Abstract

Analytical paraxial and nonparaxial propagation expressions for vectorial elegant Laguerre-Gaussian (eLG) beam together with its even and odd modes are introduced by use of the vectorial Rayleigh-Sommerfeld formulas and the relations between eLG and elegant Hermite-Gaussian (eHG) modes. The propagation features of vectorial eLG beams are studied and analyzed comparatively in the paraxial and nonparaxial regimes with vivid illustration. It is shown that the propagation behavior of nonparaxial vectorial eLG beams is notably different from that of paraxial cases.

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References

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    [CrossRef]

2008 (1)

2007 (1)

2005 (1)

2004 (2)

2002 (2)

S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. 27(21), 1872–1874 (2002).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002).
[CrossRef]

1999 (1)

H. C. Kim and Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169(1-6), 9–16 (1999).
[CrossRef]

1998 (1)

S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

1993 (1)

I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[CrossRef]

1986 (1)

1985 (1)

1973 (1)

April, A.

Bandres, M. A.

Borghi, R.

Ciattoni, A.

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002).
[CrossRef]

Crosignani, B.

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002).
[CrossRef]

Duan, K.

Elias, L. R.

I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[CrossRef]

Fukumitsu, O.

Gutiérrez-Vega, J. C.

Kim, H. C.

H. C. Kim and Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169(1-6), 9–16 (1999).
[CrossRef]

Kimel, I.

I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[CrossRef]

Lee, Y. H.

H. C. Kim and Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169(1-6), 9–16 (1999).
[CrossRef]

Lü, B.

Mei, Z.

Porto, P. D.

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002).
[CrossRef]

Saghafi, S.

S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

Santarsiero, M.

Seshadri, S. R.

Sheppard, C. J. R.

S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

Siegman, A. E.

Takenaka, T.

Wang, B.

Yokota, M.

Zauderer, E.

Zhao, D.

IEEE J. Quantum Electron. (1)

I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[CrossRef]

J. Mod. Opt. (1)

S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (2)

H. C. Kim and Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169(1-6), 9–16 (1999).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

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Figures (5)

Fig. 1
Fig. 1

Normalized intensity distribution of a vectorial nonparaxial eLG beam with n = 2, m = 3 in the plane z = 15zR for different f value.

Fig. 2
Fig. 2

Normalized intensity distribution of a fundamental Gaussian beam in the plane z = 15zR for different f value.

Fig. 3
Fig. 3

The three-dimensional intensity distribution (a) and corresponding contour graph (b) of a nonparaxial vectorial eLG beam with n = 2, m = 3 for f = 0.2 at the plane z = 15zR , and its corresponding paraxial result (c) and (d).

Fig. 4
Fig. 4

The three-dimensional intensity distribution (a) and corresponding contour graph (b) of the even mode of a nonparaxial vectorial eLG beam with n = 2, m = 3 for f = 0.2 at the plane z = 15zR , and its corresponding paraxial result (c) and (d).

Fig. 5
Fig. 5

As Fig. 4, but for the odd mode.

Equations (20)

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E(ρ,z)=12πE(ρ0,0)z[exp(ikR)R]d2ρ0,
Ez(ρ,z)=12π{Ex(ρ0,0)x[exp(ikR)R]+Ey(ρ0,0)y[exp(ikR)R]}d2ρ0,
E(ρ0,0)=(ρ0w0)mLnm(ρ02w02)exp(ρ02w02)exp(imθ0)e^x,
eimθρmLnm(ρ2)=(1)n22n+mn!t=0ns=0mis(nt)(ms)HM1(x)HN1(y),
E(ρ0,0)=(1)n22n+mn!t=0ns=0mis(nt)(ms)HM1(x0w0)HN1(y0w0)exp(x02+y02w02)e^x.
exp[(x1x2)22u]Hm(x1)dx1=2πu(12u)m2Hm(x212u),
E(ρ,z)=i(1)nzC222n+mn!kr2f2exp(ikr)t=0ns=0mis(nt)(ms)PM1+N1HM1(C1Px)                  ×HN1(C1Py)exp[C122C2(x2+y2)]e^x,
Ez(ρ,z)=i(1)nC222n+mn!kr2f2exp(ikr)t=0ns=0mis(nt)(ms)PM1+N1[xHM1(C1Px)                 P2kfHM1+1(C1Px)M1kfPHM11(C1Px)]HN1(C1Py)exp[C122C2(x2+y2)],
Ep(ρ,z)=i(1)nC2p22n+mn!kzf2exp(ikz)exp[ik2z(x2+y2)]t=0ns=0mis(nt)(ms)PpM1+N1                   ×HM1(C1pPpx)HN1(C1pPpy)exp[C1p22C2p(x2+y2)]e^x,
ρmLnm(ρ2)cos(mθ)=(1)n22n+mn!t=0ns=0[m/2](1)s(nt)(m2s)HM2(x)HN2(y),
Ε(e)(ρ0,0)=(1)n22n+mn!t=0ns=0[m/2](1)s(nt)(m2s)HM2(x0w0)HN2(y0w0)exp(x02+y02w02)e^x,
E(e)(ρ,z)=i(1)nzC222n+mn!kr2f2exp(ikr)t=0ns=0[m/2](1)s(nt)(m2s)PM2+N2                  ×HM2(C1Px)HN2(C1Py)exp[C122C2(x2+y2)]e^x,
Ez(e)(ρ,z)=i(1)nC222n+mn!kr2f2exp(ikr)t=0ns=0[m/2](1)s(nt)(m2s)PM2+N2[xHM2(C1Px)              P2kfHM2+1(C1Px)M2kfPHM21(C1Px)]HN2(C1Py)exp[C122C2(x2+y2)].
Ep(e)(ρ,z)=i(1)nC2p22n+mn!kzf2exp(ikz)exp[ik2z(x2+y2)]t=0ns=0[m/2](1)s(nt)(m2s)                   ×PpM2+N2HM2(C1pPpx)HN2(C1pPpy)exp[C1p22C2p(x2+y2)]e^x.
ρmLnm(ρ2)sin(mθ)=(1)n22n+mn!t=0ns=0[(m1)/2](1)s(nt)(m2s+1)HM3(x)HN3(y),
E(o)(ρ0,0)=(1)n22n+mn!t=0ns=0[(m1)/2](1)s(nt)(m2s+1)HM3(x0w0)HN3(y0w0)exp(x02+y02w02)e^x,
E(o)(ρ,z)=i(1)nzC222n+mn!kr2f2exp(ikr)t=0ns=0[(m1)/2](1)s(nt)(m2s+1)PM3+N3                   ×HM3(C1Px)HN3(C1Py)exp[C122C2(x2+y2)]e^x,
Ez(o)(ρ,z)=i(1)nC222n+mn!kr2f2exp(ikr)t=0ns=0[(m1)/2](1)s(nt)(m2s+1)PM3+N3[xHM3(C1Px)                  P2kfHM3+1(C1Px)M3kfPHM31(C1Px)]HN3(C1Py)exp[C122C2(x2+y2)].
Ep(o)(ρ,z)=i(1)nC2p22n+mn!kzf2exp(ikz)exp[ik2z(x2+y2)]t=0ns=0[(m1)/2](1)s(nt)(m2s+1)                  ×PpM3+N3HM3(C1pPpx)HN3(C1pPpy)exp[C1p22C2p(x2+y2)]e^x.
E(x,y,z)=iw0w(z)exp[ikz(x2+y2)w2(z)+ik2z(x2+y2)],

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