Abstract

We propose a three-parameter family of asymmetric Bessel–Gauss (aBG) beams with integer and fractional orbital angular momentum (OAM). The aBG beams are described by the product of a Gaussian function by the nth-order Bessel function of the first kind of complex argument, having finite energy. The aBG beam’s asymmetry degree depends on a real parameter c0: at c=0, the aBG beam is coincident with a conventional radially symmetric Bessel–Gauss (BG) beam; with increasing c, the aBG beam acquires a semicrescent shape, then becoming elongated along the y axis and shifting along the x axis for c1. In the initial plane, the intensity distribution of the aBG beams has a countable number of isolated optical nulls on the x axis, which result in optical vortices with unit topological charge and opposite signs on the different sides of the origin. As the aBG beam propagates, the vortex centers undergo a nonuniform rotation with the entire beam about the optical axis (c1), making a π/4 turn at the Rayleigh range and another π/4 turn after traveling the remaining distance. At different values of the c parameter, the optical nulls of the transverse intensity distribution change their position, thus changing the OAM that the beam carries. An isolated optical null on the optical axis generates an optical vortex with topological charge n. A vortex laser beam shaped as a rotating semicrescent has been generated using a spatial light modulator.

© 2014 Optical Society of America

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References

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  1. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  2. Y. Li, H. Lee, and E. Wolf, “New generalized Bessel-Gauss beams,” J. Opt. Soc. Am. A 21, 640–646 (2004).
    [CrossRef]
  3. A. P. Kisilev, “New structures in paraxial Gaussian beams,” Opt. Spectrosc. 96, 479–481 (2004).
    [CrossRef]
  4. J. C. Gutierrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005).
    [CrossRef]
  5. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
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    [CrossRef]
  10. V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Algorithm for the generation of non-diffracting Bessel beams,” J. Mod. Opt. 42, 1231–1239 (1995).
    [CrossRef]
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    [CrossRef]
  15. V. V. Kotlyar and A. A. Kovalev, “Hermite-Gaussian modal laser beams with orbital angular momentum,” J. Opt. Soc. Am. A 31, 274–282 (2014).
    [CrossRef]
  16. J. Gutiérrez-Vega and M. Bandres, “Normalization of the Mathieu-Gauss optical beams,” J. Opt. Soc. Am. A 24, 215–220 (2007).
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    [CrossRef]
  18. V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Asymmetric Bessel modes,” Opt. Lett. 39, 2395–2398 (2014).
    [CrossRef]
  19. A. P. Prudnikov, J. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Science, 1983).

2014

2013

2009

2008

B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants for electromagnetic fields with elliptic-cylindrical symmetry,” Phys. Rev. A 78, 033813 (2008).
[CrossRef]

2007

2006

K. Volke-Sepulveda and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A 8, 867–877 (2006).

2005

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
[CrossRef]

J. C. Gutierrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005).
[CrossRef]

2004

Y. Li, H. Lee, and E. Wolf, “New generalized Bessel-Gauss beams,” J. Opt. Soc. Am. A 21, 640–646 (2004).
[CrossRef]

A. P. Kisilev, “New structures in paraxial Gaussian beams,” Opt. Spectrosc. 96, 479–481 (2004).
[CrossRef]

2001

2000

1996

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

1995

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Algorithm for the generation of non-diffracting Bessel beams,” J. Mod. Opt. 42, 1231–1239 (1995).
[CrossRef]

1987

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

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

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Bandres, M.

Bandres, M. A.

Brychkov, J. A.

A. P. Prudnikov, J. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Science, 1983).

Chavez-Cedra, S.

Dennis, M. R.

Durnin, J.

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Gori, F.

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

Guattari, G.

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

Gutierrez-Vega, J. C.

Gutiérrez-Vega, J.

Hacyan, S.

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
[CrossRef]

Iturbe-Castillo, M. D.

Jáuregui, R.

B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants for electromagnetic fields with elliptic-cylindrical symmetry,” Phys. Rev. A 78, 033813 (2008).
[CrossRef]

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
[CrossRef]

Khonina, S. N.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Algorithm for the generation of non-diffracting Bessel beams,” J. Mod. Opt. 42, 1231–1239 (1995).
[CrossRef]

Kisilev, A. P.

A. P. Kisilev, “New structures in paraxial Gaussian beams,” Opt. Spectrosc. 96, 479–481 (2004).
[CrossRef]

Kotlyar, V. V.

Kovalev, A. A.

Lee, H.

Ley-Koo, E.

K. Volke-Sepulveda and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A 8, 867–877 (2006).

Li, Y.

Marichev, O. I.

A. P. Prudnikov, J. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Science, 1983).

Miller, W.

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

New, G. H. C.

Padovani, C.

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

Prudnikov, A. P.

A. P. Prudnikov, J. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Science, 1983).

Ring, J. D.

Rodríguez-Lara, B. M.

B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants for electromagnetic fields with elliptic-cylindrical symmetry,” Phys. Rev. A 78, 033813 (2008).
[CrossRef]

Santarsiero, M.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Sheppard, C. J. R.

Shirripa Spagnolo, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Soifer, V. A.

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Asymmetric Bessel modes,” Opt. Lett. 39, 2395–2398 (2014).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Algorithm for the generation of non-diffracting Bessel beams,” J. Mod. Opt. 42, 1231–1239 (1995).
[CrossRef]

Volke-Sepulveda, K.

K. Volke-Sepulveda and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A 8, 867–877 (2006).

Wolf, E.

