Abstract

We use analytical and numerical techniques to study the superposition of two vector Laguerre–Guassian (LG) beams. Vector LG beams contain rich polarization information and have many interesting intensity and polarization features. The composited electric field shows an inhomogeneous polarization distribution. We focus on looking for the distribution laws for the singularities obtained. The number and positions of singularity points are calculated and confirmed by numerical methods. We also study the fundamental cases for nonconcentric superposed vector LG beams and discover the difference distribution laws of several types of singular points in all the possible combinations.

© 2017 Optical Society of America

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References

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2016 (1)

U. Levy, S. Derevyanko, and Y. Silberberg, “Chapter four-light modes of free space,” Prog. Opt. 61, 237–281 (2016).
[Crossref]

2013 (2)

2012 (3)

2011 (1)

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

2010 (2)

E. Brasselet and S. Juodkazis, “Intangible pointlike tracers for liquid-crystal-based microsensors,” Phys. Rev. A 82, 063832 (2010).
[Crossref]

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010).
[Crossref]

2009 (1)

2008 (3)

2006 (2)

2005 (1)

2004 (1)

S. Orlov, K. Regelskis, and A. Stabinis, “Free-space propagation of second harmonic Laguerre–Gaussian beams carrying phase singularity,” J. Opt. A 6, S255–S258 (2004).
[Crossref]

2003 (2)

S. Orlov, K. Regelskis, V. Smilgevičius, and A. Stabinis, “Free-space propagation of second harmonic beams carrying optical vortices,” Opt. Commun. 215, 1–9 (2003).
[Crossref]

I. D. Maleev and G. A. Swartzlander, “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169–1176 (2003).
[Crossref]

2002 (4)

A. I. Mokhun, M. S. Soskin, and I. Freund, “Elliptic critical points: C-points, a-lines, and the sign rule,” Opt. Lett. 27, 995–997 (2002).
[Crossref]

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
[Crossref]

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
[Crossref]

I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).
[Crossref]

2001 (3)

M. V. Berry, “Geometry of phase and polarization singularities illustrated by edge diffraction and the tides,” Proc. SPIE 4403, 1–12 (2001).
[Crossref]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).
[Crossref]

1998 (1)

1983 (1)

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).
[Crossref]

Alfano, R. R.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

Alonso, M. A.

Arnold, V. I.

V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Vol. 250 of Grundlehren der Mathematischen Wissenschaften (Springer, 2012).

Beckley, A. M.

Berry, M. V.

M. V. Berry, “Geometry of phase and polarization singularities illustrated by edge diffraction and the tides,” Proc. SPIE 4403, 1–12 (2001).
[Crossref]

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).
[Crossref]

Bliokh, K. Y.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Brasselet, E.

E. Brasselet and S. Juodkazis, “Intangible pointlike tracers for liquid-crystal-based microsensors,” Phys. Rev. A 82, 063832 (2010).
[Crossref]

Brown, T. G.

Cardano, F.

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

de Lisio, C.

Dennis, M. R.

M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
[Crossref]

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402–11411 (2006).
[Crossref]

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
[Crossref]

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).
[Crossref]

Derevyanko, S.

U. Levy, S. Derevyanko, and Y. Silberberg, “Chapter four-light modes of free space,” Prog. Opt. 61, 237–281 (2016).
[Crossref]

Evans, S.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref]

Flossmann, F.

Freund, I.

A. I. Mokhun, M. S. Soskin, and I. Freund, “Elliptic critical points: C-points, a-lines, and the sign rule,” Opt. Lett. 27, 995–997 (2002).
[Crossref]

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
[Crossref]

I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).
[Crossref]

Galvez, E. J.

Hasman, E.

Juodkazis, S.

E. Brasselet and S. Juodkazis, “Intangible pointlike tracers for liquid-crystal-based microsensors,” Phys. Rev. A 82, 063832 (2010).
[Crossref]

Karimi, E.

Khadka, S.

Kleiner, V.

Kozawa, Y.

Levy, U.

U. Levy, S. Derevyanko, and Y. Silberberg, “Chapter four-light modes of free space,” Prog. Opt. 61, 237–281 (2016).
[Crossref]

Li, Y. P.

