and propagation variations of Laguerre–Gaussian beams reflected and transmitted at a dielectric interface,” J. Opt. Soc. Am. A 25, 881–890 (2008).
    [CrossRef]
  15. M. Merano, N. Hermosa, and J. P. Woerdman, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
    [CrossRef]
  16. N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE 7950, 79500F (2011).
  17. N. Hermosa, A. Aiello, and J. P. Woerdman, “Radial mode dependence of optical beam shifts,” Opt. Lett. 37, 1044–1046 (2012).
    [CrossRef]
  18. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  19. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).
  20. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  21. G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. A 69, 575–578 (1979).
    [CrossRef]
  22. F. W. J. Olver and L. C. Maximon, “Bessel functions,” in NIST Handbook of Mathematical Functions, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds. (Cambridge University, 2010), p. 265.
  23. A. Cerjan and C. Cerjan, “Orbital angular momentum of Laguerre-Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 28, 2253–2260 (2011).
    [CrossRef]
  24. A. Aiello and J. P. Woerdman, “Theory of angular Goos-Hanchen shift near Brewster incidence,” arXiv:0903.3730v2 (2009).
  25. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201 (1972).
    [CrossRef]
  26. K. Y. Bliokh, I. V. Shadrivov, and Y. S. Kivshar, “Goos-Hanchen and Imbert-Fedorov shifts of polarized vortex beams,” Opt. Lett. 34, 389–391 (2009).
    [CrossRef]
  27. M. Merano, N. Hermosa, A. Aiello, and J. P. Woerdman, “Demonstration of a quasi-scalar angular Goos-Hanchen effect,” Opt. Lett. 35, 3562–3564 (2010).
    [CrossRef]
  28. J. Ou, Y. Jiang, F. Li, and L. Liu, “Shift of beam centroid of Laguerre-Gaussian beams reflected and refracted at a dielectric interface,” Acta Phys. Sin. 60, 114203 (2011) (in Chinese).
  29. L. Kong, X. Wang, S. Li, Y. Li, J. Chen, B. Gu, and H. Wang, “Spin Hall effect of reflected light from an air-glass interface around the Brewster’s angle,” Appl. Phys. Lett. 100, 071109 (2012).
    [CrossRef]

2012

N. Hermosa, A. Aiello, and J. P. Woerdman, “Radial mode dependence of optical beam shifts,” Opt. Lett. 37, 1044–1046 (2012).
[CrossRef]

L. Kong, X. Wang, S. Li, Y. Li, J. Chen, B. Gu, and H. Wang, “Spin Hall effect of reflected light from an air-glass interface around the Brewster’s angle,” Appl. Phys. Lett. 100, 071109 (2012).
[CrossRef]

2011

J. Ou, Y. Jiang, F. Li, and L. Liu, “Shift of beam centroid of Laguerre-Gaussian beams reflected and refracted at a dielectric interface,” Acta Phys. Sin. 60, 114203 (2011) (in Chinese).

N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE 7950, 79500F (2011).

A. Cerjan and C. Cerjan, “Orbital angular momentum of Laguerre-Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 28, 2253–2260 (2011).
[CrossRef]

2010

M. Merano, N. Hermosa, A. Aiello, and J. P. Woerdman, “Demonstration of a quasi-scalar angular Goos-Hanchen effect,” Opt. Lett. 35, 3562–3564 (2010).
[CrossRef]

M. Merano, N. Hermosa, and J. P. Woerdman, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[CrossRef]

2009

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3, 337–340 (2009).
[CrossRef]

K. Y. Bliokh, I. V. Shadrivov, and Y. S. Kivshar, “Goos-Hanchen and Imbert-Fedorov shifts of polarized vortex beams,” Opt. Lett. 34, 389–391 (2009).
[CrossRef]

2008

2007

K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75, 066609 (2007).
[CrossRef]

2006

K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[CrossRef]

