Abstract

We study the propagation of real-argument Laguerre–Gaussian beams beyond the paraxial approximation using the perturbation corrections to the complex-argument Laguerre–Gaussian beams derived earlier by Takenaka et al. [J. Opt. Soc. Am. A 2, 826 (1985) [CrossRef]  ]. Each higher-order correction to the amplitude of the real-argument beam (l, m) is represented as a superposition of the same-order corrections to the amplitudes of the complex-argument beams (l, q) with q=0,1,2,,m. We derive explicit expressions for the electric and magnetic fields of transversely and longitudinally polarized real-argument beams and calculate the chirality densities of these beams up to the fourth order of the smallness parameter. For the first time to the best of our knowledge, we show that essentially achiral Gaussian beams (corresponding to l=m=0) possess nonzero chirality density due to the wavefront curvature. The obtained corrections to the paraxial beams may prove useful for precise laser beam shaping and in studies of optomechanical forces.

© 2017 Optical Society of America

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References

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2017 (1)

I. A. Vovk, A. S. Baimuratov, W. Zhu, A. G. Shalkovskiy, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral nanoparticles in singular light fields,” Sci. Rep. 7, 45925 (2017).
[Crossref]

2016 (2)

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6, 36884 (2016).
[Crossref]

I. D. Rukhlenko, A. S. Baimuratov, N. V. Tepliakov, A. V. Baranov, and A. V. Fedorov, “Shape-induced optical activity of chiral nanocrystals,” Opt. Lett. 41, 2438–2441 (2016).
[Crossref]

2015 (1)

K. Y. Bliokh, F. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

2012 (1)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

2011 (2)

2010 (1)

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref]

2008 (1)

2007 (1)

2005 (1)

2003 (1)

F. M. Dickey, “Laser beam shaping,” Opt. Photon. News 14(4), 30–35 (2003).
[Crossref]

2002 (2)

1999 (1)

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

1998 (1)

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

1996 (1)

1994 (1)

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[Crossref]

1985 (1)

1979 (2)

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]

1973 (1)

Agrawal, G. P.

Ahmed, N.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Allen, L.

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[Crossref]

Andronaki, S. A.

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6, 36884 (2016).
[Crossref]

Baimuratov, A. S.

I. A. Vovk, A. S. Baimuratov, W. Zhu, A. G. Shalkovskiy, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral nanoparticles in singular light fields,” Sci. Rep. 7, 45925 (2017).
[Crossref]

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6, 36884 (2016).
[Crossref]

I. D. Rukhlenko, A. S. Baimuratov, N. V. Tepliakov, A. V. Baranov, and A. V. Fedorov, “Shape-induced optical activity of chiral nanocrystals,” Opt. Lett. 41, 2438–2441 (2016).
[Crossref]

Baranov, A. V.

I. A. Vovk, A. S. Baimuratov, W. Zhu, A. G. Shalkovskiy, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral nanoparticles in singular light fields,” Sci. Rep. 7, 45925 (2017).
[Crossref]

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6, 36884 (2016).
[Crossref]

I. D. Rukhlenko, A. S. Baimuratov, N. V. Tepliakov, A. V. Baranov, and A. V. Fedorov, “Shape-induced optical activity of chiral nanocrystals,” Opt. Lett. 41, 2438–2441 (2016).
[Crossref]

Barnett, S. M.

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, F. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Cai, Y.

Cerjan, A.

Cerjan, C.

Chen, C. G.

Cohen, A. E.

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

Deng, D.

Dickey, F. M.

F. M. Dickey, “Laser beam shaping,” Opt. Photon. News 14(4), 30–35 (2003).
[Crossref]

F. M. Dickey and T. E. Lizotte, Laser Beam Shaping Applications (CRC Press, 2017).

Dolinar, S.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Duan, K.

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Fedorov, A. V.

I. A. Vovk, A. S. Baimuratov, W. Zhu, A. G. Shalkovskiy, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral nanoparticles in singular light fields,” Sci. Rep. 7, 45925 (2017).
[Crossref]

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6, 36884 (2016).
[Crossref]

I. D. Rukhlenko, A. S. Baimuratov, N. V. Tepliakov, A. V. Baranov, and A. V. Fedorov, “Shape-induced optical activity of chiral nanocrystals,” Opt. Lett. 41, 2438–2441 (2016).
[Crossref]

Ferrera, J.