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Maier, M.

Maleev, I. D.

Marrucci, L.

Milione, G.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

Mokhun, A. I.

A. I. Mokhun, M. S. Soskin, and I. Freund, “Elliptic critical points: C-points, a-lines, and the sign rule,” Opt. Lett. 27, 995–997 (2002).
[Crossref]

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
[Crossref]

Niv, A.

Nolan, D. A.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

Nomoto, S.

Nye, J. F.

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).
[Crossref]

Orlov, S.

S. Orlov, K. Regelskis, and A. Stabinis, “Free-space propagation of second harmonic Laguerre–Gaussian beams carrying phase singularity,” J. Opt. A 6, S255–S258 (2004).
[Crossref]

S. Orlov, K. Regelskis, V. Smilgevičius, and A. Stabinis, “Free-space propagation of second harmonic beams carrying optical vortices,” Opt. Commun. 215, 1–9 (2003).
[Crossref]

Regelskis, K.

S. Orlov, K. Regelskis, and A. Stabinis, “Free-space propagation of second harmonic Laguerre–Gaussian beams carrying phase singularity,” J. Opt. A 6, S255–S258 (2004).
[Crossref]

S. Orlov, K. Regelskis, V. Smilgevičius, and A. Stabinis, “Free-space propagation of second harmonic beams carrying optical vortices,” Opt. Commun. 215, 1–9 (2003).
[Crossref]

Santamato, E.

Sato, S.

Schubert, W. H.

Schwarz, U. T.

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Silberberg, Y.

U. Levy, S. Derevyanko, and Y. Silberberg, “Chapter four-light modes of free space,” Prog. Opt. 61, 237–281 (2016).
[Crossref]

Slussarenko, S.

Smilgevicius, V.

S. Orlov, K. Regelskis, V. Smilgevičius, and A. Stabinis, “Free-space propagation of second harmonic beams carrying optical vortices,” Opt. Commun. 215, 1–9 (2003).
[Crossref]

Soskin, M. S.

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
[Crossref]

A. I. Mokhun, M. S. Soskin, and I. Freund, “Elliptic critical points: C-points, a-lines, and the sign rule,” Opt. Lett. 27, 995–997 (2002).
[Crossref]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

Stabinis, A.

S. Orlov, K. Regelskis, and A. Stabinis, “Free-space propagation of second harmonic Laguerre–Gaussian beams carrying phase singularity,” J. Opt. A 6, S255–S258 (2004).
[Crossref]

S. Orlov, K. Regelskis, V. Smilgevičius, and A. Stabinis, “Free-space propagation of second harmonic beams carrying optical vortices,” Opt. Commun. 215, 1–9 (2003).
[Crossref]

Strogatz, S. H.

S. H. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering (Westview, 2014).

Swartzlander, G. A.

Sztul, H. I.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

Tovar, A. A.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

Vyas, S.

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Zhan, Q.

Zhang, Y.

Zhao, Y.

Adv. Opt. Photon. (1)

Appl. Opt. (2)

J. Opt. A (1)

S. Orlov, K. Regelskis, and A. Stabinis, “Free-space propagation of second harmonic Laguerre–Gaussian beams carrying phase singularity,” J. Opt. A 6, S255–S258 (2004).
[Crossref]

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

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

Nat. Photonics (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Opt. Commun. (4)

S. Orlov, K. Regelskis, V. Smilgevičius, and A. Stabinis, “Free-space propagation of second harmonic beams carrying optical vortices,” Opt. Commun. 215, 1–9 (2003).
[Crossref]

I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).
[Crossref]

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
[Crossref]

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
[Crossref]

Opt. Express (5)

Opt. Lett. (4)

Phys. Rev. A (1)

E. Brasselet and S. Juodkazis, “Intangible pointlike tracers for liquid-crystal-based microsensors,” Phys. Rev. A 82, 063832 (2010).
[Crossref]

Phys. Rev. Lett. (2)

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

Proc. R. Soc. London A (2)

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).
[Crossref]

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).
[Crossref]

Proc. SPIE (1)

M. V. Berry, “Geometry of phase and polarization singularities illustrated by edge diffraction and the tides,” Proc. SPIE 4403, 1–12 (2001).
[Crossref]

Prog. Opt. (2)

U. Levy, S. Derevyanko, and Y. Silberberg, “Chapter four-light modes of free space,” Prog. Opt. 61, 237–281 (2016).
[Crossref]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

Other (3)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Vol. 250 of Grundlehren der Mathematischen Wissenschaften (Springer, 2012).

