Abstract

The free-propagation features of light beams whose transverse electric field lines are logarithmic spirals (namely, spirally polarized beams) are investigated in both the paraxial and the nonparaxial regime. The complete propagated electric field is considered, and some general properties are obtained regardless of the specific transverse distribution. Simple and significant analytical results are obtained when the transverse intensity profile is chosen as that pertinent to an axially symmetric Laguerre–Gaussian beam of order 1 (namely, spirally polarized donut beams). In particular, it is found that for such beams, the propagated longitudinal electric field can be expressed as a simple superposition of elegant Laguerre–Gaussian beams. Numerical results are presented for different values of the beam parameters and are compared with recently obtained experimental results.

© 2004 Optical Society of America

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  29. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
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    [CrossRef]

2003 (5)

2002 (6)

2001 (4)

1998 (2)

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998).
[CrossRef]

1996 (2)

J. Weeber, E. Bourillot, A. Dereux, J. Goudonnet, Y. Chen, C. Girard, “Observation of light confinement effects with a near-field optical microscope,” Phys. Rev. Lett. 77, 5332–5335 (1996).
[CrossRef] [PubMed]

M. Stalder, M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948–1950 (1996).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

1992 (1)

1990 (1)

1986 (2)

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1961 (1)

Allen, L.

L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1999), pp. 291–372.

Armstrong, D. J.

Babiker, M.

L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1999), pp. 291–372.

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Biener, G.

Biss, D. P.

Bomzon, Z.

Borghi, R.

Bourillot, E.

J. Weeber, E. Bourillot, A. Dereux, J. Goudonnet, Y. Chen, C. Girard, “Observation of light confinement effects with a near-field optical microscope,” Phys. Rev. Lett. 77, 5332–5335 (1996).
[CrossRef] [PubMed]

Braat, J. J. M.

Brown, T. G.

D. P. Biss, T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express 9, 490–497 (2001).
[CrossRef] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Chen, Y.

J. Weeber, E. Bourillot, A. Dereux, J. Goudonnet, Y. Chen, C. Girard, “Observation of light confinement effects with a near-field optical microscope,” Phys. Rev. Lett. 77, 5332–5335 (1996).
[CrossRef] [PubMed]

Ciattoni, A.

R. Borghi, A. Ciattoni, M. Santarsiero, “Axial amplitude of nonparaxial Gaussian beams,” J. Opt. Soc. Am. A 19, 1207–1211 (2002).
[CrossRef]

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
[CrossRef]

Courjon, D.

T. Grosjean, D. Courjon, M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203, 1–5 (2002).
[CrossRef]

Crosignani, B.

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
[CrossRef]

Dereux, A.

J. Weeber, E. Bourillot, A. Dereux, J. Goudonnet, Y. Chen, C. Girard, “Observation of light confinement effects with a near-field optical microscope,” Phys. Rev. Lett. 77, 5332–5335 (1996).
[CrossRef] [PubMed]

Di Porto, P.

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
[CrossRef]

Dirksen, P.

Dorn, R.

R. Dorn, S. Quabis, G. Leuchs, “The focus of light. Linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917–1926 (2003).

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 2339011–2339014 (2003).
[CrossRef]

Ford, D. H.

Girard, C.

J. Weeber, E. Bourillot, A. Dereux, J. Goudonnet, Y. Chen, C. Girard, “Observation of light confinement effects with a near-field optical microscope,” Phys. Rev. Lett. 77, 5332–5335 (1996).
[CrossRef] [PubMed]

Gori, F.

Goudonnet, J.

J. Weeber, E. Bourillot, A. Dereux, J. Goudonnet, Y. Chen, C. Girard, “Observation of light confinement effects with a near-field optical microscope,” Phys. Rev. Lett. 77, 5332–5335 (1996).
[CrossRef] [PubMed]

Grosjean, T.

T. Grosjean, D. Courjon, M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203, 1–5 (2002).
[CrossRef]

Hall, G. D.

Hasman, E.

Jackel, S.

Janssen, A. J. E. M.

Jordan, R. H.

Juskaitis, R.

Kim, G. H.

Kimura, W. D.

Kleiner, V.

Laabs, H.

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 2339011–2339014 (2003).
[CrossRef]

R. Dorn, S. Quabis, G. Leuchs, “The focus of light. Linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917–1926 (2003).

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1966).

Massoumian, F.

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Meir, A.

Moshe, I.

Mukunda, N.

Neil, M.

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Ostenberg, H.

Padgett, M. J.

L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1999), pp. 291–372.