R. Dasgupta and P. K. Gupta, “Experimental observation of spin-independent transverse shift of the centre of gravity of a reflected Laguerre-Gaussian light beam,” Opt. Commun. 257, 91–96 (2006).
[CrossRef]

H. Okuda and H. Sasada, “Huge transverse deformation in nonspecular reflection of a light beam possessing orbital angular momentum near critical incidence,” Opt. Express 14, 8393–8402 (2006).
[CrossRef]

2001

V. G. Fedoseyev, “Spin-independent transverse shift of the centre of gravity of a reflected and of a refracted light beam,” Opt. Commun. 193, 9–18 (2001).
[CrossRef]

1992

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

1979

G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. A 69, 575–578 (1979).
[CrossRef]

1975

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1972

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[CrossRef]

W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201 (1972).
[CrossRef]

1955

F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465–467 (1955).

1947

F. Goos and H. Hanchen, “Ein neuer und fundamentaler Versuch zur total reflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. A 69, 575–578 (1979).
[CrossRef]

Aiello, A.

N. Hermosa, A. Aiello, and J. P. Woerdman, “Radial mode dependence of optical beam shifts,” Opt. Lett. 37, 1044–1046 (2012).
[CrossRef]

N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE 7950, 79500F (2011).

M. Merano, N. Hermosa, A. Aiello, and J. P. Woerdman, “Demonstration of a quasi-scalar angular Goos-Hanchen effect,” Opt. Lett. 35, 3562–3564 (2010).
[CrossRef]

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3, 337–340 (2009).
[CrossRef]

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hanchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008).
[CrossRef]

A. Aiello and J. P. Woerdman, “Theory of angular Goos-Hanchen shift near Brewster incidence,” arXiv:0903.3730v2 (2009).

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, E. Wolf, ed., Vol. 39 (Elsevier, 1999), pp. 291–372.

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, E. Wolf, ed., Vol. 39 (Elsevier, 1999), pp. 291–372.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

Bliokh, K. Y.

K. Y. Bliokh, I. V. Shadrivov, and Y. S. Kivshar, “Goos-Hanchen and Imbert-Fedorov shifts of polarized vortex beams,” Opt. Lett. 34, 389–391 (2009).
[CrossRef]

K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75, 066609 (2007).
[CrossRef]

K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[CrossRef]

Bliokh, Y. P.

K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75, 066609 (2007).
[CrossRef]

K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[CrossRef]

Carter, W. H.

Cerjan, A.

Cerjan, C.

Chen, J.

L. Kong, X. Wang, S. Li, Y. Li, J. Chen, B. Gu, and H. Wang, “Spin Hall effect of reflected light from an air-glass interface around the Brewster’s angle,” Appl. Phys. Lett. 100, 071109 (2012).
[CrossRef]

Dasgupta, R.

R. Dasgupta and P. K. Gupta, “Experimental observation of spin-independent transverse shift of the centre of gravity of a reflected Laguerre-Gaussian light beam,” Opt. Commun. 257, 91–96 (2006).
[CrossRef]

Fedorov, F. I.

F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465–467 (1955).

Fedoseyev, V. G.

V. G. Fedoseyev, “Spin-independent transverse shift of the centre of gravity of a reflected and of a refracted light beam,” Opt. Commun. 193, 9–18 (2001).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Goos, F.

F. Goos and H. Hanchen, “Ein neuer und fundamentaler Versuch zur total reflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Gu, B.

L. Kong, X. Wang, S. Li, Y. Li, J. Chen, B. Gu, and H. Wang, “Spin Hall effect of reflected light from an air-glass interface around the Brewster’s angle,” Appl. Phys. Lett. 100, 071109 (2012).
[CrossRef]

Gupta, P. K.

R. Dasgupta and P. K. Gupta, “Experimental observation of spin-independent transverse shift of the centre of gravity of a reflected Laguerre-Gaussian light beam,” Opt. Commun. 257, 91–96 (2006).
[CrossRef]

Hanchen, H.

