Abstract

We demonstrate a method for generation of beams of light with large angular momenta. The method utilizes whispering gallery mode resonators that transform a plane electromagnetic wave into high order Bessel beams. Interference pattern among the beams as well as shadow pictures induced by the beams are observed and studied.

© 2006 Optical Society of America

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References

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  1. L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Progress in Optics 39, 291-372 (1999).
    [CrossRef]
  2. L. Allen, "Introduction to the atoms and angular momentum of light special issue," J. Opt. B 4, S1-S6 (2002).
    [CrossRef]
  3. E. Santamato, "Photon orbital angular momentum: problems and perspectives," Progress in Physics 52, 1141-1153 (2004).
  4. D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
    [CrossRef]
  5. J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997).
    [CrossRef]
  6. J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
    [CrossRef]
  7. S. Sundbeck, I. Gruzberg, and D. G. Grier, "Structure and scaling of helical modes of light," Opt. Lett. 30, 477-479 (2005).
    [CrossRef] [PubMed]
  8. A. B. Matsko, A. A. Savchenkov, D. Strekalov, and L. Maleki, "Whispering gallery resonators for studying orbital angular momentum of a photon," Phys. Rev. Lett. 95, 143904 (2005).
    [CrossRef] [PubMed]
  9. E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
    [CrossRef] [PubMed]
  10. F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  11. L. Landau and E. Lifshitz, Classical Theory of Fields (Reed International Educational and Professional Publishing, Oxford, 1980).

2005 (3)

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

S. Sundbeck, I. Gruzberg, and D. G. Grier, "Structure and scaling of helical modes of light," Opt. Lett. 30, 477-479 (2005).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, and L. Maleki, "Whispering gallery resonators for studying orbital angular momentum of a photon," Phys. Rev. Lett. 95, 143904 (2005).
[CrossRef] [PubMed]

2004 (1)

E. Santamato, "Photon orbital angular momentum: problems and perspectives," Progress in Physics 52, 1141-1153 (2004).

2003 (1)

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

2002 (2)

L. Allen, "Introduction to the atoms and angular momentum of light special issue," J. Opt. B 4, S1-S6 (2002).
[CrossRef]

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Progress in Optics 39, 291-372 (1999).
[CrossRef]

1997 (1)

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997).
[CrossRef]

1987 (1)

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Allen, L.

L. Allen, "Introduction to the atoms and angular momentum of light special issue," J. Opt. B 4, S1-S6 (2002).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Progress in Optics 39, 291-372 (1999).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997).
[CrossRef]

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Progress in Optics 39, 291-372 (1999).
[CrossRef]

Courtial, J.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997).
[CrossRef]

Crawford, P. R.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

Curtis, J. E.

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Dholakia, K.

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997).
[CrossRef]

Galvez, E. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Grier, D. G.

S. Sundbeck, I. Gruzberg, and D. G. Grier, "Structure and scaling of helical modes of light," Opt. Lett. 30, 477-479 (2005).
[CrossRef] [PubMed]

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Gruzberg, I.

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Haglin, P. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Maleki, L.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, and L. Maleki, "Whispering gallery resonators for studying orbital angular momentum of a photon," Phys. Rev. Lett. 95, 143904 (2005).
[CrossRef] [PubMed]

Matsko, A. B.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, and L. Maleki, "Whispering gallery resonators for studying orbital angular momentum of a photon," Phys. Rev. Lett. 95, 143904 (2005).
[CrossRef] [PubMed]

McGloin, D.

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

Padgett, M. J.

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Progress in Optics 39, 291-372 (1999).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Pysher, M. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

Santamato, E.

E. Santamato, "Photon orbital angular momentum: problems and perspectives," Progress in Physics 52, 1141-1153 (2004).

Savchenkov, A. A.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, and L. Maleki, "Whispering gallery resonators for studying orbital angular momentum of a photon," Phys. Rev. Lett. 95, 143904 (2005).
[CrossRef] [PubMed]

Strekalov, D.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, and L. Maleki, "Whispering gallery resonators for studying orbital angular momentum of a photon," Phys. Rev. Lett. 95, 143904 (2005).
[CrossRef] [PubMed]

Sundbeck, S.