Phillips, M. C.

Quabis, S.

R. Dorn, S. Quabis, G. Leuchs, “The focus of light. Linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917–1926 (2003).

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 2339011–2339014 (2003).
[CrossRef]

Santarsiero, M.

Schadt, M.

Seshadri, S. R.

Sheppard, C. J. R.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Simon, R.

Smith, A. V.

Smith, L. W.

Spajer, M.

T. Grosjean, D. Courjon, M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203, 1–5 (2002).
[CrossRef]

Stalder, M.

Sudarshan, E. C. G.

Tidwell, S. C.

Tovar, A. A.

van de Nes, A. S.

Weeber, J.

J. Weeber, E. Bourillot, A. Dereux, J. Goudonnet, Y. Chen, C. Girard, “Observation of light confinement effects with a near-field optical microscope,” Phys. Rev. Lett. 77, 5332–5335 (1996).
[CrossRef] [PubMed]

Wilson, T.

Wünsche, A.

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Zauderer, E.

Appl. Opt. (3)

J. Mod. Opt. (1)

R. Dorn, S. Quabis, G. Leuchs, “The focus of light. Linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917–1926 (2003).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (9)

Opt. Commun. (3)

T. Grosjean, D. Courjon, M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203, 1–5 (2002).
[CrossRef]

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
[CrossRef]

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Phys. Rev. A (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Phys. Rev. Lett. (3)

J. Weeber, E. Bourillot, A. Dereux, J. Goudonnet, Y. Chen, C. Girard, “Observation of light confinement effects with a near-field optical microscope,” Phys. Rev. Lett. 77, 5332–5335 (1996).
[CrossRef] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 2339011–2339014 (2003).
[CrossRef]

Other (5)

L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, Amsterdam, 1999), pp. 291–372.

The following identity has been used: ∫02πdψ sin ψ exp[ik(r2+r′2-2rr′cos ψ+z2)1/2](r2+r′2-2rr′cos ψ+z2)1/2=0.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1966).

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Figures (6)

Fig. 1
Fig. 1

Transverse electric field for a spirally polarized beam. The reference frame (u^α, v^α) is also shown.

Fig. 2
Fig. 2

Behavior of the logarithmic spiral defined by Eq. (14) for α=1.2.

Fig. 3
Fig. 3

Modulus of the normalized longitudinal on-axis electric field for a SPDB. Curves refer to different values of β=πw/λ and are drawn as functions of z/zR, where zR=πw2/λ. Dots represent the paraxial solution.

Fig. 4
Fig. 4

Modulus of the relative difference between the exact expression of the longitudinal on-axis electric field for a SPDB and its paraxial approximation. Curves refer to different values of β=πw/λ and are plotted as functions of z/zR, where zR=πw2/λ.

Fig. 5
Fig. 5

Behaviors of |E| and |Ez|, for the case of a radially polarized beam (α=0), as functions of the normalized transverse coordinate r/w, evaluated at the transverse plane z=zR through the paraxial approximation (m=0) (circles), by adding the first-order (m=1) nonparaxial correction (triangles) and by adding also the second-order (m=2) nonparaxial correction (solid curve) for a SPDB with (a), (b) β=2; (c), (d) β=3; (e, f) β=4; and (g), (h) β=5.

Fig. 6
Fig. 6

Intensity profiles, evaluated across the plane z=0 as functions of the lateral position, pertinent to the transverse (dotted curve), longitudinal (triangles), and total (solid curve) electric fields, for the case of a radially polarized beam (α=0) with β2.3, to be compared with the experimental results shown in Fig. 3(d) of Ref. 14. Wavelength is λ=632.8 nm.

Equations (58)

Equations on this page are rendered with MathJax. Learn more.

