Abstract

It is common practice to work in the approximation that beam-like radiation fields are polarized transverse to the propagation axis. However, even in the paraxial approximation, this fails to correctly describe beam polarization and propagation characteristics. We present here the paraxial Maxwell’s equations for beams having cylindrical polarization, which describe the full vector structure of these beams in the paraxial regime. The effect that these relations have on the polarization and propagation of cylindrically polarized Laguerre–Gauss and Bessel–Gauss beams is subsequently explored.

© 2014 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2013 (1)

2012 (2)

2011 (1)

2010 (4)

2009 (3)

2007 (1)

2006 (1)

2003 (2)

A. Bouhelier, J. Reger, M. Beversluis, and L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210, 220–224 (2003).
[CrossRef]

J. Lekner, “Polarization of tightly focused laser beams,” J. Opt. A 5, 6–14 (2003).
[CrossRef]

2001 (1)

2000 (1)

1998 (2)

1996 (1)

1994 (1)

W. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[CrossRef]

1975 (1)

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

Abeysinghe, D.

W. Chen, D. Abeysinghe, R. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9, 4320–4325 (2009).
[CrossRef]

Agio, M.

Ahmed, M.

Beversluis, M.

A. Bouhelier, J. Reger, M. Beversluis, and L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210, 220–224 (2003).
[CrossRef]

Boisvert, R.

F. Olver, D. Lozier, R. Boisvert, and C. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Bouhelier, A.

A. Bouhelier, J. Reger, M. Beversluis, and L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210, 220–224 (2003).
[CrossRef]

Brown, T. G.

Cao, G. W.

Chen, W.

W. Chen, R. L. Nelson, and Q. Zhan, “Geometrical phase and surface plasmon focusing with azimuthal polarization,” Opt. Lett. 37, 581–583 (2012).
[CrossRef]

W. Chen, D. Abeysinghe, R. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9, 4320–4325 (2009).
[CrossRef]

Clark, C.

F. Olver, D. Lozier, R. Boisvert, and C. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Conry, J.

Deng, D.

Erikson, W.

W. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[CrossRef]

Gori, F.

Graf, T.

Green, P. L.

Hall, D.

Hall, D. G.

Huang, K.

Kozawa, V.

Kraus, M.

Lax, M.

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

Lekner, J.

J. Lekner, “Polarization of tightly focused laser beams,” J. Opt. A 5, 6–14 (2003).
[CrossRef]

Li, K.

Li, Y. P.

Louisell, W. H.

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

Lozier, D.

F. Olver, D. Lozier, R. Boisvert, and C. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

McKnight, W. B.

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

Michalowski, A.

Mojarad, N.

Moore, J. W.

J. W. Moore, R. Vyas, and S. Singh, “The hidden side of a laser beam,” in Proceedings of Hall Symposium, J. C. Begquist, S. A. Diddams, L. Hollberg, C. Oates, J. Ye, and L. Kaleth, eds. (World Scientific, 2006), pp. 97–99.

Nelson, R.

W. Chen, D. Abeysinghe, R. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9, 4320–4325 (2009).
[CrossRef]

Nelson, R. L.

Novotny, L.

A. Bouhelier, J. Reger, M. Beversluis, and L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210, 220–224 (2003).
[CrossRef]

Olver, F.

F. Olver, D. Lozier, R. Boisvert, and C. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Reger, J.

A. Bouhelier, J. Reger, M. Beversluis, and L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210, 220–224 (2003).
[CrossRef]

Roxworthy, B.

B. Roxworthy and K. Toussaint, “Optical trapping with p-phase cylindrical vector beams,” New J. Phys. 12, 073012 (2010).
[CrossRef]

Salamin, Y.

Y. Salamin, “Low-diffraction direct particle acceleration by a radially polarized laser beam,” Phys. Lett. A 374, 4950–4953 (2010).
[CrossRef]

Sato, S.

Shi, P.

Singh, S.

J. Conry, R. Vyas, and S. Singh, “Polarization of optical beams carrying orbital angular momentum,” J. Opt. Soc. Am. A 30, 821–824 (2013).
[CrossRef]

J. Conry, R. Vyas, and S. Singh, “Cross-polarization of linearly polarized Hermite-Gauss laser beams,” J. Opt. Soc. Am. A 29, 579–584 (2012).
[CrossRef]

W. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[CrossRef]

R. Vyas and S. Singh, “Cross polarization of Maxwell-Gaussian laser beams with orbital and spin angular momentum,” in Coherence and Quantum Optics IX, N. P. Bigelow, J. H. Eberly, and C. R. Stroud, eds. (AIP, 2008), pp. 344–345.

J. W. Moore, R. Vyas, and S. Singh, “The hidden side of a laser beam,” in Proceedings of Hall Symposium, J. C. Begquist, S. A. Diddams, L. Hollberg, C. Oates, J. Ye, and L. Kaleth, eds. (World Scientific, 2006), pp. 97–99.

Toussaint, K.

B. Roxworthy and K. Toussaint, “Optical trapping with p-phase cylindrical vector beams,” New J. Phys. 12, 073012 (2010).
[CrossRef]

Tovar, A.

