Abstract

Based on the vector Fresnel diffraction integrals, analytical expressions for the electric and magnetic components of first-order Laguerre–Gaussian beams diffracted at a half-plane screen are derived and used to study the electric and magnetic polarization singularities in the diffraction field for both two- and three-dimensional (2D and 3D) cases. It is shown that there exist 2D and 3D electric and magnetic polarization singularities in the diffraction field, which do not coincide each other in general. By suitably varying the waist width ratio, off-axis displacement parameter, amplitude ratio, or propagation distance, the motion, pair-creation, and annihilation of circular polarization singularities, and the motion of linear polarization singularities take place in 2D and 3D electric and magnetic fields. The V point, at which two circular polarization singularities with the same topological charge but opposite handedness collide, appears in the 2D electric field under certain conditions in the diffraction field and free-space propagation. A comparison with the free-space propagation is also made.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic waves,” Proc. R. Soc. London Ser. A 409, 21–36 (1987).
    [CrossRef]
  2. J. V. Hajnal, “Observation of singularities in the electric and magnetic fields of freely propagating microwaves,” Proc. R. Soc. London Ser. A 430, 413–421 (1990).
    [CrossRef]
  3. J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP, 1999).
  4. M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London Ser. A 457, 141–155 (2001).
    [CrossRef]
  5. M. R. Dennis, “Polarization singularities in paraxial vector fields morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
    [CrossRef]
  6. 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]
  7. I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27, 545–547 (2002).
    [CrossRef]
  8. O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
    [CrossRef]
  9. O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57–65 (2002).
    [CrossRef]
  10. M. S. Soskin, V. G. Denisenko, and R. I. Egorov, “Singular Stokes-polarimetry as new technique for metrology and inspection of polarized speckle fields,” Proc. SPIE 5458, 79–85 (2004).
    [CrossRef]
  11. M. V. Berry, “The electric and magnetic polarization singularities of paraxial waves,” J. Opt. A 6, 475–481 (2004).
    [CrossRef]
  12. F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
    [CrossRef]
  13. R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express 14, 5733–5745 (2006).
    [CrossRef]
  14. K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express 16, 695–709 (2008).
    [CrossRef]
  15. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
    [CrossRef]
  16. A. A. Chernyshov, C. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A 11, 094010(8) (2009).
    [CrossRef]
  17. V. Savaryn, Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “Polarization singularities of optical fields caused by structural dislocations in crystals,” J. Opt. 15, 044023(9) (2013).
    [CrossRef]
  18. I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
    [CrossRef]

2013

V. Savaryn, Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “Polarization singularities of optical fields caused by structural dislocations in crystals,” J. Opt. 15, 044023(9) (2013).
[CrossRef]

2009

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

A. A. Chernyshov, C. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A 11, 094010(8) (2009).
[CrossRef]

2008

2006

2005

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

2004

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, “Singular Stokes-polarimetry as new technique for metrology and inspection of polarized speckle fields,” Proc. SPIE 5458, 79–85 (2004).
[CrossRef]

M. V. Berry, “The electric and magnetic polarization singularities of paraxial waves,” J. Opt. A 6, 475–481 (2004).
[CrossRef]

2002

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

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57–65 (2002).
[CrossRef]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27, 545–547 (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]

2001

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

1994

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef]

1990

J. V. Hajnal, “Observation of singularities in the electric and magnetic fields of freely propagating microwaves,” Proc. R. Soc. London Ser. A 430, 413–421 (1990).
[CrossRef]

1987

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic waves,” Proc. R. Soc. London Ser. A 409, 21–36 (1987).
[CrossRef]

Angelsky, O. V.

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27, 545–547 (2002).
[CrossRef]

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57–65 (2002).
[CrossRef]

Berry, M. V.

M. V. Berry, “The electric and magnetic polarization singularities of paraxial waves,” J. Opt. A 6, 475–481 (2004).
[CrossRef]

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

Bliokh, K. Y.

Bogatyryova, H. V.

A. A. Chernyshov, C. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A 11, 094010(8) (2009).
[CrossRef]

Chernyshov, A. A.

A. A. Chernyshov, C. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A 11, 094010(8) (2009).
[CrossRef]

Denisenko, V. G.

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, “Singular Stokes-polarimetry as new technique for metrology and inspection of polarized speckle fields,” Proc. SPIE 5458, 79–85 (2004).
[CrossRef]

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[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 Ser. A 457, 141–155 (2001).
[CrossRef]

Egorov, R. I.