Sztul, H. I.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

Williams, R. E.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

Contemp. Phys. (1)

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

J. Opt. B (1)

L. Allen, "Introduction to the atoms and angular momentum of light special issue," J. Opt. B 4, S1-S6 (2002).
[CrossRef]

Opt. Commun. (3)

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997).
[CrossRef]

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (2)

A. B. Matsko, A. A. Savchenkov, D. Strekalov, and L. Maleki, "Whispering gallery resonators for studying orbital angular momentum of a photon," Phys. Rev. Lett. 95, 143904 (2005).
[CrossRef] [PubMed]

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef] [PubMed]

Progress in Optics (1)

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Progress in Optics 39, 291-372 (1999).
[CrossRef]

Progress in Physics (1)

E. Santamato, "Photon orbital angular momentum: problems and perspectives," Progress in Physics 52, 1141-1153 (2004).

Other (1)

L. Landau and E. Lifshitz, Classical Theory of Fields (Reed International Educational and Professional Publishing, Oxford, 1980).

Supplementary Material (1)

» Media 1: MOV (708 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

Scheme of the experiment.

Fig. 2.
Fig. 2.

A low contrast whispering gallery mode resonator for a Bessel beam generation. The resonator is created by the local increase of the waveguide radius. A tapered waveguide is used to generate a Bessel beam in the free space. Increasing the waveguide radius changes the wave vector of the light, guaranteeing escape of the light from the waveguide into free space. (A) A low contrast fused silica WGM resonator attached to the tapered waveguide. (B) Coupling light into the WGMs of the resonator using cleaved fiber. (C) Tapered fiber used to release generated Bessel beam into free space.

Fig. 3.
Fig. 3.

(A) Far field interference pattern of multiple copropagating Bessel beams. (B) Far field shadow of a 250 μm-thick needle placed to the beam emerging the fiber taper.

Fig. 4.
Fig. 4.

(0.7 Mb) Movie. A shadow of a straight wire illuminated with the light possessing large angular momentum. The distance between the wire and the waveguide surface changes from 12.5 mm to zero. Two mixed interference patterns come from two polarizations of the light.

Fig. 5.
Fig. 5.

To the explanation of the influence of the dispersion of the Bessel waves on az-imuthal localization of light. Let us assume that the fiber coupler excites two Bessel waves shown with red and green curves in a cylindrical waveguide. The initially localized waves become azimuthally delocalized during the propagation. They cross the plane with coordinate Z at different points described by different azimuthal angles.

Fig. 6.
Fig. 6.

Azimuthal angle distribution of the real part of the optical field at different distances from the fiber coupler. The angular width of the distribution is increasing monotonically with the distance because of the different propagation constants of the Bessel waves supported by the waveguide and having different spacial distributions and the same frequency.

Fig. 7.
Fig. 7.

Numerical simulation of the interference pattern obtained for the Bessel beams with l = 3000, Δl = 1000, wavelength λ = 650 nm, and zn = 20 mm (see text for the explanation).

Fig. 8.
Fig. 8.

Intensity pattern for the truncated Bessel beams leaving the fiber taper with 10 μm radius. The ordinate axis coincides with the symmetry axis of the taper. The pictures correspond to beams with numbers (A) m = 17 and l = 30, and (B) m = 10 and l = 15.

Fig. 9.
Fig. 9.

Formation of the shadow (inset) of a straight rod AA´ illuminated with a Bessel beam.

Equations (11)

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k z ( z ) k 0 2 ( l nr ( z ) ) 2 ,
f ( r , z , ϕ , l ) = A 0 l r exp [ j ( ϕ + ϕ 0 ( l ) ) l + j 0 z k z 1 z 1 l d z 1 ]
F ( r , z , ϕ , l 0 , Δ l ) = l = l 0 Δ l l 0 f ( r , z , ϕ , l ) = l 0 Δ l l = l 0 A 0 l r exp ( j ( ϕ + ϕ 0 ( l ) ) l + j 0 z k 0 2 ( l nr ( z 1 ) ) 2 d z 1 ) .
B x y = C S F x ' y ' e ik ( xx ' + yy ' ) H dx ' dy ' ,
B ϕ θ = C 1 0 Δ ϕ F ( r = r n Δ r 2 , z = z n ϕ + ϕ 1 , l 0 , Δ l ) e j k 0 r ϕ 1 sin θ k 0 rd ϕ 1 ,
B ( x , y , z ) = C 2 S f e j k 0 R R ds ,
B r z = C 2 0 r 0 0 2 π J m ( k l r ' ) R e j k 0 R r ' ' dr ' ; R = ( r ' cos ϕ ' r ) 2 + ( r ' sin ϕ ' ) 2 + z 2
FC = r n 2 + ( h tgα ) 2 .
FB = r n 2 + ( H tgα ) 2 .
CB = H h tgα .
cos γ = FC 2 + FB 2 CB 2 2 FC FB tgγ = r n H h tgα r n 2 + Hh ( tgα ) 2 .

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