F. Goos and H. Hanchen, “Ein neuer und fundamentaler Versuch zur total reflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Hermosa, N.

N. Hermosa, A. Aiello, and J. P. Woerdman, “Radial mode dependence of optical beam shifts,” Opt. Lett. 37, 1044–1046 (2012).
[CrossRef]

N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE 7950, 79500F (2011).

M. Merano, N. Hermosa, A. Aiello, and J. P. Woerdman, “Demonstration of a quasi-scalar angular Goos-Hanchen effect,” Opt. Lett. 35, 3562–3564 (2010).
[CrossRef]

M. Merano, N. Hermosa, and J. P. Woerdman, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[CrossRef]

Hosten, O.

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef]

Imbert, C.

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[CrossRef]

Jiang, Y.

J. Ou, Y. Jiang, F. Li, and L. Liu, “Shift of beam centroid of Laguerre-Gaussian beams reflected and refracted at a dielectric interface,” Acta Phys. Sin. 60, 114203 (2011) (in Chinese).

Kivshar, Y. S.

Kong, L.

L. Kong, X. Wang, S. Li, Y. Li, J. Chen, B. Gu, and H. Wang, “Spin Hall effect of reflected light from an air-glass interface around the Brewster’s angle,” Appl. Phys. Lett. 100, 071109 (2012).
[CrossRef]

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Li, F.

J. Ou, Y. Jiang, F. Li, and L. Liu, “Shift of beam centroid of Laguerre-Gaussian beams reflected and refracted at a dielectric interface,” Acta Phys. Sin. 60, 114203 (2011) (in Chinese).

Li, S.

L. Kong, X. Wang, S. Li, Y. Li, J. Chen, B. Gu, and H. Wang, “Spin Hall effect of reflected light from an air-glass interface around the Brewster’s angle,” Appl. Phys. Lett. 100, 071109 (2012).
[CrossRef]

Li, Y.

L. Kong, X. Wang, S. Li, Y. Li, J. Chen, B. Gu, and H. Wang, “Spin Hall effect of reflected light from an air-glass interface around the Brewster’s angle,” Appl. Phys. Lett. 100, 071109 (2012).
[CrossRef]

Liu, L.

J. Ou, Y. Jiang, F. Li, and L. Liu, “Shift of beam centroid of Laguerre-Gaussian beams reflected and refracted at a dielectric interface,” Acta Phys. Sin. 60, 114203 (2011) (in Chinese).

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Maximon, L. C.

F. W. J. Olver and L. C. Maximon, “Bessel functions,” in NIST Handbook of Mathematical Functions, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds. (Cambridge University, 2010), p. 265.

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Merano, M.

N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE 7950, 79500F (2011).

M. Merano, N. Hermosa, A. Aiello, and J. P. Woerdman, “Demonstration of a quasi-scalar angular Goos-Hanchen effect,” Opt. Lett. 35, 3562–3564 (2010).
[CrossRef]

M. Merano, N. Hermosa, and J. P. Woerdman, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[CrossRef]

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3, 337–340 (2009).
[CrossRef]

Okuda, H.

Olver, F. W. J.

F. W. J. Olver and L. C. Maximon, “Bessel functions,” in NIST Handbook of Mathematical Functions, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds. (Cambridge University, 2010), p. 265.

Ou, J.

J. Ou, Y. Jiang, F. Li, and L. Liu, “Shift of beam centroid of Laguerre-Gaussian beams reflected and refracted at a dielectric interface,” Acta Phys. Sin. 60, 114203 (2011) (in Chinese).

Padgett, M. J.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, E. Wolf, ed., Vol. 39 (Elsevier, 1999), pp. 291–372.

Pattanayak, D. N.

G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. A 69, 575–578 (1979).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Sasada, H.

Shadrivov, I. V.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

van Exter, M. P.

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3, 337–340 (2009).
[CrossRef]

Wang, H.

