Abstract

We derive a general expression for the field distribution of Hermite- and Laguerre-Gaussian (LG) beams reflected at a dielectric interface. The intensity distributions of the reflected LG light beam at the beam waist are also observed experimentally in the vicinity of the critical incidence. They are greatly deformed because of the nonspecular transverse effect induced by the orbital angular momentum of the LG beam. The observed and calculated intensity distributions agree well and indicate that a large fraction of the electromagnetic energy flows as much as the transverse beam size.

© 2006 Optical Society of America

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  1. B. R. Horowitz and T. Tamir, "Lateral displacement of a light beam at a dielectric interface," J. Opt. Soc. Am. 61, 586-594 (1971).
    [CrossRef]
  2. C. C. Chan and T. Tamir, "Beam phenomena at and near critical incidence upon a dielectric interface," J. Opt. Soc. Am. A 4, 655-663 (1987).
    [CrossRef]
  3. W. Nasalski, "Longitudinal and transverse effects of nonspecular reflection," J. Opt. Soc. Am. A 13, 172-181 (1996).
    [CrossRef]
  4. W. Nasalski, "Three-dimensional beam reflection at dielectric interfaces," Opt. Commun. 197, 217-233 (2001).
    [CrossRef]
  5. F. Goos and H. Hänchen, "Ein neue und fundamentaler Versuch zur total Reflection," Ann. Phys.  1, 333-345 (1947).
    [CrossRef]
  6. F. I. Fedorov, "K teorii polnovo otrazenija," Dokl. Akad. Nauk SSSR 105, 465-467 (1955).
  7. 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]
  8. M. Onoda, S. Murakami, and N. Nagaosa, "Hall effect of light," Phys. Rev. Lett. 93, 083901-1-4 (2004).
    [CrossRef] [PubMed]
  9. A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601-1-4 (2002).
    [CrossRef]
  10. 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] [PubMed]
  11. L. Allen, S. M. Barnett, and M. J. Padgett, ed. Optical angular momentum (Institute of Physics Publishing, Bristol and Philadelphia, 2003).
    [CrossRef]
  12. 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]
  13. 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]
  14. M. McGuirk and C. K. Carniglia, "An angular spectrum representation approach to the Goos- Hänchen shift," J. Opt. Soc. Am. 67, 103-107 (1975).
    [CrossRef]
  15. P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988), Chap. 14.
  16. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, corrected and enlarged edition (Academic, 1980), Chap. 7.
  17. M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
    [CrossRef]
  18. V. Yu. Bazhenov, M. V. Vasnetov, and M. S. Soskin, "Laser beams with screw dislocations in their wavefronts," JETP Lett. 52, 429-431 (1990).
  19. M. A. Clifford, J. Arlt, J. Courtial, and K. Dholakia, "High-order Laguerre-Gaussian laser modes for studies of cold atoms," Opt. Commun. 156, 300-306 (1998).
    [CrossRef]

2006 (1)

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]

2001 (2)

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]

W. Nasalski, "Three-dimensional beam reflection at dielectric interfaces," Opt. Commun. 197, 217-233 (2001).
[CrossRef]

1998 (1)

M. A. Clifford, J. Arlt, J. Courtial, and K. Dholakia, "High-order Laguerre-Gaussian laser modes for studies of cold atoms," Opt. Commun. 156, 300-306 (1998).
[CrossRef]

1996 (1)

1992 (1)

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] [PubMed]

1990 (1)

V. Yu. Bazhenov, M. V. Vasnetov, and M. S. Soskin, "Laser beams with screw dislocations in their wavefronts," JETP Lett. 52, 429-431 (1990).