Voss, A.

Vyas, R.

J. Conry, R. Vyas, and S. Singh, “Polarization of optical beams carrying orbital angular momentum,” J. Opt. Soc. Am. A 30, 821–824 (2013).
[CrossRef]

J. Conry, R. Vyas, and S. Singh, “Cross-polarization of linearly polarized Hermite-Gauss laser beams,” J. Opt. Soc. Am. A 29, 579–584 (2012).
[CrossRef]

R. Vyas and S. Singh, “Cross polarization of Maxwell-Gaussian laser beams with orbital and spin angular momentum,” in Coherence and Quantum Optics IX, N. P. Bigelow, J. H. Eberly, and C. R. Stroud, eds. (AIP, 2008), pp. 344–345.

J. W. Moore, R. Vyas, and S. Singh, “The hidden side of a laser beam,” in Proceedings of Hall Symposium, J. C. Begquist, S. A. Diddams, L. Hollberg, C. Oates, J. Ye, and L. Kaleth, eds. (World Scientific, 2006), pp. 97–99.

Weber, R.

Yan, S.

Yao, B.

Youngworth, K. S.

Zhan, Q.

Zhang, X. B.

Adv. Opt. Photon. (1)

J. Microsc. (1)

A. Bouhelier, J. Reger, M. Beversluis, and L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210, 220–224 (2003).
[CrossRef]

J. Opt. A (1)

J. Lekner, “Polarization of tightly focused laser beams,” J. Opt. A 5, 6–14 (2003).
[CrossRef]

J. Opt. Soc. Am. A (5)

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

Nano Lett. (1)

W. Chen, D. Abeysinghe, R. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9, 4320–4325 (2009).
[CrossRef]

New J. Phys. (1)

B. Roxworthy and K. Toussaint, “Optical trapping with p-phase cylindrical vector beams,” New J. Phys. 12, 073012 (2010).
[CrossRef]

Opt. Express (4)

Opt. Lett. (4)

Phys. Lett. A (1)

Y. Salamin, “Low-diffraction direct particle acceleration by a radially polarized laser beam,” Phys. Lett. A 374, 4950–4953 (2010).
[CrossRef]

Phys. Rev. A (1)

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

Phys. Rev. E (1)

W. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[CrossRef]

Other (3)

J. W. Moore, R. Vyas, and S. Singh, “The hidden side of a laser beam,” in Proceedings of Hall Symposium, J. C. Begquist, S. A. Diddams, L. Hollberg, C. Oates, J. Ye, and L. Kaleth, eds. (World Scientific, 2006), pp. 97–99.

R. Vyas and S. Singh, “Cross polarization of Maxwell-Gaussian laser beams with orbital and spin angular momentum,” in Coherence and Quantum Optics IX, N. P. Bigelow, J. H. Eberly, and C. R. Stroud, eds. (AIP, 2008), pp. 344–345.

F. Olver, D. Lozier, R. Boisvert, and C. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

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Figures (9)

Fig. 1.
Fig. 1.

(a) Plot of the maximum difference Δi=Di/Dmax between the corrected and uncorrected intensity distributions of a LGexp,0,0 mode. (b) Plot of F versus 1/kw0 for a number of LGexp,0, beams. (c) Plot of wM(z) of several LGexp,0, modes.

Fig. 2.
Fig. 2.

Rows are from top to bottom the radial, azimuthal, longitudinal, and total intensity distributions of an LGtrig,2,2 mode. Notice that the longitudinal intensity is not scale invariant. It should also be noted that Iz is O(1/(kw0)2) smaller than Is and Iϕ. In scaled units, each box is of dimension 10×10.

Fig. 3.
Fig. 3.

Ratio b/a for LGtrig,0,2 beam as a function of ϕ for scaled radius S=2 and different values of propagation distance Z.

Fig. 4.
Fig. 4.

Each pair of rows shows time evolution of the vector plot for the transverse polarization and density plot for the longitudinal polarization of an LGtrig,0,2 beam. The points lie on angles ϕ=0,π/8,π/4 and the radius S=2. In the focal plane, transverse field vanishes at ωt=π/2 and 3π/2, whereas the longitudinal field vanishes at ωt=0 and π. In scaled units, each box is of dimension 6×6.

Fig. 5.
Fig. 5.

Each pair of rows shows time evolution of the vector plot for the transverse polarization and density plot for the longitudinal polarization of an LGexp,0,2 beam for different values of Z. The points lie on angles ϕ=0,π/8,π/4 at the radius S=2 and have identical values of b/a. In scaled units each box is of dimension 6×6.

Fig. 6.
Fig. 6.

Plot of b/a versus Z for a number of LGexp,,0 beams.

Fig. 7.
Fig. 7.

(a) Plot of Δ versus Z for a BGexp,2 beam with β=2, a2=b2. (b) Plot of F versus 1/kw0 for a BGtrig,m beam with β=1, am=bm and several values of m. (c) Plot of wM(z) of several BGtrig,m modes with am=bm and β=1. Note that the curve for the m=2 is almost indistinguishable from the m=0 curve in (b) and (c).