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, “Singular Stokes-polarimetry as new technique for metrology and inspection of polarized speckle fields,” Proc. SPIE 5458, 79–85 (2004).
[CrossRef]

Felde, C. V.

A. A. Chernyshov, C. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A 11, 094010(8) (2009).
[CrossRef]

Flossmann, F.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Freund, I.

Hajnal, J. V.

J. V. Hajnal, “Observation of singularities in the electric and magnetic fields of freely propagating microwaves,” Proc. R. Soc. London Ser. A 430, 413–421 (1990).
[CrossRef]

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic waves,” Proc. R. Soc. London Ser. A 409, 21–36 (1987).
[CrossRef]

Hasman, E.

Kleiner, V.

Krupych, O.

V. Savaryn, Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “Polarization singularities of optical fields caused by structural dislocations in crystals,” J. Opt. 15, 044023(9) (2013).
[CrossRef]

Maier, M.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[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]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57–65 (2002).
[CrossRef]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27, 545–547 (2002).
[CrossRef]

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

Mokhun, I. I.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27, 545–547 (2002).
[CrossRef]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57–65 (2002).
[CrossRef]

Niv, A.

Nye, J. F.

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic waves,” Proc. R. Soc. London Ser. A 409, 21–36 (1987).
[CrossRef]

J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP, 1999).

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

Polyanskii, P. V.

A. A. Chernyshov, C. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A 11, 094010(8) (2009).
[CrossRef]

Savaryn, V.

V. Savaryn, Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “Polarization singularities of optical fields caused by structural dislocations in crystals,” J. Opt. 15, 044023(9) (2013).
[CrossRef]

Schoonover, R. W.

Schwarz, U. T.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Shvartsman, N.

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef]

Skab, I.

V. Savaryn, Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “Polarization singularities of optical fields caused by structural dislocations in crystals,” J. Opt. 15, 044023(9) (2013).
[CrossRef]

Soskin, M. S.

A. A. Chernyshov, C. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A 11, 094010(8) (2009).
[CrossRef]

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, “Singular Stokes-polarimetry as new technique for metrology and inspection of polarized speckle fields,” Proc. SPIE 5458, 79–85 (2004).
[CrossRef]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57–65 (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]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27, 545–547 (2002).
[CrossRef]

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

Vasylkiv, Y.

V. Savaryn, Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “Polarization singularities of optical fields caused by structural dislocations in crystals,” J. Opt. 15, 044023(9) (2013).
[CrossRef]

Visser, T. D.

Vlokh, R.

V. Savaryn, Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “Polarization singularities of optical fields caused by structural dislocations in crystals,” J. Opt. 15, 044023(9) (2013).
[CrossRef]

J. Opt.

V. Savaryn, Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “Polarization singularities of optical fields caused by structural dislocations in crystals,” J. Opt. 15, 044023(9) (2013).
[CrossRef]

J. Opt. A

A. A. Chernyshov, C. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A 11, 094010(8) (2009).
[CrossRef]

M. V. Berry, “The electric and magnetic polarization singularities of paraxial waves,” J. Opt. A 6, 475–481 (2004).
[CrossRef]

Opt. Commun.

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57–65 (2002).
[CrossRef]

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

Opt. Express

Opt. Lett.

Phys. Rev. A

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef]

Phys. Rev. E

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

Phys. Rev. Lett.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Proc. R. Soc. London Ser. A

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic waves,” Proc. R. Soc. London Ser. A 409, 21–36 (1987).
[CrossRef]

J. V. Hajnal, “Observation of singularities in the electric and magnetic fields of freely propagating microwaves,” Proc. R. Soc. London Ser. A 430, 413–421 (1990).
[CrossRef]

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

Proc. SPIE

M. S. Soskin, V. G. Denisenko, and R. I. Egorov, “Singular Stokes-polarimetry as new technique for metrology and inspection of polarized speckle fields,” Proc. SPIE 5458, 79–85 (2004).
[CrossRef]

Prog. Opt.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

Other

J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP, 1999).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Contour lines of the real and imaginary parts of Eqs. (7a) and (7b) for different values of the off-axis displacement parameter: (a), (d) a=4λ; (b), (e) a=24λ; (c) a=26.36λ; and (f) a=26.34λ.