L. Kong, X. Wang, S. Li, Y. Li, J. Chen, B. Gu, and H. Wang, “Spin Hall effect of reflected light from an air-glass interface around the Brewster’s angle,” Appl. Phys. Lett. 100, 071109 (2012).
[CrossRef]

Wang, X.

L. Kong, X. Wang, S. Li, Y. Li, J. Chen, B. Gu, and H. Wang, “Spin Hall effect of reflected light from an air-glass interface around the Brewster’s angle,” Appl. Phys. Lett. 100, 071109 (2012).
[CrossRef]

Woerdman, J. P.

N. Hermosa, A. Aiello, and J. P. Woerdman, “Radial mode dependence of optical beam shifts,” Opt. Lett. 37, 1044–1046 (2012).
[CrossRef]

N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE 7950, 79500F (2011).

M. Merano, N. Hermosa, A. Aiello, and J. P. Woerdman, “Demonstration of a quasi-scalar angular Goos-Hanchen effect,” Opt. Lett. 35, 3562–3564 (2010).
[CrossRef]

M. Merano, N. Hermosa, and J. P. Woerdman, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[CrossRef]

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3, 337–340 (2009).
[CrossRef]

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hanchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

A. Aiello and J. P. Woerdman, “Theory of angular Goos-Hanchen shift near Brewster incidence,” arXiv:0903.3730v2 (2009).

Acta Phys. Sin.

J. Ou, Y. Jiang, F. Li, and L. Liu, “Shift of beam centroid of Laguerre-Gaussian beams reflected and refracted at a dielectric interface,” Acta Phys. Sin. 60, 114203 (2011) (in Chinese).

Ann. Phys.

F. Goos and H. Hanchen, “Ein neuer und fundamentaler Versuch zur total reflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Appl. Phys. Lett.

L. Kong, X. Wang, S. Li, Y. Li, J. Chen, B. Gu, and H. Wang, “Spin Hall effect of reflected light from an air-glass interface around the Brewster’s angle,” Appl. Phys. Lett. 100, 071109 (2012).
[CrossRef]

Dokl. Akad. Nauk SSSR

F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465–467 (1955).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nat. Photonics

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3, 337–340 (2009).
[CrossRef]

Opt. Commun.

V. G. Fedoseyev, “Spin-independent transverse shift of the centre of gravity of a reflected and of a refracted light beam,” Opt. Commun. 193, 9–18 (2001).
[CrossRef]

R. Dasgupta and P. K. Gupta, “Experimental observation of spin-independent transverse shift of the centre of gravity of a reflected Laguerre-Gaussian light beam,” Opt. Commun. 257, 91–96 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

M. Merano, N. Hermosa, and J. P. Woerdman, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

Phys. Rev. D

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[CrossRef]

Phys. Rev. E

K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75, 066609 (2007).
[CrossRef]

Phys. Rev. Lett.

K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[CrossRef]

Proc. SPIE

N. Hermosa, M. Merano, A. Aiello, and J. P. Woerdman, “Orbital angular momentum induced beam shifts,” Proc. SPIE 7950, 79500F (2011).

Science

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef]

Other

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, E. Wolf, ed., Vol. 39 (Elsevier, 1999), pp. 291–372.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

A. Aiello and J. P. Woerdman, “Theory of angular Goos-Hanchen shift near Brewster incidence,” arXiv:0903.3730v2 (2009).

F. W. J. Olver and L. C. Maximon, “Bessel functions,” in NIST Handbook of Mathematical Functions, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds. (Cambridge University, 2010), p. 265.

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Figures (6)

Fig. 1.
Fig. 1.

Geometry of beam reflection at a dielectric interface. In the local coordinates (xa,ya,za), a=i or r, the axis zi is coincident with the central wave vector of the incident beam, k0=k0z^i. The axis zr is coincident with the geometric-optical axis of the reflected beam. The reflected wave vector k˜ is the mirror image of the incident wave vector k.

Fig. 2.
Fig. 2.

