Abstract

We observe TM and TE Laguerre–Gaussian (LG) light beams reflected and transmitted at a dielectric interface near critical incidence. The intensity distribution of the reflected beam is transversely deformed near the beam waist, and that of the transmitted beam is similar to that of a diagonal Hermite–Gaussian beam. The former rotates around the optical axis by approximately π2 with propagation, and the latter returns to that of the incident LG beam. These observations agree well with numerical calculations based on an angular spectral analysis and are attributable to the helical wavefront of the LG beams, the sharp incidence-angle dependence of the Fresnel reflection and transmission coefficients, and the Gouy phase.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  28. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123-132 (1993).
    [CrossRef]
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  30. H. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinstein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951-S962 (1992).
    [CrossRef]
  31. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67-71 (2000).
    [CrossRef]
  32. V. V. Kottlyar, V. A. Soifer, and S. N. Konina, “Rotation of multimode Gauss-Laguerre light beams in free space,” Tech. Phys. Lett. 23, 657-658 (1997).
    [CrossRef]
  33. A. T. O'Neil, I. M. Vicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
    [CrossRef] [PubMed]
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  36. L.Allen, S.M.Barnett, and M.J.Padgett, eds. Optical Angular Momentum (IOP, 2003).
    [CrossRef]
  37. H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003).
    [CrossRef]
  38. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448-5456 (2004).
    [CrossRef] [PubMed]
  39. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
    [CrossRef] [PubMed]

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

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] [PubMed]

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 and Y. Pagani, “Excitation and cancellation of higher-order beam modes at isotropic interfaces,” J. Opt. A, Pure Appl. Opt. 8, 21 (2006).
[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] [PubMed]

W. Nasalski, “Polarization versus spatial characteristics of optical beams at a planar isotropic interface,” Phys. Rev. E 74, 056613 (2006).
[CrossRef]

2005

A. Köházi-Kis, “Cross-polarization effects of light beams at interfaces of isotropic media,” Opt. Commun. 253, 28 (2005).
[CrossRef]

2004

2003

H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003).
[CrossRef]

2002

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

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]

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

2000

F. I. Baida, D. van Labeke, and J-M. Vigoureux, “Numerical study of the displacement of a three dimensional Gaussian beam transmitted at total internal reflection. Near-field applications,” J. Opt. Soc. Am. A 17, 858-866 (2000).

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67-71 (2000).
[CrossRef]

1997

V. V. Kottlyar, V. A. Soifer, and S. N. Konina, “Rotation of multimode Gauss-Laguerre light beams in free space,” Tech. Phys. Lett. 23, 657-658 (1997).
[CrossRef]

1996

1993

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123-132 (1993).
[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] [PubMed]

H. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinstein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

1990

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

1977

O. Costa de Beauregard, C. Imbert, and Y. Lévy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3553-3562 (1977).
[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]

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]

1974

Y. M. Antar and W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962-972 (1974).

1973

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]

1971

1955

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

1947

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

Ann. Phys.

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

Appl. Opt.

Can. J. Phys.

Y. M. Antar and W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962-972 (1974).

Dokl. Akad. Nauk SSSR

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

J. Opt. A, Pure Appl. Opt.

W. Nasalski and Y. Pagani, “Excitation and cancellation of higher-order beam modes at isotropic interfaces,” J. Opt. A, Pure Appl. Opt. 8, 21 (2006).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

JETP Lett.

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.

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67-71 (2000).
[CrossRef]

A. Köházi-Kis, “Cross-polarization effects of light beams at interfaces of isotropic media,” Opt. Commun. 253, 28 (2005).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123-132 (1993).
[CrossRef]

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

Opt. Express

Opt. Quantum Electron.

H. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinstein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

Phys. Rev. A

H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003).
[CrossRef]

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

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]

O. Costa de Beauregard, C. Imbert, and Y. Lévy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3553-3562 (1977).
[CrossRef]

Phys. Rev. E

W. Nasalski, “Polarization versus spatial characteristics of optical beams at a planar isotropic interface,” Phys. Rev. E 74, 056613 (2006).
[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]

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] [PubMed]

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

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

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Tech. Phys. Lett.

