Abstract

We studied a novel family of paraxial laser beams forming an overcomplete yet nonorthogonal set of modes. These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. The intensity profile is characterized by a single brilliant ring with the singularity at its center, where the field amplitude vanishes. The complex amplitude is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric-Gaussian (HyGG) modes. Unlike the recently introduced hypergeometric modes [Opt. Lett. 32, 742 (2007) ], the HyGG modes carry a finite power and have been generated in this work with a liquid-crystal spatial light modulator. We briefly consider some subfamilies of the HyGG modes as the modified Bessel Gaussian modes, the modified exponential Gaussian modes, and the modified Laguerre–Gaussian modes.

© 2007 Optical Society of America

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References

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  1. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Mod. Opt. 42, 217 (1995).
    [CrossRef]
  2. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
    [CrossRef] [PubMed]
  3. J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. Lett. 83, 1171 (1999).
    [CrossRef]
  4. J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
    [CrossRef]
  5. J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, Opt. Lett. 25, 99 (2000).
    [CrossRef]
  6. G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
    [CrossRef] [PubMed]
  7. G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
    [CrossRef]
  8. W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, 1977).
  9. V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, Opt. Lett. 32, 742 (2007).
    [CrossRef] [PubMed]
  10. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).
  11. The Hilbert inner product is defined according to ⟨u∣v⟩=∫u*vρdρdphiv.

2007 (2)

2002 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

1995 (1)

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Mod. Opt. 42, 217 (1995).
[CrossRef]

1987 (2)

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
[CrossRef]

1977 (1)

W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, 1977).

1970 (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Campos, J.

Cottrell, D. M.

Davis, J. A.

Durnin, J.

J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Fagerholm, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

Friberg, A. T.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

He, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Mod. Opt. 42, 217 (1995).
[CrossRef]

Heckenberg, N. R.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Mod. Opt. 42, 217 (1995).
[CrossRef]

Khonina, S. N.

Kotlyar, V. V.

McNamara, D. E.

Miceli, J. J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Miller, W.

W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, 1977).

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

Rubinsztein-Dunlop, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Mod. Opt. 42, 217 (1995).
[CrossRef]

Salo, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

Salomaa, M. M.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

Skidanov, R. V.

Soifer, V. A.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, J. Mod. Opt. 42, 217 (1995).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (3)

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

Other (3)

W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, 1977).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

The Hilbert inner product is defined according to ⟨u∣v⟩=∫u*vρdρdphiv.

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Figures (3)

Fig. 1
Fig. 1

Experimentally observed intensity distributions of the HyGG m , m mode for m = 1 , , 6 in the transverse plane z = 0.18 z 0 . The Rayleigh range was z 0 = 747.4 cm .

Fig. 2
Fig. 2

Ring diameter d of the HyGG m , m mode for m = 1 , , 10 measured at plane z = 0.18 z 0 . The Rayleigh range was z 0 = 747.4 cm . The reported values of the diameters were scaled with respect to the value d 1 for m = 1 . 엯, theory; ●, experiment.

Fig. 3
Fig. 3

Ratio between the diameter d ( z ) of the HyGG 3 , 3 mode and the 1 e 2 intensity radius w ( z ) of the generating TEM 00 Gaussian beam as a function of z. The Rayleigh range was z 0 = 7.8 cm .

Equations (4)

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HyGG p m = u p m ( ρ , ϕ ; ζ ) = C p m Γ ( 1 + m + p 2 ) Γ ( m + 1 ) × i m + 1 ζ p 2 ( ζ + i ) [ 1 + m + ( p 2 ) ] × ρ m e [ i ρ 2 ( ζ + i ) ] e i m ϕ × F 1 1 ( p 2 , m + 1 ; ρ 2 ζ ( ζ + i ) ) ,
lim ζ 0 u p m ( ρ , ϕ ; ζ ) = C p m ρ p + m e ρ 2 + i m ϕ .
A p q = ( q + m ) ! q ! Γ ( p + m + 1 ) Γ ( q p 2 ) Γ ( p 2 + m + 1 ) Γ ( p 2 ) Γ ( q + m + 1 ) .
A p q = ( 1 ) q ( p 2 ) ! ( p 2 + m ) ! ( p 2 q ) ! q ! ( p + m ) ! ( q + m ) ! ,

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