Fukumitsu, O.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2014).

Gun’ko, Y. K.

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6, 36884 (2016).
[Crossref]

Guo, Q.

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

Heilmann, R. K.

Huang, H.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Kim, H.-C.

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

Konkola, P. T.

Laabs, H.

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

Landau, L. D.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, 1971).

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]

Lee, Y. H.

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

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, 1971).

Liu, J.

Lizotte, T. E.

F. M. Dickey and T. E. Lizotte, Laser Beam Shaping Applications (CRC Press, 2017).

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]

Lü, B.

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]

Mei, Z.

Nori, F.

K. Y. Bliokh, F. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Pattanayak, D. N.

Ren, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Rodrguez-Fortuño, F.

K. Y. Bliokh, F. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Rukhlenko, I. D.

I. A. Vovk, A. S. Baimuratov, W. Zhu, A. G. Shalkovskiy, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral nanoparticles in singular light fields,” Sci. Rep. 7, 45925 (2017).
[Crossref]

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6, 36884 (2016).
[Crossref]

I. D. Rukhlenko, A. S. Baimuratov, N. V. Tepliakov, A. V. Baranov, and A. V. Fedorov, “Shape-induced optical activity of chiral nanocrystals,” Opt. Lett. 41, 2438–2441 (2016).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2014).

Saleh, B. E.

B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Vol. 22.

Schattenburg, M. L.

Shalkovskiy, A. G.

I. A. Vovk, A. S. Baimuratov, W. Zhu, A. G. Shalkovskiy, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral nanoparticles in singular light fields,” Sci. Rep. 7, 45925 (2017).
[Crossref]

Siegman, A.

Taghizadeh, M.

Takenaka, T.

Tang, Y.

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref]

Teich, M. C.

B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Vol. 22.

Tepliakov, N. V.

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6, 36884 (2016).
[Crossref]

I. D. Rukhlenko, A. S. Baimuratov, N. V. Tepliakov, A. V. Baranov, and A. V. Fedorov, “Shape-induced optical activity of chiral nanocrystals,” Opt. Lett. 41, 2438–2441 (2016).
[Crossref]

Török, P.

Tur, M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Varga, P.

Vovk, I. A.

I. A. Vovk, A. S. Baimuratov, W. Zhu, A. G. Shalkovskiy, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral nanoparticles in singular light fields,” Sci. Rep. 7, 45925 (2017).
[Crossref]

Wang, B.

Wang, J.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Willner, A. E.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yan, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yang, J.-Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yokota, M.

Yue, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Zayats, A. V.

K. Y. Bliokh, F. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Zhao, C.

Zhao, D.

Zhu, W.

I. A. Vovk, A. S. Baimuratov, W. Zhu, A. G. Shalkovskiy, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral nanoparticles in singular light fields,” Sci. Rep. 7, 45925 (2017).
[Crossref]

J. Opt. Soc. Am. (2)

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

Nat. Photonics (2)

K. Y. Bliokh, F. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Opt. Commun. (3)

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

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[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)

Opt. Photon. News (1)

F. M. Dickey, “Laser beam shaping,” Opt. Photon. News 14(4), 30–35 (2003).
[Crossref]

Phys. Rev. A (2)

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

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

Phys. Rev. Lett. (1)

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref]

Sci. Rep. (2)

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6, 36884 (2016).
[Crossref]

I. A. Vovk, A. S. Baimuratov, W. Zhu, A. G. Shalkovskiy, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral nanoparticles in singular light fields,” Sci. Rep. 7, 45925 (2017).
[Crossref]

Other (6)

F. M. Dickey and T. E. Lizotte, Laser Beam Shaping Applications (CRC Press, 2017).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Vol. 22.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1972).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2014).

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, 1971).

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Figures (1)

Fig. 1.
Fig. 1. Chirality density [in units of (8cez0)1P/(πw02)] of transversely polarized Gaussian beam (l=m=0) for z=z0.