V. V. Kottlyar, V. A. Soifer, and S. N. Konina, “Rotation of multimode Gauss-Laguerre light beams in free space,” Tech. Phys. Lett. 23, 657-658 (1997).
[CrossRef]

Other

L.Allen, S.M.Barnett, and M.J.Padgett, eds. Optical Angular Momentum (IOP, 2003).
[CrossRef]

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, 1997), pp. 30-33.

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

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000), Chap. 7.

O. Hosten and P. G. Kwiat, “Observing the spin Hall effect of light via quantum weak measurements,” presented at Frontiers in Optics 2007, OSA's Annual Meeting, San Jose, Calif., USA, September 16-20, 2007.

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Figures (5)

Fig. 1
Fig. 1

Coordinate systems for incident, reflected, and transmitted beams and those based on the dielectric interface. The beam waists are located at the reflection/refraction point, where the origins of the four coordinate systems coincide with each other. Arrows of the beam coordinates indicate only the direction of the axis.

Fig. 2
Fig. 2

Experimental setup. Intensity distributions on CCD plates correspond to the LG p = 0 , l = ± 1 beams.

Fig. 3
Fig. 3

Observed and calculated IDs of reflected and transmitted TM quasi-LG beams of (a) l = + 1 and (b) l = 3 , where θ 0 is the incident angle and θ c is the critical angle. The horizontal and vertical axes of the images correspond to the x ref tra and y ref tra directions. The observation plane is 6 cm from the reflection/refraction point.

Fig. 4
Fig. 4

Propagation variation in the IDs of reflected and transmitted TM-polarized quasi-LG beams with l = + 2 . The incidence angle is the critical angle for the reflected beam, and θ c 0.15 ° for the transmitted beam. The distances from the reflection/refraction point to the observation planes are (a) 6, (b) 12, (c) 18, and (d) 24 cm .

Fig. 5
Fig. 5

Schematic diagram of the incident LG beam and the dielectric interface.

Equations (32)