Simulated intensity distribution of reflected TM LGp=0,l=2 beam from an air–glass interface at the Brewster angle. In the four panels, the incidence beam waist w0 is (a) 10 μm, (b) 30 μm, (c) 50 μm, and (d) 100 μm, respectively. It clearly shows that the reflected beam’s intensity distributions are asymmetrical in the xryr plane, and the distortion is more apparent as the beam waist decreases.

Fig. 3.
Fig. 3.

Numerical simulated and analytical results of the shifts of the reflected LGp=0,l=2 beam with w0=100μm (upper) and w0=10μm (lower) as a function of θi, where θi is the incident angle. xp and yp represent the longitudinal and transverse coordinates of the CBIG for p-polarization, respectively. Each panel takes N=5, 10, and 20 for example.

Fig. 4.
Fig. 4.

Longitudinal and transverse shifts of the reflected LGp=0,l=1 beam with w0=80μm for 45° polarization.

Fig. 5.
Fig. 5.

Transverse shifts of the reflected LG beams with (a) w0=100μm and (b) w0=10μm for different topological charge near the Brewster angle. The topological charge takes 0, 1, 2, 3 in sequence, where l=0 represents the case of the incident Gaussian beam.

Fig. 6.
Fig. 6.

(a), (c) Longitudinal and (b), (d) transverse shifts of the reflected LGp=0,l=2 beams with different beam waist, corresponding to the paraxial regime (upper) and the nonparaxial regime (lower), respectively. For comparative analysis, the plot of beam waist w0=100μm is given in the nonparaxial regime.

Equations (64)