S. H. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering (Westview, 2014).

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Figures (19)

Fig. 1.
Fig. 1. Azimuthal system used in this paper. Different colors show different azimuthal values ϕ.
Fig. 2.
Fig. 2. Numerically calculated intensity distributions and polarization elliptic distributions of vector LG beams with p=0. (a1)–(a4) m=1; from left to right: type-1, type-2, type-3, and type-4. (b1)–(b4) m=2; from left to right: type-1, type-2, type-3, and type-4.
Fig. 3.
Fig. 3. Intensity distributions of two coaxial superposed LG beams in different cases when δ=0. (a1) m1=1, type-1; m2=2, type-2. (b1) m1=1, type-1; m2=2, type-1. (a2) m1=1, type-1; m2=4, type-1. (b2) m1=1, type-1; m2=4, type-2. (a3) m1=2, type-1; m2=2, type-1. (b3) m1=2, type-1; m2=2, type-2.
Fig. 4.
Fig. 4. Two vector LG beams separated along the x axis by a distance of a+b.
Fig. 5.
Fig. 5. Relationship between intensity and σ, δ in case 1 and case 4. In case 1, the type-1 beams are located at the points (s,0) and (s,0). In case 4, the type-1 beam is located at the point (s,0) and the type-2 beam is located at the point (s,0).
Fig. 6.
Fig. 6. Relationship between intensity and σ, δ in cases 2, 3, 5, and 6. In case 2, the type-1 beam is located at the point (s,0) and the type-3 beam is located at the point (s,0). In case 3, the type-2 beam is located at the point (s,0) and the type-4 beam is located at the point (s,0). In case 5, the type-2 beam is located at the point (s,0) and the type-3 beam is located at the point (s,0). In case 6, the type-1 beam is located at the point (s,0) and the type-4 beam is located at the point (s,0).
Fig. 7.
Fig. 7. Phase ϕ12 distributions for vector LG beams with different azimuthal indices and types: (a1) m=1, type-1; (a2) m=2, type-1; (a3) m=3, type-1; (b1) m=1, type-2; (b2) m=2, type-2; (b3) m=3, type-2. The state of polarization is also shown in all the figures.
Fig. 8.
Fig. 8. Calculated polarization field distributions of two superposed vector LG beams. m1=1, type-1; m2=2, type-1. δ=0. (a1)–(a3) Intensity and polarization ellipse distributions at z=zR, z=0, and z=zR, respectively. (b1)–(b3) The corresponding ϕ12 distributions, where the red points represent right-handed polarized C-points and the blue points represent left-handed polarized C-points.
Fig. 9.
Fig. 9. Calculated polarization field distributions of two superposed vector LG beams. m1=1, type-1; m2=2, type-1. (a1, b1) z=0, δ=π/4; (a2, b2) z=zR, δ=π/4; (a3, b3) z=zR, δ=π/4.
Fig. 10.
Fig. 10. Calculated polarization field distributions of two superposed vector LG beams. Beam 1: m1=1, type-1; beam 2: m2=2, type-2, δ=0. (a1)–(a3) Intensity distributions and polarization ellipse distributions of the composite fields at z=zR, z=0, and z=zR, respectively. (b1)–(b3) The corresponding ϕ12 distributions, where the red points represent right-handed polarized C-points and the blue points represent left-handed polarized C-points.
Fig. 11.
Fig. 11. Calculated polarization field distributions of two superposed vector LG beams. Beam 1: m1=1, type-1; beam 2: m2=2, type-2. (a1, b1) z=0, δ=π/4; (a2, b2) z=zR, δ=π/4; (a3, b3) z=zR, δ=π/4.
Fig. 12.
Fig. 12. Intensity and polarization distribution of two superposed vector LG beams in case 1 (case 1’) at the beam waist.
Fig. 13.
Fig. 13. Intensity and polarization distribution of two superposed vector LG beams in case 1 (case 1’) at z=zR. Red and blue dots imply right-handed and left-handed polarized C-points, respectively.
Fig. 14.
Fig. 14. Intensity and polarization distribution of two superposed vector LG beams in case 4 (case 5’) at the beam waist. The type-1 beam is located at (s,0), and the type-2 beam is located at (s,0).
Fig. 15.
Fig. 15. Intensity and polarization distribution of two superposed vector LG beams in case 4 (case 5’) at z=zR. The type-1 beam is located at (s,0), and the type-2 beam is located at (s,0). Red and blue dots imply right-handed and left-handed polarized C-points, respectively.
Fig. 16.
Fig. 16. Intensity and polarization distribution of two superposed vector LG beams in case 2 (case 3’) at the beam waist. The type-1 beam is located at (s,0), and the type-3 beam is located at (s,0).
Fig. 17.
Fig. 17. Intensity and polarization distribution of two superposed vector LG beams in case 2 (case 3’) at z=zR. The type-1 beam is located at (s,0), and the type-3 beam is located at (s,0). Red and blue dots imply right-handed and left-handed polarized C-points, respectively.
Fig. 18.
Fig. 18. Intensity and polarization distribution of two superposed vector LG beams in case 6 (case 7’) at the beam waist. The type-1 beam is located at (s,0), and the type-4 beam is located at (s,0).
Fig. 19.
Fig. 19. Intensity and polarization distribution of two superposed vector LG beams in case 6 (case 7’) at z=zR. The type-1 beam is located at (s,0), and the type-4 beam is located at (s,0). Red and blue dots imply right-handed and left-handed polarized C-points, respectively.

