Abstract

We report a series of experimental, analytical, and numerical studies demonstrating strong circular dichroism for the HE11-like core mode in helically twisted hollow-core single-ring photonic crystal fiber (SR-PCF), formed by spinning the preform during fiber drawing. In the SR-PCFs studied, the hollow core is surrounded by a single ring of nontouching capillaries. Coupling between these capillaries results in the formation of helical Bloch modes carrying orbital angular momentum. When twisted, strong circular birefringence appears in the ring, so that coupling to the core mode is possible for only one circular polarization state. The result is a SR-PCF that acts as a circular polarizer, offering 1.4 dB/m for the low-loss polarization state and 9.7 dB/m for the high-loss state over a 25 nm band centered at 1593 nm wavelength. In addition, we report for the first time that the vector fields of the helical Bloch modes are perfectly periodic when evaluated in cylindrical coordinates. Such fibers have many potential applications, such as generating circularly polarized light in gas-filled SR-PCF and realizing polarizing elements in the deep and vacuum ultraviolet.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

S.-F. Gao, Y.-Y. Wang, W. Ding, D.-L. Jiang, S. Gu, X. Zhang, and P. Wang, “Hollow-core conjoined-tube negative-curvature fibre with ultralow loss,” Nat. Commun. 9, 2828 (2018).
[Crossref]

2017 (5)

2016 (2)

2015 (2)

S. Li and J. Li, “Chiral photonic crystal fibers with single mode and single polarization,” Opt. Commun. 356, 96–102 (2015).
[Crossref]

C. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Optical activity in multihelicoidal optical fibers,” Phys. Rev. A 92, 033809 (2015).
[Crossref]

2014 (1)

P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

2013 (3)

2012 (2)

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. St. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref]

F. Yu, W. J. Wadsworth, and J. C. Knight, “Low loss silica hollow core fibers for 3–4  μm spectral region,” Opt. Express 20, 11153–11158 (2012).
[Crossref]

2011 (1)

2010 (1)

2009 (1)

2008 (1)

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” COMPEL 27, 806–819 (2008).
[Crossref]

2007 (1)

S. Longhi, “Bloch dynamics of light waves in helical optical waveguide arrays,” Phys. Rev. B 76, 195119 (2007).
[Crossref]

2004 (1)

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182–184 (2004).
[Crossref]

2003 (1)

K. Saitoh and M. Koshiba, “Single-polarization single-mode photonic crystal fibers,” IEEE Photon. Technol. Lett. 15, 1384–1386 (2003).
[Crossref]

1996 (1)

1989 (1)

R. I. Laming and D. N. Payne, “Electric-current sensors employing spun highly birefringent optical fibers,” J. Lightwave Technol. 7, 2084–2094 (1989).
[Crossref]

1986 (1)

C. D. Hussey, R. D. Birch, and Y. Fujii, “Circularly birefringent single-mode optical fibers,” Electron. Lett. 22, 129–130 (1986).
[Crossref]

1964 (1)

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

Abdolvand, A.

P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

Agha, Y. O.

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” COMPEL 27, 806–819 (2008).
[Crossref]

Ahmed, G.

Alexeyev, C. N.

C. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Optical activity in multihelicoidal optical fibers,” Phys. Rev. A 92, 033809 (2015).
[Crossref]

Amsanpally, A.

Auguste, J.-L.

Barnett, S. M.

Baz, A.

Benabid, F.

Beravat, R.

Biancalana, F.

T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, X. M. Xi, and P. St. J. Russell, “Topological Zeeman effect and circular birefringence in twisted photonic crystal fibers,” J. Opt. Soc. Am. B 30, 2921–2927 (2013).
[Crossref]

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. St. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref]

Birch, R. D.

C. D. Hussey, R. D. Birch, and Y. Fujii, “Circularly birefringent single-mode optical fibers,” Electron. Lett. 22, 129–130 (1986).
[Crossref]

Biriukov, A. S.