Fig. 2.
Fig. 2.

Contour lines of the real and imaginary parts of Eqs. (7c) and (7d) for different values of the off-axis displacement parameter: (a), (d) a=4λ; (b), (e) a=24λ; (c) a=26.32λ; and (f) a=26.28λ.

Fig. 3.
Fig. 3.

Distance Δ/λ versus the off-axis displacement parameter a.

Fig. 4.
Fig. 4.

Contour lines of the real and imaginary parts of Eq. (7a) at b=a: (a) a=2λ and (b) a=10λ.

Fig. 5.
Fig. 5.

Linear polarization singularities for different values of the propagation distance z: (a) z=1000λ and (b) z=1020λ.

Fig. 6.
Fig. 6.

(a) Contour lines the real and imaginary parts of Eq. (7a) at a=4λ, z=1000λ in free space and (b) distance Δ/λ versus the off-axis displacement parameter a.

Equations (27)

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

T(x0,y0)={1,y000y0<0,
Ex(x0,y0,0)=T(x0,y0)E0x2w0x(x0a+iy0)exp[x02+y02w0x2],
Ey(x0,y0,0)=T(x0,y0)E0y2w0y(x0b+iy0)exp[x02+y02w0y2],
Ex(x,y,z)=eikzλziEx(x0,y0,0)exp{ik2z[(xx0)2+(yy0)2]}dx0dy0,
Ey(x,y,z)=eikzλziEy(x0,y0,0)exp{ik2z[(xx0)2+(yy0)2]}dx0dy0,
Ex(x,y,z)=E0x(kw0x2+2iz)2(πw0xzek(x2+y2+z(ikw0x2+2z)kw0x2+2iz)(1z(1i)πw0x2w0x2ikz×((kw0x2+2iz)z)3/2(akw0x2kw0x2x+2iaz)ik2w0x2y2kw0x2z+2iz2erfi(kw0xy(1+i)2z(kw0x2+2iz)))+(kw0x2+2iz)((kw0x2+2iz)ik2w0x2y2kw0x2z+2iz2(a2π(kw0x2+2iz)+w0x2(k2π(x+iy)2ze(ik2w0x2y22kw0x2+4iz2)2w0x2ikz))+ik2w0x4y24πw0x22ikπzerfi(ik2w0x2y22kw0x2z+4iz2))),
Ey(x,y,z)=E0y(kw0y2+2iz)2(πw0yzek(x2+y2+z(ikw0y2+2z)kw0y2+2iz)(1z(1i)πw0y2w0y2ikz×((kw0y2+2iz)z)3/2(bkw0y2kw0y2x+2ibz)ik2w0y2y2kw0y2z+2iz2erfi(kw0yy(1+i)2z(kw0y2+2iz)))+(kw0y2+2iz)((kw0y2+2iz)ik2w0y2y2kw0y2z+2iz2(b2π(kw0y2+2iz)+w0y2(k2π(x+iy)2ze(ik2w0y2y22kw0y2+4iz2)2w0y2ikz))+ik2w0y4y24πw0y22ikπzerfi(ik2w0y2y22kw0y2z+4iz2))),
Ezikt·Et,Bzikt·Bt,
Bx(x,y,z)=1c{Ey12k2[(2x22y2)Ey22xyEx]},
By(x,y,z)=1c{Ex12k2[(2x22y2)Ex+22xyEy]},
Et·Et=0(cElines),
Bt·Bt=0(cBlines),
E·E=0(CElines),
B·B=0(CBlines),
ER=Ex+iEy2,EL=ExiEy2,
ImEt*×Et=0(lEsurfaces),
ImBt*×Bt=0(lBsurfaces),
ImE*×E=0(LElines),
ImB*×B=0(LBlines),
|ER|2|EL|2=0(lElines),
|BR|2|BL|2=0(lBlines),
Re(Et*t·Et)=0(LBpoints),
Re(Bt*t·Bt)=0(LBpoints),
Ex(x,y,z)=22E0x[kπw0x4(x+iya)2iaπzw0x2]λw0x(kw0x2+2iz)2exp{ik(z+i(x2+y2)kw0x2+2iz)},
Ey(x,y,z)=22E0y[kπw0y4(x+iyb)2ibπzw0y2]λw0y(kw0y2+2iz)2exp{ik(z+i(x2+y2)kw0y2+2iz)}.
x=a,
y=2azkw02,

Metrics