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

x α = x cos θ α z sin θ α ,
y α = y ,
z α = x sin θ α + z cos θ α ,
E n , m HG , inc , j ( x , y , z ) = E 0 inc , j v n HG ( x cos θ inc z sin θ inc , x sin θ inc + z cos θ inc ) v m HG ( y , x sin θ inc + z cos θ inc ) exp [ i k 1 ( x sin θ inc + z cos θ inc ) ] ,
v n HG ( ξ , ζ ) = C n HG w 0 w ( ζ ) H n ( 2 ξ w ( ζ ) ) exp [ ξ 2 w ( ζ ) 2 + i k α ξ 2 2 R ( ζ ) i ( n + 1 2 ) ψ ( ζ ) ] ,
C n HG = 2 1 2 n π n !
w ( ζ ) = w 0 1 + ( ζ z R α ) 2 ,
R ( ζ ) = ζ [ 1 + ( z R α ζ ) 2 ] ,
ψ ( ζ ) = arctan ( ζ z R α ) ,
z R α = k α w 0 2 2 .
E ̃ n , m HG , inc , j ( k x , k y ) = E n , m HG , inc , j ( x , y , z = 0 ) exp [ i ( k x x + k y y ) ] d x d y E 0 inc , j v n HG ( x cos θ inc , z inc = 0 ) v m HG ( y , z inc = 0 ) exp ( i k 1 x sin θ inc ) exp [ i ( k x x + k y y ) ] d x d y .
E n , m HG , ref , j ( x , y , z ) = 1 ( 2 π ) 2 r j ( k x , k y ) E ̃ n , m HG , inc , j ( k x , k y ) exp [ i k 1 2 k x 2 k y 2 z + i ( k x x + k y y ) ] d k x d k y .
E n , m HG , ref , j ( x , y , z ) = 1 2 π v m HG ( y , x sin θ inc z cos θ inc ) r ̂ j ( k x ) E ̃ n HG , inc , j ( k x ) exp [ i k 1 2 k x 2 z + i k x x ] d k x .
E ̃ n HG , inc , j ( k x ) = E 0 inc , j v n HG ( x cos θ inc , z inc = 0 ) exp ( i k 1 x sin θ inc ) exp ( i k x x ) d x = E 0 inc , j C n HG π w 0 ( i ) n cos θ inc H n [ ( k x k 1 sin θ inc ) w 0 2 cos θ inc ] exp [ ( k x k 1 sin θ inc ) 2 w 0 2 4 cos 2 θ inc ] ,
r ̂ p ( k x ) = k 1 2 k x 2 ( n 1 n 2 ) 2 k 2 2 k x 2 k 1 2 k x 2 + ( n 1 n 2 ) 2 k 2 2 k x 2 ,
r ̂ s ( k x ) = k 1 2 k x 2 k 2 2 k x 2 k 1 2 k x 2 + k 2 2 k x 2 ,
E n , m HG , tra , j ( x , y , z ) = 1 ( 2 π ) 2 t j ( k x , k y ) E ̃ n , m HG , inc , j ( k x , k y ) exp [ i k 2 2 k x 2 k y 2 z + i ( k x x + k y y ) ] d k x d k y ,
E n , m HG , tra , j ( x , y , z ) = 1 2 π v m HG ( y , z cos θ t + x sin θ t ) t ̂ j ( k x ) E ̃ n HG , inc , j ( k x ) exp [ i k 2 2 k x 2 z + i k x x ] d k x ,
t ̂ p ( k x ) = 2 k 1 2 k x 2 n 2 n 1 k 1 2 k x 2 + n 1 n 2 k 2 2 k x 2 ,
t ̂ s ( k x ) = 2 k 1 2 k x 2 k 1 2 k x 2 + k 2 2 k x 2 .
E p , l LG , inc , j ( ρ inc , ϕ inc , z inc ) = E 0 inc , j u p , l LG ( ρ inc , ϕ inc , z inc ) exp ( i k 1 z inc ) ,
u p , l LG ( ρ , ϕ , ζ ) = C p , l LG ( 1 ) p w 0 w ( ζ ) ( 2 ρ w ( ζ ) ) l L p l ( 2 ρ 2 w ( ζ ) 2 ) × exp ( i l ϕ ) exp [ ρ 2 w ( ζ ) 2 + i k α ρ 2 2 R ( ζ ) i ( 2 p + l + 1 ) ψ ( ζ ) ]
C p , l LG = 2 p ! π ( p + l ) ! ,
u p , l LG ( ρ inc , ϕ inc , z inc ) = k = 0 N i k b k p , l v N k HG ( x inc , z inc ) v k HG ( y inc , z inc ) ,
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 p , l quasi - LG , inc , j ( ρ , ϕ , ζ ) = E 0 inc , j exp ( i k ζ ζ ) p = 0 a p , l u p , l LG ( ρ , ϕ , ζ ) ,
a p , l = p ! ( p + l ) ! l 2 Γ ( p + l 2 ) p ! exp ( i 2 p ψ ) ,
E n , m HG , ref , j ( x , y , z ) = E 0 inc , j exp ( i k 1 z ref ) v m HG ( y ref , z ref ) × { r ̂ j ( k 1 sin θ 0 ) v n HG ( x ref , z ref ) + i 1 w 0 r ̂ j k x k x = k 1 sin θ 0 [ n + 1 v n + 1 HG ( x ref , z ref ) n v n 1 HG ( x ref , z ref ) ] } .
E p , ± l LG , ref , j ( ρ ref , ϕ ref , z ref ) = E 0 inc , j exp ( i k 1 z ref ) { r ̂ j ( k 1 sin θ 0 ) u p , ± l LG ( ρ ref , ϕ ref , z ref ) + i 2 w 0 r ̂ j k x k x = k 1 sin θ 0 [ p + l + 1 u p , ± ( l + 1 ) LG ( ρ ref , ϕ ref , z ref ) + p + 1 u p + 1 , ± ( l 1 ) LG ( ρ ref , ϕ ref , z ref ) p + l u p , ± ( l 1 ) LG ( ρ ref , ϕ ref , z ref ) p u p 1 , ± ( l + 1 ) LG ( ρ ref , ϕ ref , z ref ) ] } .
ϕ ref = B ψ ( z ref ) ,
B = ( 2 p + l ) ( 2 p + l ) l l .

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