Fig. 8.
Fig. 8.

BGtrig,3 beam with m=3, am=bm, and β=3. The rows are from top to bottom the radial, azimuthal, longitudinal, and total intensity distributions. Note the change in transverse scale in moving from column 3 to 4. Also note that upon propagation, the dominant component shifts from azimuthal in the first column to radial in the last column. In scaled units each box is of dimension 8×8.

Fig. 9.
Fig. 9.

(a) Ratio b/a as a function of ϕ at several values of Z for an BGtrig,2 beam with a2=1, b2=0, and β=1. (b) The ratio b/a as a function of ϕ at several values of Z for an BGtrig,2 beam with a2=b2 and β=1.

Tables (2)

Tables Icon

Table 1. Values of Mode Parameters Associated with TE and TM LG Beams

Tables Icon

Table 2. Values of Mode and Beam Parameters Associated with TE and TM BG Beams

Equations (47)

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

E(r,t)=E(r)exp[i(kzωt)][Es(r)s+Eϕ(r)ϕ+Ez(r)z]exp[i(kzωt)],
B(r,t)=B(r)exp[i(kzωt)][Bs(r)s+Bϕ(r)ϕ+Bz(r)z]exp[i(kzωt)].
·E+ikEz=0,
·B+ikBz=0,
×E+ikz×E=ikcB,
×B+ikz×B=ikcE,
Ez=ik[1ss(sEs)+1sEϕϕ],
Bz=ik[1ss(sBs)+1sBϕϕ].
Esc=Bϕ12k2[1s22Bϕϕ2s(1ss(sBϕ))]1k2s2Bsϕs,
Eϕc=Bs12k2[1s22Bsϕ2s(1ss(sBs))]+1k2s2Bϕϕs,
Ezc=ik[1ss(sBϕ)1sBsϕ],
cBs=Eϕ+12k2[1s22Eϕϕ2s(1ss(sEϕ))]+1k2s2Esϕs,
cBϕ=Es+12k2[1s22Esϕ2s(1ss(sEs))]1k2s2Eϕϕs,
cBz=ik[1ss(sEϕ)1sEsϕ].
1ss(sψss)+1s2(2ψsϕ22ψϕϕψs)+2ikψsz=0,
1ss(sψϕs)+1s2(2ψϕϕ2+2ψsϕψϕ)+2ikψϕz=0.
Es=ψs+14k2[s(1ss(sψs))1s22ψsϕ2]+12k2s2ψϕϕs,
Eϕ=ψϕ14k2[s(1ss(sψϕ))1s22ψϕϕ2]+12k2s2ψsϕs,
Ez=ik[1ss(sψs)+1sψϕϕ],
cBs=ψϕ14k2[s(1ss(sψϕ))1s22ψϕϕ2]+12k2s2ψsϕs,
cBϕ=ψs14k2[s(1ss(sψs))1s22ψsϕ2]12k2s2ψϕϕs,
cBz=ik[1ss(sψϕ)1sψsϕ]
ψs±(r,t)=g(s,z)fp,±(s,z)Ts±(ϕ),
ψϕ±(r,t)=g(s,z)fp,±(s,z)Tϕ±(ϕ).
g(s,z)=Aw0w(z)exp{i[ks2/(2q(z))Φ(z)+P0]},
fp,±(s,z)=[2s2w2(z)]|±1|2Lp|±1|(2s2w2(z))×exp[i(2p+|±1|)Φ(z)],
Ts±(ϕ)={sin(ϕ)cos(ϕ)iexp(±iϕ)},
Tϕ±(ϕ)={cos(ϕ)±sin(ϕ)exp(±iϕ)}.
Φ(z)=arctan(z/zR)
1q(z)=1R(z)+i2kw2(z),
w(z)=w01+(z/zR)2,
R(z)=z+zR2/z,
Atrig=2πp!w0Γ(p+(±1)+1)=2Aexp.
ψs±(r)=g(s,z)hs(s,z)tms±(ϕ),
ψϕ±(r)=g(s,z)hϕ(s,z)tmϕ±(ϕ).
hs(s,z)=Q(z)[amJm1(u)+bmJm+1(u)],
hϕ(s,z)=Q(z)[amJm1(u)bmJm+1(u)],
Q(z)=exp{iβ2z/[2k(1+iz/zR)]},
u=βs1+iz/zR,
tms±(ϕ)={cos(mϕ)sin(mϕ)exp(±imϕ)},
tmϕ±(ϕ)={sin(mϕ)cos(mϕ)±iexp(±imϕ)}.
Atrig=2πexp(β2w02/8)w0[am2Im1(β2w024)+bm2Im+1(β2w024)]1/2=2Aexp,
β=βw0/2,S=2s/w(z),Z=z/zR.
Di(z)=max(s,ϕ)[0,)×[0,2π)||Ei|2|Ei0|2|,
F=|R2(SzSz0)dA|R2SzdA,
02π0wM(z)SzdA=(1e2)Ptot.
LGexp±=LGtrig(upper)±+iLGtrig(lower)±,

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