E(r, z)=-12πE(r, 0) zexp(ikR)Rd2r,
Ez(r, z)=12πEx(r, 0) xexp(ikR)R+Ey(r, 0) yexp(ikR)Rd2r,
x []=x-xzz [];y []=y-yzz [],
E(r, z)=-12πzE(r, 0) exp(ikR)Rd2r,
Ez(r, z)=12πzz(r-r)E(r, 0)×exp(ikR)Rd2r.
××E=k2E,
E=0.
E(r; z)=F(r; z)exp(ikz)=[F(r; z)+e^zFz(r; z)]exp(ikz),
2F+2ikzF=-z2F,
F+zFz+ikFz=0,
F=m=0F(2m)(r; z),
Fz=m=0Fz(2m+1)(r; z),
F(2m)(r, z)=i2kmp=1mcp(m)zpzm+pF(0)(r, z),
cp(m)=(2m)!m(p-1)!(m-p)!(m+p)!,
Fz(1)(r, z)=ik F(0)(r, z),
Fz(2m+1)(r, z)=ik F(2m)(r, z)+ik zFz(2m-1)(r, z).
Fz(2m+1)(r, z)=ik zFz(2m-1)+i2kmp=1mcp(m)zpzm+pFz(1)(r, z).
Ex(r; 0)=cos(θ+α)f(r),
Ey(r; 0)=sin(θ+α)f(r),
r=r0expθtan α,
Ex(r; z)=cos(θ+α)g(r; z),
Ey(r; z)=sin(θ+α)g(r; z),
E(r; 0)=f(r)(cos θu^α+sin θv^α),
E(r; z)=Eu(r; z)u^α+Ev(r; z)v^α,
E{uv}(r; z)=g(r; z)cos θsin θ.
E{vu}(r; z)
=-12πzf(r)cos θsin θexp(ikR)Rd2r=-12πz0rf(r)dr02πcos θsin θ×exp{ik[r2+r2-2rrcos(θ-θ)+z2]1/2}[r2+r2-2rrcos(θ-θ)+z2]1/2×dθ,
E{vu}(r; z)=-12πz0rf(r)dr02πcos(θ-ψ)sin(θ-ψ)×exp[ik(r2+r2-2rrcos ψ+z2)1/2](r2+r2-2rrcos ψ+z2)1/2dψ
g(r; z)=-12πz0rdrf(r)02πcos ψ×exp[ik(r2+r2-2rrcos ψ+z2)1/2](r2+r2-2rrcos ψ+z2)1/2dψ.
E+z Ez=0,
E=g(r, z)r+g(r, z)rcos α.
f(r)=Arwexp-r2w2,
E(r; 0)=exp-r2w2xwu^α+ywv^α.
Epar(r; z)=Fpar(r; z)exp(ikz)=exp(ikz)(1+iz/zR)2exp-r2/w21+iz/zR×xwu^α+ywv^α,
Fpar+ikFzpar=0,
Fzpar(r, z)=ik Fpar(r, z),
Fzpar(r, z)=2ikwL1(r; z)cos α.
Ln(r; z)=1(1+iz/zR)n+1 Lnr2/w21+iz/zR×exp-r2/w21+iz/zR,
E(0, z)=0.
Ez(0; z)=-cos α2wzz0r2exp-r2w2×exp[ik(r2+z2)1/2](r2+z2)1/2d(r2).
Ez(0; z)=cos αw G1w2, z2,
G(p, η)=2pη0exp(-pξ) exp[ik(ξ+η)1/2](ξ+η)1/2dξ.
0exp(-pξ) exp[ik(ξ+η)1/2](ξ+η)1/2dξ=πp1/2exp-k24p+pηerfcpη1-ik2pη,
G(p, η)=exp(ikη)pπ2exppη-ik2p2×erfcpη-ik2p×1+k22p+2pη-pη+ik2p.
Ez(0; z)=π2exp(ζ2)erfc(ζ)(1+2|ζ|2)-ζ*×cos α exp(ikz),
ζ=z-izRw=zw-iπwλ.
Ezpar(0; z)=iI(ζ)ζ2cos α exp(ikz),
exp(ζ2)erfc(ζ)1π1ζ-12ζ3,
Ez(0; z)12ζ1-ζ*ζ-1ζ2cos α exp(ikz)12ζ1-ζ*ζcos α exp(ikz)=iI(ζ)ζ2cos α exp(ikz),
F(0)(r; z)=11+izzR2exp-r2/w21+izzR  ×xwu^α+ywv^α,
Fz(1)(r; z)=iβL1(r; z)cos α.
F(2m)(r; z)=-w2 (u^αx+v^αy)G(2m)(r; z),
G(2m)(r; z)=1(2kzR)mp=1m(m+p)!cp(m)×zizRpLm+p(r, z).
Ln(r, z)=(izR)nn!nznL0(r, z).
Fz(1)(r; z)=-w cos α zL0(r; z),
Fz(3)(r; z)=icos αβ3L2(r, z)-3i2zzRL3(r, z),
Fz(5)(r; z)=3icos α2β5L3(r, z)-4izzRL4(r, z)-52zzR2L5(r, z).
02πdψ sin ψ exp[ik(r2+r2-2rrcos ψ+z2)1/2](r2+r2-2rrcos ψ+z2)1/2=0.

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