1987 (1)

1975 (2)

M. McGuirk and C. K. Carniglia, "An angular spectrum representation approach to the Goos- Hänchen shift," J. Opt. Soc. Am. 67, 103-107 (1975).
[CrossRef]

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

1972 (1)

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]

1971 (1)

1955 (1)

F. I. Fedorov, "K teorii polnovo otrazenija," Dokl. Akad. Nauk SSSR 105, 465-467 (1955).

1947 (1)

F. Goos and H. Hänchen, "Ein neue und fundamentaler Versuch zur total Reflection," Ann. Phys.  1, 333-345 (1947).
[CrossRef]

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] [PubMed]

Arlt, J.

M. A. Clifford, J. Arlt, J. Courtial, and K. Dholakia, "High-order Laguerre-Gaussian laser modes for studies of cold atoms," Opt. Commun. 156, 300-306 (1998).
[CrossRef]

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. V. Vasnetov, and M. S. Soskin, "Laser beams with screw dislocations in their wavefronts," JETP Lett. 52, 429-431 (1990).

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] [PubMed]

Carniglia, C. K.

Chan, C. C.

Clifford, M. A.

M. A. Clifford, J. Arlt, J. Courtial, and K. Dholakia, "High-order Laguerre-Gaussian laser modes for studies of cold atoms," Opt. Commun. 156, 300-306 (1998).
[CrossRef]

Courtial, J.

M. A. Clifford, J. Arlt, J. Courtial, and K. Dholakia, "High-order Laguerre-Gaussian laser modes for studies of cold atoms," Opt. Commun. 156, 300-306 (1998).
[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]

Dholakia, K.

M. A. Clifford, J. Arlt, J. Courtial, and K. Dholakia, "High-order Laguerre-Gaussian laser modes for studies of cold atoms," Opt. Commun. 156, 300-306 (1998).
[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]

Goos, F.

F. Goos and H. Hänchen, "Ein neue und fundamentaler Versuch zur total Reflection," Ann. Phys.  1, 333-345 (1947).
[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]

Hänchen, H.

F. Goos and H. Hänchen, "Ein neue und fundamentaler Versuch zur total Reflection," Ann. Phys.  1, 333-345 (1947).
[CrossRef]

Horowitz, B. R.

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]

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]

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]

McGuirk, M.

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]

Nasalski, W.

W. Nasalski, "Three-dimensional beam reflection at dielectric interfaces," Opt. Commun. 197, 217-233 (2001).
[CrossRef]

W. Nasalski, "Longitudinal and transverse effects of nonspecular reflection," J. Opt. Soc. Am. A 13, 172-181 (1996).
[CrossRef]

Soskin, M. S.

V. Yu. Bazhenov, M. V. Vasnetov, and M. S. Soskin, "Laser beams with screw dislocations in their wavefronts," JETP Lett. 52, 429-431 (1990).

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] [PubMed]

Tamir, T.

Vasnetov, M. V.

V. Yu. Bazhenov, M. V. Vasnetov, and M. S. Soskin, "Laser beams with screw dislocations in their wavefronts," JETP Lett. 52, 429-431 (1990).

Woerdman, J. P.

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] [PubMed]

Ann. Phys. (1)

F. Goos and H. Hänchen, "Ein neue und fundamentaler Versuch zur total Reflection," Ann. Phys.  1, 333-345 (1947).
[CrossRef]

Dokl. Akad. Nauk SSSR (1)

F. I. Fedorov, "K teorii polnovo otrazenija," Dokl. Akad. Nauk SSSR 105, 465-467 (1955).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

JETP Lett. (1)

V. Yu. Bazhenov, M. V. Vasnetov, and M. S. Soskin, "Laser beams with screw dislocations in their wavefronts," JETP Lett. 52, 429-431 (1990).

