Abstract

We propose a new, three-parameter family of diffraction-free asymmetric elegant Bessel modes (aB-modes) with an integer and fractional orbital angular momentum (OAM). The aB-modes are described by the nth-order Bessel function of the first kind with complex argument. The asymmetry degree of the nonparaxial aB-mode is shown to depend on a real parameter c0: when c=0, the aB-mode is identical to a conventional radially symmetric Bessel mode; with increasing c, the aB-mode starts to acquire a crescent form, getting stretched along the vertical axis and shifted along the horizontal axis for c1. On the horizontal axis, the aB-modes have a denumerable number of isolated intensity zeros that generate optical vortices with a unit topological charge of opposite sign on opposite sides of 0. At different values of the parameter c, the intensity zeros change their location on the horizontal axis, thus changing the beam’s OAM. An isolated intensity zero on the optical axis generates an optical vortex with topological charge n. The OAM per photon of an aB-mode depends near-linearly on c, being equal to (n+cI1(2c)/I0(2c)), where is the Planck constant and In(x) is a modified Bessel function.

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References

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2014

2013

2011

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrkiewicz, P. Di Trapani, and M. J. Padgett, Eur. J. Phys. Special Topics 199, 159 (2011).
[CrossRef]

2001

2000

1995

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, J. Mod. Opt. 42, 1231 (1995).
[CrossRef]

1987

Abramovitz, M.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Math Series (National Bureau of Standards, 1965).

Bowman, R.

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrkiewicz, P. Di Trapani, and M. J. Padgett, Eur. J. Phys. Special Topics 199, 159 (2011).
[CrossRef]

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series: Special Functions (Gordon & Breach, 1986).

Chavez-Cedra, S.

Dennis, M. R.

Di Trapani, P.

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrkiewicz, P. Di Trapani, and M. J. Padgett, Eur. J. Phys. Special Topics 199, 159 (2011).
[CrossRef]

Durnin, J.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1965).

Gutierrez-Vega, J. C.

Iturbe-Castillo, M. D.

Jedrkiewicz, O.

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrkiewicz, P. Di Trapani, and M. J. Padgett, Eur. J. Phys. Special Topics 199, 159 (2011).
[CrossRef]

Khonina, S. N.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, J. Mod. Opt. 42, 1231 (1995).
[CrossRef]

Kotlyar, V. V.

V. V. Kotlyar and A. A. Kovalev, J. Opt. Soc. Am. A 31, 274 (2014).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, J. Mod. Opt. 42, 1231 (1995).
[CrossRef]

Kovalev, A. A.

Marichev, O. I.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series: Special Functions (Gordon & Breach, 1986).

Miller, W.

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

Muller, N.

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrkiewicz, P. Di Trapani, and M. J. Padgett, Eur. J. Phys. Special Topics 199, 159 (2011).
[CrossRef]

New, G. H. C.

Padgett, M. J.

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrkiewicz, P. Di Trapani, and M. J. Padgett, Eur. J. Phys. Special Topics 199, 159 (2011).
[CrossRef]

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series: Special Functions (Gordon & Breach, 1986).

Ring, J. D.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1965).

Soifer, V. A.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, J. Mod. Opt. 42, 1231 (1995).
[CrossRef]

Stegun, I. A.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Math Series (National Bureau of Standards, 1965).

Zambrana-Puyalto, X.

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrkiewicz, P. Di Trapani, and M. J. Padgett, Eur. J. Phys. Special Topics 199, 159 (2011).
[CrossRef]

Eur. J. Phys. Special Topics

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrkiewicz, P. Di Trapani, and M. J. Padgett, Eur. J. Phys. Special Topics 199, 159 (2011).
[CrossRef]

J. Mod. Opt.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, J. Mod. Opt. 42, 1231 (1995).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Other

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series: Special Functions (Gordon & Breach, 1986).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1965).

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Math Series (National Bureau of Standards, 1965).

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

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Figures (3)

Fig. 1.
Fig. 1.

(a) Intensity and (b) phase patterns of the zero-order light beam in Eq. (8) (n=0) in the initial plane and intensity profiles at (c) z=y=0 and (d) z=x=0.

Fig. 2.
Fig. 2.

(a) Intensity and (b) phase patterns of the third-order (n=3) light beam of Eq. (8) in the initial plane and intensity profiles at (c) z=y=0 and (d) z=x=0.

Fig. 3.
Fig. 3.

(a)–(d) Intensity and (e)–(h) phase of the third-order aB-mode in Eq. (8) (n=3) for different values of c: (a), (e) 0.1; (b), (f) 1; (c), (g) 2; (d), (h) 10.

Equations (19)

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(2+k2)E(x,y,z)=0,
En(r,ϕ,z)=exp(ikzcosθ0+inφ)Jn(krsinθ0),
E(r,φ,z=0)=n=0Cnexp(inφ)Jn(krsinθ0),
En(ξ,η,z)=Cen(ξ,q)cen(η,q)exp(ikzcosθ0),
E2n(ξ,η,z)=exp(ikzcosθ0)×m=0(1)mA2m2n(q)cos(2mφ)J2m(krsinθ0),
En(r,φ,z=0;c)=p=0cpexp(inφ+ipφ)p!Jn+p(αr).
k=0tkk!Jk+v(x)=xv/2(x2t)v/2Jv(x22tx),
En(r,φ,z=0;c)=[αrαr2cexp(iφ)]n/2×Jn{αr[αr2cexp(iφ)]}exp(inφ).
x+p=c+c2+γp2α,xp=cc2+γp2α.
En(r,φ)=p=0(c)pexp[i(n+p)φ]p!Jn+p(αr)=[αrαr+2cexp(iφ)]n/2Jn{αr[αr+2cexp(iφ)]}einϕ.
A(θ,φ)=p=0(ic)pexp[i(n+p)φ]p!δ(θθ0)=(i)n2πsinθ0exp[inφicexp(iφ)]δ(θθ0).
En(r,ϕ,z)=02πππA(θ,φ)×exp[ikrcos(ϕφ)sinθ+ikzcosθ]sinθdθdφ
Jz=Im{limR0R02πE*Eφrdrdφ},
I=limR0R02πE*Erdrdφ.
Jz=2πlimRp=0c2p(n+p)(p!)20RJn+p2(αr)rdr,
I=2πlimRp=0c2p(p!)20RJn+p2(αr)rdr.
Jp2(αr)rdr=r22[Jp2(αr)Jp1(αr)Jp+1(αr)].
JzI=n+p=0c2pp(p!)2[p=0c2p(p!)2]1=n+cI1(2c)I0(2c),
(Enαc,Emβd)=2πδ(αβ)α(dc)nm2Inm(2cd),

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