Abstract

An analytical solution of the scalar Helmholtz equation to describe the propagation of a laser light beam in the positive direction of the optical axis is derived. The complex amplitude of such a beam is found to be in direct proportion to the product of two linearly independent solutions of Kummer’s differential equation. Relationships for a particular case of such beams—namely, the Hankel–Bessel (HB) beams—are deduced. The focusing of the HB beams is studied.

© 2012 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).
  2. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef]
  3. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chavez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
    [CrossRef]
  4. M. A. Bandres, J. C. Gutierrez-Vega, and S. Chavez-Cedra, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46 (2004).
    [CrossRef]
  5. V. Kotlyar, R. Skidanov, S. Khonina, and V. Soifer, “Hypergeometric modes,” Opt. Lett. 32, 742–744 (2007).
    [CrossRef]
  6. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. 32, 3053–3055 (2007).
    [CrossRef]
  7. V. Kotlyar and A. Kovalev, “Family of hypergeometric laser beams,” J. Opt. Soc. Am. A 25, 262–270 (2008).
    [CrossRef]
  8. M. Bandres and J. Gutiérrez-Vega, “Circular beams,” Opt. Lett. 33, 177–179 (2008).
    [CrossRef]
  9. V. V. Kotlyar and A. A. Kovalev, “Nonparaxial hypergeometric beams,” J. Opt. A 11, 045711 (2009).
    [CrossRef]
  10. M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, 1965).
  11. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Special Functions,Vol. 2 of Integrals and Series (Gordon & Breach Science, 1990).
  12. W. Miller, Symmetry and Separation of Variables (Addison-Wesley, 1977).

2009 (1)

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial hypergeometric beams,” J. Opt. A 11, 045711 (2009).
[CrossRef]

2008 (2)

2007 (2)

2004 (1)

2000 (1)

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Abramovitz, M.

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

Bandres, M.

Bandres, M. A.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Special Functions,Vol. 2 of Integrals and Series (Gordon & Breach Science, 1990).

Chavez-Cedra, S.

Chavez-Cerda, S.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Gutierrez-Vega, J. C.

Gutiérrez-Vega, J.

Gutiérrez-Vega, J. C.

Iturbe-Castillo, M. D.

Karimi, E.

Khonina, S.

Kotlyar, V.

Kotlyar, V. V.

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial hypergeometric beams,” J. Opt. A 11, 045711 (2009).
[CrossRef]

Kovalev, A.

Kovalev, A. A.

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial hypergeometric beams,” J. Opt. A 11, 045711 (2009).
[CrossRef]

Marichev, O. I.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Special Functions,Vol. 2 of Integrals and Series (Gordon & Breach Science, 1990).

Marrucci, L.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Miller, W.

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

Piccirillo, B.

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Special Functions,Vol. 2 of Integrals and Series (Gordon & Breach Science, 1990).

Santamato, E.

Skidanov, R.

Soifer, V.

Stegun, I. A.

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).

Zito, G.

J. Opt. A (1)

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial hypergeometric beams,” J. Opt. A 11, 045711 (2009).
[CrossRef]

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

Opt. Lett. (5)

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Other (4)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).

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

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Special Functions,Vol. 2 of Integrals and Series (Gordon & Breach Science, 1990).

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

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Figures (8)

Fig. 1.
Fig. 1.

(a) Squared modulus of the function Eq. (21) in the Orz plane for n=0 and A0=100; the intensity as a function of (b) the longitudinal axis at r=0 and (c) the radial axis at z=0.

Fig. 2.
Fig. 2.

(a) Intensity in the Orz plane at n=1, A0=100 (λxλ, 0zλ; white denotes zero, black denotes maximum intensity). (b) Radial sections of the squared modulus of the function Eq. (21) in different planes: (b) z=0.1λ, (c) z=0.01λ, and (d) z=0.001λ.

Fig. 3.
Fig. 3.

Simulation results for the beam Eq. (30) for wavelength λ=633nm: (a) intensity and (b) phase in the transverse plane z=2λ.

Fig. 4.
Fig. 4.

Intensity profiles in the planes z=2λ, z=4λ, and z=6λ derived using Eq. (21) at n=1, A0=100.

Fig. 5.
Fig. 5.

Simulation of the propagation of the HB beam for n=1 using the FDTD method (TE polarization, Ex0): (a) time-averaged intensity in the plane z=2λ, (b) its horizontal and (c) vertical profiles.

Fig. 6.
Fig. 6.

FDTD-method-based simulation of focusing the HB beam (n=0, f=4λ): (a) the modulus of the amplitude Ex at t=20λ/c, (b) intensity I=|Ex|2+|Ey|2+|Ez|2 in the Oxz plane, (c) time-averaged intensity in the plane z=4λ.

Fig. 7.
Fig. 7.

FDTD-method-based simulation of focusing the HB beam (n=3, f=4λ): the modulus of the amplitude Ex at t=20λ/c.

Fig. 8.
Fig. 8.