Opt. Commun. (4)

M. A. Clifford, J. Arlt, J. Courtial, and K. Dholakia, "High-order Laguerre-Gaussian laser modes for studies of cold atoms," Opt. Commun. 156, 300-306 (1998).
[CrossRef]

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]

W. Nasalski, "Three-dimensional beam reflection at dielectric interfaces," Opt. Commun. 197, 217-233 (2001).
[CrossRef]

Phys. Rev. A (2)

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] [PubMed]

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

Phys. Rev. D (1)

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]

Other (5)

M. Onoda, S. Murakami, and N. Nagaosa, "Hall effect of light," Phys. Rev. Lett. 93, 083901-1-4 (2004).
[CrossRef] [PubMed]

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601-1-4 (2002).
[CrossRef]

L. Allen, S. M. Barnett, and M. J. Padgett, ed. Optical angular momentum (Institute of Physics Publishing, Bristol and Philadelphia, 2003).
[CrossRef]

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988), Chap. 14.

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, corrected and enlarged edition (Academic, 1980), Chap. 7.

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Figures (5)

Fig. 1.
Fig. 1.

Geometry and coordinate systems of incident and reflected beams.

Fig. 2.
Fig. 2.

Experimental setup. CCD: charge coupled device

Fig. 3.
Fig. 3.

Observed and calculated IDs of reflected p-polarized quasi- LG p=0,l=′1 beams. The X-and Y-axes direct rightward and upward, respectively.

Fig. 4.
Fig. 4.

Observed and calculated IDs and their intensity profiles of reflected p-polarized quasi-LG p=0,l=-1, LG p=0,l=2, and LG p=0,l=-3 beams and reflected s-polarized quasi LG p=0,l=1 beam at critical incidence. Red solid curves are the observed intensity profiles, and blue dotted curves are the calculated ones. Green dotted curves are calculated for the pure LG p=0,l beam with the different value of kw 0.

Fig. 5.
Fig. 5.

Calculated transverse intensity contrast. Black and red curves are for the quasi-LG p=0,l±1 and LG p=0,l±2 beams, and thick and thin ones correspond to the p- and s-polarizations, respectively.

Equations (18)

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E i , n , m j , HG ( x , y , z ) = E i j v n , m HG ( x , y , z ) e ikz
ν n , m HG ( x , y , z = 0 ) = u n HG ( 2 x w 0 ) u m HG ( 2 y w 0 )
u n HG ( ξ ) = N n HG H n ( ξ ) exp ( ξ 2 2 )
N n HG = 2 n + 1 2 π n !
E ˜ i , n , m j , HG k x k y = dx dy E i , n , m j , HG ( x , y , z = 0 ) exp [ i ( k x x + k y y ) ]
= E i j w 0 2 π ( i ) n + m u n HG ( k x w 0 2 ) u m HG ( k x w 0 2 )
R j ( θ ) = E r j E i j = m i cos θ n rel 2 sin 2 θ m j cos θ + n rel 2 sin 2 θ
R j ( θ ) = m i cos θ i sin 2 θ n rel 2 m j cos θ + i sin 2 θ n rel 2
E r , n , m j , HG X Y Z = 0 = E i j u m HG ( 2 Y w 0 ) w 0 ( i ) n 2 π R j ( θ 0 + k X k ) u n HG ( k X w 0 2 ) exp ( i k X X ) d k X
E i , p , l j , LG r ϕ z = E i j v p , l LG r ϕ z e ikz
ν p , l LG r ϕ z = 0 = N p , l LG ( 1 ) p ( 2 r w 0 ) l L p l ( 2 r 2 w 0 2 ) exp ( r 2 w 0 2 ) e ikϕ
N p , l LG = 2 p ! π ( p + l ) !
v p , l LG r ϕ z = 0 = k = 0 N i k b k p , l v N k , k HG x y z = 0
b k p , l = ( N k ) ! k ! 2 N p ! ( p + l ) ! 1 k ! d k d ξ k [ ( 1 ξ ) p ( 1 + ξ ) p + l ] ξ = 0
b k p , l = ( 1 ) k b k p , l
E l r ϕ z = E 0 e ikz p = 0 a p , l v p , l LG r ϕ z
a p , l = p ! ( p + l ) ! l 2 Γ ( p + l 2 ) p !
C = I 1 st I 2 nd I 1 st + I 2 nd ,

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