Tables (2)

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Table 1. Relationships between Different Cases and Beam Types

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Table 2. Relationships between Different Cases and Beam Types

Equations (29)

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Ep,m(r,ϕ,z)={Yp,m[cos(m1)ϕe^ϕsin(m1)ϕe^r]Yp,m[cos(m+1)ϕe^ϕ+sin(m+1)ϕe^r]Yp,m[sin(m1)ϕe^ϕ+cos(m1)ϕe^r]Yp,m[sin(m+1)ϕe^ϕ+cos(m+1)ϕe^r]},
Yp,m=Ap,m·r|m|ω|m|+1·G·Lpm·Up,m·exp(ikz),
ρ=m/2ω0.
E=I1E1+I2E2eiδ,
I=|E|2=I1|E1|2+I2|E2|2+2I1I2|E1||E2|Q(ϕ)S(z).
Q(ϕ)δ=0={cos(m1m2)ϕsin(m1m2)ϕsin(m1m2)ϕcos(m1+m2)ϕsin(m1+m2)ϕsin(m1+m2)ϕ}.
rzero=ω2(|m2|!|m1|!·E12E22)12(|m2||m|1).
zzero=zRtan(nπ+δ||m1||m2||),n=0,±1,±2,and|n|<||m1||m2||2.
E(r,ϕ,z)=I1E1(r1,ϕ1,z)+I2E2(r2,ϕ2,z)eiδ,
r1cos(ϕ1)=rcos(ϕ)+a=x+a,r2cos(ϕ2)=rcos(ϕ)b=xb,r1sin(ϕ1)=rsin(ϕ)=y,r2sin(ϕ2)=rsin(ϕ)=y.
Ep,m(r,ϕ,z)={Yp,m[sin(mϕ)e^x+cos(mϕ)e^y]Yp,m[sin(mϕ)e^x+cos(mϕ)ϕe^y]Yp,m[cos(mϕ)e^x+sin(mϕ)ϕe^y]Yp,m[cos(mϕ)e^xsin(mϕ)ϕe^y]}.
E(x,y,z)=I1E1(x,y,z)exp[i(ϕ1(x,y,z))]+I2E2(x,y,z)exp[i(ϕ2(x,y,z)+δ)].
I1E1(x,y,z)=E012yω012(1+z2/zR12)×exp[(x+a)2+y2ω012(1+z2/zR12)]e^x+E012(x+a)ω012(1+z2/zR12)×exp[(x+a)2+y2ω012(1+z2/zR12)]e^yϕ1(x,y,z)=kzk[(x+a)2+y2]2R1+2arctan(z/zR1),I2E2(x,y,z)=E022yω022(1+z2/zR22)×exp[(xb)2+y2ω022(1+z2/zR22)]e^x+E022(xb)ω022(1+z2/zR22)×exp[(xb)2+y2ω022(1+z2/zR22)]e^yϕ2(x,y,z)=kzk[(xb)2+y2]2R2+2arctan(z/zR2).