Intensity distribution in the plane z=f: (a) two-dimensional pattern and its (b) vertical and (c) horizontal profiles.

Equations (37)

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(2r2+1rr+1r22φ2+2z2+k2)E(r,φ,z)=0,
E(r,φ,z)=E(r,z)r|p|exp(inφ+ikz),
2Er2+2Ez2+(2|p|+1r)Er+2ikEz+(p2n2r2)E=0.
{u=r2+z2+z,v=r2+z2z,
u2Eu2+v2Ev2+(n+1+iku)Eu+(n+1ikv)Ev=0.
[uPP+(n+1+iku)PP]=[vQQ(n+1ikv)QQ]=C,
{ud2Pdu2+(n+1+iku)dPduCP=0,vd2Qdv2+(n+1ikv)dQdv+CQ=0.
{ξd2Pdξ2+(n+1ξ)dPdξDP=0,ηd2Qdη2+(n+1η)dQdηDQ=0,
{P(ξ)=F11(D,n+1,ξ),Q(η)=F11(D,n+1,η).
E(r,φ,z)=A0(kr)nexp(inφ+ikz)F11(D,n+1,iku)F11(D,n+1,ikv),
F11(v+12,2v+1,2iz)=Γ(1+v)exp(iz)(z2)vJv(z),
E(r,φ,z)=A0Γ2(1+n2)4nexp(inφ)Jn2[k2(z+r2+z2)]Jn2[k2(r2+z2z)].
E(r,φ,z)=A0Γ(1+n2)exp(inφ)2πkz(2kr2z)n2cos(kzπn4π4)
U(a,b,z)=πsinπb[M(a,b,z)Γ(1+ab)Γ(b)z1bM(1+ab,2b,z)Γ(a)Γ(2b)].
F11(a,b,z)=AU(a,b,z)+BU(ba,b,z),
A=Γ(b)Γ(ba)T,B=Γ(b)Γ(a)(1)bezT,T=sinπbsinπ(ba)+(1)bsinπa.
E(r,φ,z)=A0(kr)nexp(inφ)[exp(ikz)Γ(n+1D)U(D,n+1,iku)F11(D,n+1,ikv)+(1)n+1exp(ikz)Γ(D)U(n+1D,n+1,iku)F11(n+1D,n+1,ikv)].
E(r,φ,z)=i2n+1A0(kr)nexp(inφ)×[iΓ(n+β+12)exp(ikz)U(n+β+12,n+1,iku)F11(n+β+12,n+1,ikv)+iΓ(nβ+12)exp(ikz)U(nβ+12,n+1,iku)F11(nβ+12,n+1,ikv)].
E+(r,φ,z)=(1)nΓ(n+β+12)A0(kr)nexp(inφ+ikz)×U(n+β+12,n+1,iku)F11(n+β+12,n+1,ikv).
E0(r=0,φ,z)=Γ(β+12)A0U(β+12,1,2ikz)exp(ikz).
E+(r,φ,z)=i3n+1π2n!A0exp(inφ)Hn2(1)[k2(z+r2+z2)]×Jn2[k2(r2+z2z)].
E+(r0,φ,z=0)=(i)n(n1)!A0exp(inφ).
Hv(1)(z)2πzexp[i(zπv2π4)].
I(r=0,φ,z)=|iπ2A0H0(1)(kz)|2=π24A02[J02(kz)+Y02(kz)],
J0(kz)J1(kz)+Y0(kz)Y1(kz)=0.
E+(r,φ,z)=(i)n(n1)!πA0k1n4exp(inφ)r×[Jn12(kv2)+iJn+12(kv2)][iHn12(1)(ku2)+Hn+12(1)(ku2)].
E+(r,ϕ,z)A0exp(inϕ)Hn/2(1)(kz)Jn/2(kr22z).
r=2γzk,
E+(r,φ,z)(i)n(n1)!A0exp(inφ)tann(ξ2),
E1(r,φ,z)=2ikrA0exp(iφ)sin[k2(r2+z2z)]exp[ik2(z+r2+z2)].
limr0E1(r,φ,z)=2ikA0exp(iφ+ikz)limr0{1rsin(kr24z)}=0.
E1(r,φ,z=0)=2ikrA0sin(kr2)exp(ikr2+iφ).
E1(r,φ,zr)=2ikrA0sin(kr24z)exp(iφ+ikz).
E+(r,φ,z)=i3n+1π2n!A0exp(inφ)×Hn2(1){k2[fz+r2+(zf)2]}Jn2{k2[r2+(zf)2f+z]}.
E0(r,φ,z)=iπA02H0(1){k2[fz+r2+(zf)2]}J0{k2[r2+(zf)2f+z]}.
E0(r,φ,z=0)=iπA02H0(1)[k2(f+r2+f2)]J0[k2(r2+f2f)].
E0(rzf,φ,z)iπA02H0(1)[kr24(zf)]J0[k2(r2+(zf)2f+z)]A0ln[kr24(zf)]J0[k2(r2+(zf)2f+z)],

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