Blondy, J. M.

Blondy, J.-M.

Chafer, M.

Chang, W.

P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

Ciddor, P. E.

Conti, C.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. St. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref]

Cubillas, A. M.

A. M. Cubillas, S. Unterkofler, T. G. Euser, B. J. M. Etzold, A. C. Jones, P. J. Sadler, P. Wasserscheid, and P. St. J. Russell, “Photonic crystal fibres for chemical sensing and photochemistry,” Chem. Soc. Rev. 42, 8629–8800 (2013).
[Crossref]

Debord, B.

Dianov, E. M.

Ding, W.

S.-F. Gao, Y.-Y. Wang, W. Ding, D.-L. Jiang, S. Gu, X. Zhang, and P. Wang, “Hollow-core conjoined-tube negative-curvature fibre with ultralow loss,” Nat. Commun. 9, 2828 (2018).
[Crossref]

Edavalath, N. N.

Etzold, B. J. M.

A. M. Cubillas, S. Unterkofler, T. G. Euser, B. J. M. Etzold, A. C. Jones, P. J. Sadler, P. Wasserscheid, and P. St. J. Russell, “Photonic crystal fibres for chemical sensing and photochemistry,” Chem. Soc. Rev. 42, 8629–8800 (2013).
[Crossref]

Euser, T. G.

A. M. Cubillas, S. Unterkofler, T. G. Euser, B. J. M. Etzold, A. C. Jones, P. J. Sadler, P. Wasserscheid, and P. St. J. Russell, “Photonic crystal fibres for chemical sensing and photochemistry,” Chem. Soc. Rev. 42, 8629–8800 (2013).
[Crossref]

Frosz, M. H.

Fujii, Y.

C. D. Hussey, R. D. Birch, and Y. Fujii, “Circularly birefringent single-mode optical fibers,” Electron. Lett. 22, 129–130 (1986).
[Crossref]

Gao, S.-F.

S.-F. Gao, Y.-Y. Wang, W. Ding, D.-L. Jiang, S. Gu, X. Zhang, and P. Wang, “Hollow-core conjoined-tube negative-curvature fibre with ultralow loss,” Nat. Commun. 9, 2828 (2018).
[Crossref]

Gérôme, F.

Goto, R.

Gu, S.

S.-F. Gao, Y.-Y. Wang, W. Ding, D.-L. Jiang, S. Gu, X. Zhang, and P. Wang, “Hollow-core conjoined-tube negative-curvature fibre with ultralow loss,” Nat. Commun. 9, 2828 (2018).
[Crossref]

Guenneau, S.

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” COMPEL 27, 806–819 (2008).
[Crossref]

Günendi, M. C.

Hand, D. P.

Hölzer, P.

P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

Hu, J.

Hugonnot, E.

Humbert, G.

Hussey, C. D.

C. D. Hussey, R. D. Birch, and Y. Fujii, “Circularly birefringent single-mode optical fibers,” Electron. Lett. 22, 129–130 (1986).
[Crossref]

Jackson, S. D.

Jamier, R.

Jaworski, P.

Jiang, D.-L.

S.-F. Gao, Y.-Y. Wang, W. Ding, D.-L. Jiang, S. Gu, X. Zhang, and P. Wang, “Hollow-core conjoined-tube negative-curvature fibre with ultralow loss,” Nat. Commun. 9, 2828 (2018).
[Crossref]

Jones, A. C.

A. M. Cubillas, S. Unterkofler, T. G. Euser, B. J. M. Etzold, A. C. Jones, P. J. Sadler, P. Wasserscheid, and P. St. J. Russell, “Photonic crystal fibres for chemical sensing and photochemistry,” Chem. Soc. Rev. 42, 8629–8800 (2013).
[Crossref]

Joseph Weiblen, R.

Kang, M. S.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. St. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref]

Kawanishi, S.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182–184 (2004).
[Crossref]

Knight, J. C.

Koshiba, M.