E(x,y,0)={E012ω012yexp((x+a)2+y2ω012)E022ω022yexp((xb)2+y2ω022+iδ)}e^x+{E012ω012(x+a)exp((x+a)2+y2ω012)+E022ω022(xb)exp((xb)2+y2ω022+iδ)}e^y=CX1e^x+CY1e^y.
x=stanh[2σ2x/s+12lnE01E02+i2δ],ycosh[2σ2x/s+12lnE01E02+i2δ]=0.
x=stanh[2σ2x/s+12lnE01E02+i2δ],ysinh[2σ2x/s+12lnE01E02+i2δ]=0.
x=stanh[2σ2x/s+12lnE01E02+i2(δ+π2)],ycosh[2σ2x/s+12lnE01E02+i2(δ+π2)]=0.
x=stanh[2σ2x/s+12lnE01E02+i2(δπ2)],ycosh[2σ2x/s+12lnE01E02+i2(δπ2)]=0.
x=stanh[2σ2x/s+12lnE01E02+i2(δ+π2)],ysinh[2σ2x/s+12lnE01E02+i2(δ+π2)]=0.
x=stanh[2σ2x/s+12lnE01E02+i2(δπ2)],ysinh[2σ2x/s+12lnE01E02+i2(δπ2)]=0.
s0=|Ex|2+|Ey|2,s1=|Ex|2|Ey|2,s2=2Re(Ex*Ey),s3=2Im(Ex*Ey).
ψ12=2IC,ψ12=2η.
I1|E1|2+I2|E2|2+2I1I2|E1||E2|C(ϕ,z)=0.
C+(ϕ,z)={cos[(m1m2)ϕ+(|m1||m2|)tan1(z/zR)δ]cos[(m1m2)ϕ(|m1||m2|)tan1(z/zR)+δ]sin[(m1m2)ϕ+(|m1||m2|)tan1(z/zR)δ]sin[(m1m2)ϕ(|m1||m2|)tan1(z/zR)+δ]cos[(m1+m2)ϕ+(|m1||m2|)tan1(z/zR)δ]sin[(m1+m2)ϕ(|m1||m2|)tan1(z/zR)+δ]sin[(m1+m2)ϕ+(|m1||m2|)tan1(z/zR)δ]}.
C(ϕ,z)={cos[(m1m2)ϕ(|m1||m2|)tan1(z/zR)+δ]cos[(m1m2)ϕ+(|m1||m2|)tan1(z/zR)δ]sin[(m1m2)ϕ(|m1||m2|)tan1(z/zR)+δ]sin[(m1m2)ϕ+(|m1||m2|)tan1(z/zR)δ]cos[(m1+m2)ϕ(|m1||m2|)tan1(z/zR)+δ]sin[(m1+m2)ϕ+(|m1||m2|)tan1(z/zR)δ]sin[(m1+m2)ϕ(|m1||m2|)tan1(z/zR)+δ]}.
rc=ω2(|m2|!|m1|!·E12E22)12(|m2||m|1)=rzero.
C+(ϕ,z)=cos[(m1m2)ϕ(|m1||m2|)tan1(z/zR)+δ]=1.
C(ϕ,z)=cos[(m1m2)ϕ+(|m1||m2|)tan1(z/zR)δ]=1.
ϕc+=(2n+1)π+(|m1||m2|)tan1(z/zR)δm1m2,ϕc=(2n+1)π(|m1||m2|)tan1(z/zR)+δm1m2,n=0,±1,±2,.

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