Equations (41)

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Δψ+k2ψ=0,
ψ(r)=u(r)eikz.
Δu+2ikuz=0.
Δtu+4uζ4f22uζ2=0,
u=s=0f2su(2s),
(Δt+4ζ)u(0)+s=1f2s[(Δt+4ζ)u(2s)42u(2s2)ζ2]=0.
(Δt+4ζ)u(0)=0.
ulm(0)(r,ϕ,z)=almw0w(rw)|l|Lm|l|(2r2w2)e(r/w)2eiϕlm(r,ϕ,z),
ϕlm(r,ϕ,z)=kr22R+lϕ(|l|+2m+1)arctanzz0,
ulm(0)upq(0)*dS=alm2(m+1)|l|πw022|l|+1δlpδmq,
u^lm(0)(ρ,ϕ,σ)=(1)mm!σ|l|+m+1ρ|l|Lm|l|(σρ2)eσρ2eilϕ,
u^lm(0)v^pq*dS=δlpδmq,
v^pq*(ρ,ϕ,σ)=(1)q(|p|+q)!πw02σqρ|p|Lq|p|(σρ2)eipϕ.
u^lm(2s)=p=1s(1)s+ps(p1)!(2ssp)(1σ1)pu^l,m+s+p(0).
ulm=p,qAlmpqu^pq.
ulm=s=0f2sulm(2s),u^pq=s=0f2su^pq(2s),
ulm(2s)=p,qAlmpqu^pq(2s).
ulm(0)(r,ϕ,0)=p,qAlmpqu^pq(0)(ρ,ϕ,1).
Almpq=ulm(0)(r,ϕ,0)v^pq*(ρ,ϕ,1)dS.
Almpq=δlp(1)qalm(|l|+q)!0ρ2|l|Lm|l|(2ρ2)Lq|l|(ρ2)eρ22ρdρ,
0κ|l|Lm|l|(2κ)Lq|l|(κ)eκdκ=(1)m+q2qq!(|l|+m)!Γ(mq+1).
ulm(2s)=(1)malmq=0m2qq!(|l|+mmq)u^lq(2s).
P=(Π·ez)dS,
E=(i/k)(·A)+ikA,H=Z01×A,
E=ikeikz{t[f2e·tu+i(e·ez)(fu2f3uζ)]+i[e·t(fu2f3uζ)+i(e·ez)(u4f2uζ+4f42uζ2)]ez+ue},
H=keikzZ0[i(u2f2uζ)ez×e+ftu×e],
E=ikeikz[uex+f2tuξ+i(fuξ2f32uξζ)ez],
H=ikeikzZ0[(u2f2uζ)ey+ifuηez].
E=ikeikz{[u(0)+f2(u(2)+2u(0)ξ2)+]ex+(f22u(0)ξη+f42u(2)ξη+)ey+i[fu(0)ξ+f3(u(2)ξ22u(0)ξζ)+]ez},
H=ikeikzZ0{[u(0)+f2(u(2)2u(0)ζ)+]ey+i(fu(0)η+f3u(2)η+)ez}.
(Π·ez)|z=0=k22Z0[|u(0)|2+f2Re(2u(0)u(0)ξ+2u(0)ξ2u(0)*)+2f4Re(2u(0)ξ2u(0)*ζ)].
alm=2f2|l|πZ0P(m+1)|l|.
E=eikzw0[t(u2f2uζ)+4i(fuζf32uζ2)ez],
H=(Z0w0)1eikztu×ez.
E=eikzw0{[u(0)ξ+f2(u(2)ξ22u(0)ξζ)+]ex[u(0)η+f2(u(2)η22u(0)ηζ)+]ey+4i[fu(0)ζ+f3(u(2)ζ2u(0)ζ2)+]ez},
H=eikzZ0w0[(u(0)η+f2u(2)η+)ex(u(0)ξ+f2u(2)ξ+)ey].
(Π·ez)|z=0=f2k22Z0{|u(0)ξ|2+|u(0)η|2f2Re(2u(0)ξζu(0)*η+2u(0)ηζu(0)*ξ)}.
alm=2|l|+1π(|l|+2m+1)Z0P(m+1)|l|.
K=ϵ0k32Im{f2(u(0)ξu(0)*η+2u(0)ξηu(0)*)+f4[2u(0)ξη(u(2)*+2u(0)*ζ)+u(0)ξu(2)*η+2u(2)ξηu(0)*+(u(2)ξ22u(0)ξζ)u(0)*η]+}.
K=ϵ0k32Im{2f2u(0)ξu(0)*η+f4[u(0)ξu(2)*η(u(2)ξ22u(0)ξζ)u(0)*η+u(0)ηu(2)*ξ+(u(2)η22u(0)ηζ)u(0)*ξ]+}.
K=1cRxyw2I,

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