K. Saitoh and M. Koshiba, “Single-polarization single-mode photonic crystal fibers,” IEEE Photon. Technol. Lett. 15, 1384–1386 (2003).
[Crossref]

Kosolapov, A. F.

Koyanagi, S.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182–184 (2004).
[Crossref]

Kubota, H.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182–184 (2004).
[Crossref]

Laming, R. I.

R. I. Laming and D. N. Payne, “Electric-current sensors employing spun highly birefringent optical fibers,” J. Lightwave Technol. 7, 2084–2094 (1989).
[Crossref]

Lapin, B. P.

C. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Optical activity in multihelicoidal optical fibers,” Phys. Rev. A 92, 033809 (2015).
[Crossref]

Lee, H. W.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. St. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref]

Li, J.

S. Li and J. Li, “Chiral photonic crystal fibers with single mode and single polarization,” Opt. Commun. 356, 96–102 (2015).
[Crossref]

Li, S.

S. Li and J. Li, “Chiral photonic crystal fibers with single mode and single polarization,” Opt. Commun. 356, 96–102 (2015).
[Crossref]

Longhi, S.

S. Longhi, “Bloch dynamics of light waves in helical optical waveguide arrays,” Phys. Rev. B 76, 195119 (2007).
[Crossref]

Maier, R. R. J.

Marcatili, E. A. J.

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

Maurel, M.

Ménard, J.-M.

Menyuk, C. R.

Milione, G.

C. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Optical activity in multihelicoidal optical fibers,” Phys. Rev. A 92, 033809 (2015).
[Crossref]

Nicolet, A.

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” COMPEL 27, 806–819 (2008).
[Crossref]

Payne, D. N.

R. I. Laming and D. N. Payne, “Electric-current sensors employing spun highly birefringent optical fibers,” J. Lightwave Technol. 7, 2084–2094 (1989).
[Crossref]

Plotnichenko, V. G.

Pryamikov, A. D.

Roth, P.

Russell, P. St. J.

M. H. Frosz, P. Roth, M. C. Günendi, and P. St. J. Russell, “Analytical formulation for the bend loss in single-ring hollow-core photonic crystal fibers,” Photon. Res. 5, 88–91 (2017).
[Crossref]

N. N. Edavalath, M. C. Günendi, R. Beravat, G. K. L. Wong, M. H. Frosz, J.-M. Ménard, and P. St. J. Russell, “Higher-order mode suppression in twisted single-ring hollow-core photonic crystal fibers,” Opt. Lett. 42, 2074–2077 (2017).
[Crossref]

P. St. J. Russell, R. Beravat, and G. K. L. Wong, “Helically twisted photonic crystal fibres,” Philos. Trans. R. Soc. London A 375, 20150440 (2017).
[Crossref]

P. Uebel, M. C. Günendi, M. H. Frosz, G. Ahmed, N. N. Edavalath, J.-M. Ménard, and P. St. J. Russell, “Broadband robustly single-mode hollow-core PCF by resonant filtering of higher-order modes,” Opt. Lett. 41, 1961–1964 (2016).
[Crossref]

R. Beravat, G. K. L. Wong, X. M. Xi, M. H. Frosz, and P. St. J. Russell, “Current sensing using circularly birefringent twisted solid-core photonic crystal fiber,” Opt. Lett. 41, 1672–1675 (2016).
[Crossref]

P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

A. M. Cubillas, S. Unterkofler, T. G. Euser, B. J. M. Etzold, A. C. Jones, P. J. Sadler, P. Wasserscheid, and P. St. J. Russell, “Photonic crystal fibres for chemical sensing and photochemistry,” Chem. Soc. Rev. 42, 8629–8800 (2013).
[Crossref]

T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, X. M. Xi, and P. St. J. Russell, “Topological Zeeman effect and circular birefringence in twisted photonic crystal fibers,” J. Opt. Soc. Am. B 30, 2921–2927 (2013).
[Crossref]

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. St. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337, 446–449 (2012).
[Crossref]

Sadler, P. J.