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x^a=x^cosθaz^sinθa,
y^a=y^,
z^a=x^sinθa+z^cosθa,
k=kxx^+kyy^+z^k02kx2ky2=k0(Ux^i+Vy^i+Wz^i),
kx=k0(Ucosθi+Wsinθi),
ky=k0V,
kz=k0(UsinθiWcosθi).
Eji(ρi,ϕi,zi=0)=Ej,0(2ρiw0)lLp(l)(2ρi2w02)×exp(ρi2w02)exp(ilϕi),
E˜ji(Ki,φi)=002πEji(ρi,ϕi,zi=0)×exp[ik0ρiKicos(ϕiφi)]ρidρidϕi.
exp[ik0ρiKicos(ϕiφi)]=n=(i)nJn(k0ρiKi)exp[in(ϕiφi)],
E˜ji(Ki,φi)=2πEj,0(1)p+lilexp(ilφi)(w02)2×(Ki2f)lexp(Ki24f2)Lp(l)(Ki22f2),
Lp(l)(Ki22f2)=α=0p(1)α(p+lpα)1α!(Ki2f)2α.
Ejr(ρr,ϕr,zr)=k02(2π)2002πE˜jr(Kr,φr)×exp[ik0ρrKrcos(ϕrφr)]×exp(ik01Kr2zr)KrdKrdφr.
rj(k)=εjkzkztnr2kz+kzt,
rj(U,W)=12n=1(εj)n2n=1k=1(n/2k)(1)n+kεjn2n=1k=1m=1(n/2k)(km)(1)n+kεjn(nr21)m(WcosθiUsinθi)2m.
cos2θk=(WcosθiUsinθi)2<min(nr21,1nr2+1).
(WcosθiUsinθi)2=q=02m(2mq)Krq(1Kr2)mq2×(1)q(cosφr)qΘm,q(θi),
(1Kr2)ξexp(ik01Kr2zr)=(1)ξN=01N!(Kr22)N(k0zr)NHN,ξ(k0zr),
HN,ξ(k0zr)={t=1ξ+32c(ξ12)t(ξ)(k0zr)tξ12hN(ξ12)t(1)(k0zr)if2ξisodd,k0zrhN1(1)(k0zr)ifξ=0,t=1ξ+1aξt(ξ)(k0zr)tξhNξt(1)(k0zr)if2ξiseven andξ0,
Ejr(ρr,ϕr,zr)=Ejr0(ρr,ϕr,zr)+Ejrc(ρr,ϕr,zr).
Ejr0(ρr,ϕr,zr)=Ej,0C(εj)N=01N!(k0zr)N+1×hN1(1)(k0zr)α=0p(1)p+α×(p+lpα)1α!IN,l,α(ρr),
C(εj)=12n=1(εj)n2n=1k=1(n/2k)(1)n+kεjn,
IN,l,α(ρr)=12f2α+l+201(Kr22)N+α+l/2×exp(Kr24f2)Jl(k0ρrKr)KrdKr.
IN,l,α(ρr)=12f2α+l+2(0(Kr22)N+α+l/2×exp(Kr24f2)Jl(k0ρrKr)KrdKr1(Kr22)N+α+l/2exp(Kr24f2)×Jl(k0ρrKr)KrdKr).
Ejr0(ρr,ϕr,zr)=Ej,0C(εj)N=0f2N(k0zr)N+1×hN1(1)(k0zr)α=0p(1)p+α2N+α×(p+lpα)(N+αN)(2ρrw0)l×exp(ρr2w02)LN+α(l)(ρr2w02)exp(ilϕr).
Ejrc(ρr,ϕr,zr)=Ej,02πf2ilα=0p(1)p+l+α(p+lpα)1α!×n=1k=1m=1q=02m(n/2k)(km)(2mq)×(1)n+k+q(εj)n(nr21)mΘm,q(θi)×0102πKrq+1(1Kr2)mq2×exp(ik01Kr2zr)(Kr2f)2α+l×exp(Kr24f2)(cosφr)qexp(ilφr)×exp[ik0ρrKrcos(ϕrφr)]dKrdφr.
exp(ir1r2cosθ)=l=ilJl(r1r2)exp(ilθ),
0ex2x2n+σ+1Jσ(2xz)dx=n!2ezz12σLnσ(z),
Ejrc(ρr,ϕr,zr)=Ej,0α=0p(1)p+α(p+lpα)×n=1k=1m=1q=02m(n/2k)(km)(2mq)×(1)n+k(εj)n(nr21)mΘm,q(θi)×N=02N+α+l/2+1f2N+q(k0zr)N×HN,ξ(k0zr)s=0q(qs)(1)m+qs×exp[i(l+q2s)ϕr](N+α+s)!N!α!×(ρrw0)l+q2sexp(ρr2w02)×LN+α+s(l+q2s)(ρr2w02),