J. Mod. Opt.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Algorithm for the generation of non-diffracting Bessel beams,” J. Mod. Opt. 42, 1231–1239 (1995).
[CrossRef]

J. Opt. A

K. Volke-Sepulveda and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A 8, 867–877 (2006).

J. Opt. Soc. Am. A

Opt. Commun.

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

Opt. Express

Opt. Lett.

Opt. Spectrosc.

A. P. Kisilev, “New structures in paraxial Gaussian beams,” Opt. Spectrosc. 96, 479–481 (2004).
[CrossRef]

Phys. Rev. A

B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants for electromagnetic fields with elliptic-cylindrical symmetry,” Phys. Rev. A 78, 033813 (2008).
[CrossRef]

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005).
[CrossRef]

Other

A. P. Prudnikov, J. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Science, 1983).

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

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Figures (7)

Fig. 1.
Fig. 1.

(a), (c), (e) Intensity and (b), (d), (f) phase patterns from the third-order beam of Eq. (5) at n=3, z=0, and different values of the asymmetry parameter c: (a), (b) 0.1; (c), (d) 1; and (e), (f) 10.

Fig. 2.
Fig. 2.

(a), (c), (d) Intensity and (b), (d), (f) phase patterns from the third-order aBG beam of Eq. (5) (n=3,c=10) at different distances: (a), (b) z=0; (c), (d) z=z0; and (e), (f) z=10z0.

Fig. 3.
Fig. 3.

(a), (c) Intensity and (b), (d) phase patterns from the third-order aBG beam (n=3 and c=1) at different distance z: 0 (a), (b) and z0 (c), (d).

Fig. 4.
Fig. 4.

(a) Intensity and (b) phase patterns of the zero-order aBG beam of Eq. (5) (n=0, c=10) at z=0 and intensity profiles at (c) z=y=0 and (d) z=x=0.

Fig. 5.
Fig. 5.

OAM as a function of asymmetry parameter c at n=3, ω0=10λ, and α=1/(10λ).

Fig. 6.
Fig. 6.

Experimental setup. 1, solid-state laser of wavelength 532 nm; 2,3, collimators; 4, diaphragm; 5, light-splitting cube; 6, spatial light modulator PLUTO VIS; 7, mirror; 8, CCD camera.

Fig. 7.
Fig. 7.

(a) Computer-generated phase distribution on the modulator PLUTO VIS, and intensity patterns measured at distances of (b) 850 mm, (c) 900 mm, and (d) 950 mm from the modulator.

Equations (26)

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BGn(r,φ,z=0)=exp(r2ω02+inφ)Jn(αr),
BGn(r,φ,z)=q1(z)exp(ikziα2z2kq(z))×exp(r2ω02q(z)+inφ)Jn[αrq(z)],
En(r,φ,z;c)=p=0cpp!BGn+p(r,φ,z)=q1exp(ikziα2z2kqr2qω02)×p=0cpexp(inφ+ipφ)p!Jn+p(αrq).
k=0tkk!Jk+v(x)=xv/2(x2t)v/2Jv(x22tx).
En(r,φ,z;c)=1qexp(ikziα2z2kqr2qω02+inφ)×[αrαr2cqexp(iφ)]n/2Jn{1qαr[αr2cqexp(iφ)]}.
α2r22αcqrexp(iφ)=γnp2q2.
{α2r22αc|q|rcos(φ+Ψ)=γnp2|q|2cos(2Ψ),2αc|q|rsin(φ+Ψ)=γnp2|q|2sin(2Ψ),
{φnp=12arccos[cos(2Ψ)γnp22c2sin2(2Ψ)],rnp=|q|αγnp2cos(2Ψ)+2c2±2D,
γnpsinΨc.
{rp+=α1(c+c2+γnp2),φnp=2pπ,rp=α1(c2+γnp2c),φnp=(2p+1)π.
φ=arctg(zz0).
En(r,ϕ,z=0;c)=exp(r2ω02)×p=0(c)pexp[i(n+p)ϕ]p!Jn+p(αr)=exp(r2ω02)[αrαr+2cexp(iϕ)]n/2×Jn{αr[αr+2cexp(iϕ)]}exp(inϕ).
z=z0tg[arcsin(cγ1)].
E0(r,φ,z=0;c)=exp(r2ω02)J0{αr[αr2cexp(iφ)]},
E0(r=x,ϕ=0)=exp(x2ω02)I0{αx(2cαx)},
An(ρ,ϕ)=(i)nexp[(kρω02f)2+inϕ]In(kαρω022f),
An(ρ,ϕ)=exp[(kρω02f)2]×p=0(ic)pexp[i(n+p)ϕ]p!Ip(kαρω022f).
k=0tkk!Ik+v(x)=xv/2(x+2t)v/2Iv(x2+2tx),
An(ρ,ϕ)=exp[(kρω02f)2+inϕ]×(ξξ+2cei(ϕπ/2))n/2Ip{ξ[ξ+2cei(ϕπ/2)]},
Jz=Im{R2E*Eφrdrdφ},
I¯=R2E*Erdrdφ.
Jz=2πp=0c2p(n+p)(p!)20exp(2r2ω02)Jn+p2(αr)rdr,
I¯=2πp=0c2p(p!)20exp(2r2ω02)Jn+p2(αr)rdr.
0xexp(px2)Jν(bx)Jν(cx)dx=(2p)1exp(b2+c24p)Iν(bc2p).
JzI¯=n+p=0c2ppIn+p(y)(p!)2[p=0c2pIn+p(y)(p!)2]1,
(Enαc,Emβd)=πω022(d*c)nm2exp[ω028(α2+β2)]×p=0(cd*)p+|nm|2p!(p+|nm|)!Ip+max(m,n)(ω02αβ4),

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