A. M. Cubillas, S. Unterkofler, T. G. Euser, B. J. M. Etzold, A. C. Jones, P. J. Sadler, P. Wasserscheid, and P. St. J. Russell, “Photonic crystal fibres for chemical sensing and photochemistry,” Chem. Soc. Rev. 42, 8629–8800 (2013).
[Crossref]

Saitoh, K.

K. Saitoh and M. Koshiba, “Single-polarization single-mode photonic crystal fibers,” IEEE Photon. Technol. Lett. 15, 1384–1386 (2003).
[Crossref]

Schmeltzer, R. A.

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

Scol, F.

Semjonov, S. L.

Shephard, J. D.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Takenaga, K.

Tanaka, M.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182–184 (2004).
[Crossref]

Travers, J. C.

P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

Uebel, P.

Unterkofler, S.

A. M. Cubillas, S. Unterkofler, T. G. Euser, B. J. M. Etzold, A. C. Jones, P. J. Sadler, P. Wasserscheid, and P. St. J. Russell, “Photonic crystal fibres for chemical sensing and photochemistry,” Chem. Soc. Rev. 42, 8629–8800 (2013).
[Crossref]

Vincetti, L.

Wadsworth, W. J.

Wang, P.

S.-F. Gao, Y.-Y. Wang, W. Ding, D.-L. Jiang, S. Gu, X. Zhang, and P. Wang, “Hollow-core conjoined-tube negative-curvature fibre with ultralow loss,” Nat. Commun. 9, 2828 (2018).
[Crossref]

Wang, Y.-Y.

S.-F. Gao, Y.-Y. Wang, W. Ding, D.-L. Jiang, S. Gu, X. Zhang, and P. Wang, “Hollow-core conjoined-tube negative-curvature fibre with ultralow loss,” Nat. Commun. 9, 2828 (2018).
[Crossref]

Wasserscheid, P.

A. M. Cubillas, S. Unterkofler, T. G. Euser, B. J. M. Etzold, A. C. Jones, P. J. Sadler, P. Wasserscheid, and P. St. J. Russell, “Photonic crystal fibres for chemical sensing and photochemistry,” Chem. Soc. Rev. 42, 8629–8800 (2013).
[Crossref]

Wei, C.

Weiss, T.

T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, X. M. Xi, and P. St. J. Russell, “Topological Zeeman effect and circular birefringence in twisted photonic crystal fibers,” J. Opt. Soc. Am. B 30, 2921–2927 (2013).
[Crossref]

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Figures (7)