IN,l,α,q,s(ρr)=2q/2f2α+l+2[0(Kr22)N+α+q/2+l/2×exp(Kr24f2)Jl+q2s(K˜r)KrdKr1(Kr22)N+α+q/2+l/2×exp(Kr24f2)Jl+q2s(K˜r)KrdKr],
Ierror(ρr)=12N+p+l/2+21f2p+l+2ENpl/2(14f2),
Ei(xi,yi,zi)=(β1x^i+β2y^i)Ei(xi,yi,zi)+Ezi(xi,yi,zi)z^i,
Ejr(ρr,ϕr,zr)=k02(2π)2002παj(Kr,φr)E˜jr(Kr,φr)×exp[ik0ρrKrcos(ϕrφr)]×exp(ik01Kr2zr)KrdKrdφr,
αx(Kr,φr)=β1β2Krsinφrcotθi,
αy(Kr,φr)=β2+β1Krsinφrcotθi.
Vλ,m,N=E0α=0p(1)p+α2N+α+l/2(p+lpα)×q=02m(2mq)fqHN,ξ(k0zr)Θm,q×s=0q(qs)(1)m+qs(ρrw0)(l+q2s+λ)×(N+α+s)!N!α!exp[i(q2s+λ)ϕr]×cλ,sLN+α+s+λ(l+q2s+λ)(ρr2w02)exp(ρr2w02)
M^jVλ,m,N=n=1k=1m=1(n/2k)(km)(1)n+k×(εj)n(nr21)mVλ,m,N,
Exr(ρr,ϕr,zr)=exp(ilϕr)N=0f2N(k0zr)N×{C(εp)[β1V0,0,Nfβ2cotθi(V1,0,N+V1,0,N)]2M^p[β1V0,m,Nfβ2cotθi(V1,m,N+V1,m,N)]},
Eyr(ρr,ϕr,zr)=exp(ilϕr)N=0f2N(k0zr)N×{C(εs)[β2V0,0,N+fβ1cotθi(V1,0,N+V1,0,N)]2M^s[β2V0,m,N+fβ1cotθi(V1,m,N+V1,m,N)]},
Ezr(ρr,ϕr,zr)=iexp(ilϕr)N=0f2N+1(k0zr)N×{C(εp)[β1(V1,0,N+V1,0,N)fβ2cotθi(V2,0,NV2,0,N)]+2M^p[β1(V1,m,N+V1,m,N)fβ2cotθi(V2,m,NV2,m,N)]+C(εs)[β2(V1,0,N+V1,0,N)+fβ1cotθi(V2,0,NV2,0,N)]+2M^s[β2(V1,m,N+V1,m,N)+fβ1cotθi(V2,m,NV2,m,N)]}.
Ejr(ρr,ϕr,zr)=exp(ilϕr)N=0f2N(k0zr)N×[C(εj)V0,0,N2M^jV0,m,N],
R=rI(ρr,ϕr,zr)ρrdρrdϕrI(ρr,ϕr,zr)ρrdρrdϕr,
x=N=0N=0f2(N+N)(k0zr)(N+N)γxγ.
γ=F0κ0G^0μ0+H^0ν0,
γx=ω{1,1}(G^ωμω+H^ωνω).
F0=C2(k0zr)2(JN1JN1+YN1YN1),
Mm(ϵj)=n=1k=1(n/2k)(km)(1)n+k(εj)n(nr21)m.
T^=m=1Mm(ϵj)q=02m(2mq)fqs=0q(qs)(1)m+qsΘm,q(θi),
G^n=Ck0zrT^Pn,H^n=T^T^Qn,
P0=JN1AN,ξ+YN1BN,ξ+JN1AN,ξ+YN1BN,ξ,
Pω/w0=(JN1AN,ξ+YN1BN,ξ)×(ωN0+ωs+l+ω)+(JN1AN,ξ+YN1BN,ξ)×(ωN0+ωs+l+ω),
Q0=Qω/w0=AN,ξAN,ξ+BN,ξBN,ξ.
κ0=12Γ(L+1),
μ0=12s+1Γ(L+s+1),
μω=12s+1+ωΓ(L+s+ω),
ν0=12qs+sΓ(L+qs+s+1),
νω=ωN0+l+q+(ω2)sωsω+12qs+sω+1×Γ(L+qs+sω+1),
y=N=0N=0f2(N+N)(k0zr)(N+N)γyγ.
Pω/w0=(JN1BN,ξ+YN1AN,ξ)×(N0+s+ωl+ωω)+(JN1BN,ξ+YN1AN,ξ)×(N0+s+ωl+ωω),
Qω/w0=ω(AN,ξBN,ξBN,ξAN,ξ).
M(k)=[ck1(k+1/2)ck2(k+1/2)ck(k+1)(k+1/2)ck(k+2)(k+1/2)a(k+1)1(k+1)a(k+1)2(k+1)a(k+1)(k+1)(k+1)a(k+1)(k+2)(k+1)],
a(k+1)t(k+1)={ckt(k+1/2)+2t*ck(t+1)(k+1/2)ift=1,,k+1,ckt(k+1/2)ift=k+2,
ckt(k+1/2)={akt(k)ift=1,ak(t1)(k)+(2t+1)*akt(k)ift=2,,k+1,ak(t1)(k)ift=k+2
M(0)=[1131].

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