Fig. 1.
Fig. 1. (a) Schematic of the fiber cross-section. The hollow core with diameter D is surrounded by five evenly spaced capillaries of wall thickness h and inner diameter d. (b) Scanning electron micrograph (SEM) of the twisted SR-PCF used in the experiments. The helical twist period is 1.19 cm, d=45  μm, D=50  μm, and h=1  μm.
Fig. 2.
Fig. 2. Illustrative plots of modal refractive index versus azimuthal mode order within the first Brillouin zone for (a) untwisted and (b) twisted SR-PCF. The azimuthal mode orders allowed in the ring are marked with circles. In the twisted case the s=+1, lA=1 core mode phase-matches to the s=1, lA=1 ring mode (red arrow), while the s=1, lA=+1 core mode is not phase-matched to any ring mode. Note that the circular birefringence of the core modes is very small: 107.
Fig. 3.
Fig. 3. (a) Sketch showing the circular paths along which the Stokes parameters are calculated relative to the local (ρ^,ϕ^) axes at three different values of radius: ρ/D=0.25 (A), 0.5 (B), and 1.05 (C). (b) Stokes parameters for mode (ii) in Fig. 4, plotted against azimuthal angle ϕ over one period (72° wide). The curves are identical in each period of the ring, as expected of a helical Bloch wave. S0N is the modulus squared of the electric field, normalized to its maximum value S0M along a given path (S0M=1 on axis, 0.5043 for A, 0.0350 for B, and 0.0244 for C). When S=(S1,S2,S3)=(1,0,0) the fields are linearly polarized in the radial direction; when S=(1,0,0), the fields are linearly polarized in the azimuthal direction; when S=(0,1,0), the fields are linearly polarized at ϕ=π/4 to the radial direction; and when S=(0,1,0), the fields are linearly polarized at ϕ=π/4 to the radial direction. The light is LCP when S=(0,0,1) and RCP when S=(0,0,1).
Fig. 4.
Fig. 4. Left-hand side: Modal indices and loss in the laboratory frame as a function of wavelength for the two core + ring supermodes of the twisted fiber, calculated by finite element modeling. As one moves away from the phase-match wavelength (1593 nm), these modes evolve into core (solid curves) and lossy ring (dotted curves) modes. The modeled structure is ideal (see text) and the twist rate 529 rad/m. In this case only the ring and the RCP core modes satisfy all three conditions described in Section 2.D, resulting in resonant coupling and high loss. Right-hand side: Poynting vector distributions for the (i) high- and (ii) low-loss supermodes of core + ring at the phase-match wavelength. Note that the loss rate is higher than the coupling rate, resulting in an anti-crossing in loss, not index. (See Appendix C.)
Fig. 5.
Fig. 5. (a) Measured and (b) numerically modeled modal loss of the RCP and LCP core modes as a function of wavelength for the twisted SR-PCF. A scanning electron micrograph of the actual fiber structure was used in the modeling. As a result, the calculated RCP modal loss is about three times lower than in the ideal structure (Fig. 4), although it agrees reasonably well with the measured value.
Fig. 6.
Fig. 6. (a)–(b) Measured near-field phase distributions at the output of a 1.2 m length of fiber when 1550 nm RCP light is launched into the core (LCP light is transmitted with low loss). (a) Horizontal electric field component. (b) Vertical field component. (c) Polarization state distribution calculated from (a) and (b). (d) Measured near-field intensity distribution. (e)–(h) Corresponding numerically calculated distributions, based on a scanning electron micrograph of the actual fiber structure.
Fig. 7.
Fig. 7. Illustrating the field definitions in the vector coupled-mode theory, used to derive the dispersion relation for azimuthal Bloch waves. The angular width of an azimuthal period is ϕN=π/N.

Equations (15)

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ecyl=(eρ,eϕ)T=e02(1,is)TeiAϕ,
ecrt=(cosϕsinϕsinϕcosϕ)·ecyl=e02(1,is)Tei(A+s)ϕ,
EB(ρ,ϕ,z)=P(ρ,ϕ)exp(iβBz+iAϕ)P(ρ,ϕ)=P(ρ,ϕ+2πqN),
γ=2κcos(2(A+s)π/N),
BC(A)=(2κλ/π)sin(2π/N)sin(2πA/N).
n(Am)=n01+ρ2α2αAmλ2π+2nκcos(2πN(Am+sβ0αρ2)),
e˙n=i[D]en+i([R]1[K][R]1en+1+[R][K][R]en1),
[R]=(cosϕNsinϕNsinϕNcosϕN),[K]=(κ100κ2)
[D]=(2πB/λ000),
iγen=i([D]+[R]1[K][R]1eiϕB+[R][K][R]eiϕB)en,
kψ=Au/ρ,kζ=β0+2κcos(2(Au±1)π/N),
kζ=kzcosθ+(A/ρ)sinθAu/ρ=kzsinθ+(A/ρ)cosθ,
n±(A)=n01+ρ2α2αAλ2π+κλπcos(2πN(A±1β0αρ2))
a˙C=iγaC=iκCRaRa˙R=iγaR=iκCRaC+(αR+iϑ)aR,
iγ=αR/2±αR2/4κCR2,(aC,aR)eigen=(αR2κCR±(αR2κCR)21,i).