Abstract

The Gouy phase, sometimes called the focal phase anomaly, is the curious effect that in the vicinity of its focus a diffracted field, compared to a non-diffracted, converging spherical wave of the same frequency, undergoes a rapid phase change by an amount of π. We theoretically investigate the phase behavior and the polarization ellipse of a strongly focused, radially polarized beam. We find that the significant variation of the state of polarization in the focal region, is a manifestation of the different Gouy phases that the two electric field components undergo.

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  1. L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Comptes Rendus hebdomadaires des Séances de l’Académie des Sciences 110, 1251–1253 (1890).
  2. L. G. Gouy, “Sur la propagation anomale des ondes,” Annales des Chimie et de Physique 6eséries 24, 145–213 (1891).
  3. S. M. Baumann, D. M. Kalb, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17, 9818–9827 (2009).
    [CrossRef] [PubMed]
  4. G. M. Philip, V. Kumar, G. Milione, and N. K. Viswanathan, “Manifestation of the Gouy phase in vector-vortex beams,” Opt. Lett. 37, 2667–2669 (2012).
    [CrossRef] [PubMed]
  5. W. Zhu, A. Agrawal, and A. Nahata, “Direct measurement of the Gouy phase shift for surface plasmon-polaritons,” Opt. Express 15, 9995–10001 (2007).
    [CrossRef] [PubMed]
  6. T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
    [CrossRef]
  7. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  8. G. Lamouche, M. L. Dufour, B. Gauthier, and J.-P. Monchalin, “Gouy phase anomaly in optical coherence tomography,” Opt. Commun. 239, 297–301 (2004).
    [CrossRef]
  9. T. Klaassen, A. Hoogeboom, M. P. van Exter, and J. P. Woerdman, “Gouy phase of nonparaxial eigenmodes in a folded resonator,” J. Opt. Soc. Am. A 21, 1689–1693 (2004).
    [CrossRef]
  10. X. Pang, D. G. Fischer, and T. D. Visser, “A generalized Gouy phase for focused partially coherent light and its implications for interferometry,” J. Opt. Soc. Am. A 29, 989–993. (2012).
    [CrossRef]
  11. X. Pang, T. D. Visser, and E. Wolf, “Phase anomaly and phase singularities of the field in the focal region of high-numerical aperture systems,” Opt. Commun. 284, 5517–5522 (2011).
    [CrossRef]
  12. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
    [CrossRef] [PubMed]
  13. R. Martínez-Herrero and P. M. Mejías, “Propagation and parametric characterization of the polarization structure of paraxial radially and azimuthally polarized beams,” Opt. Laser Techn. 44, 482–485 (2012).
    [CrossRef]
  14. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
  15. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [CrossRef] [PubMed]
  16. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004).
    [CrossRef] [PubMed]
  17. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
    [CrossRef] [PubMed]
  18. T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33, 122–124 (2008).
    [CrossRef] [PubMed]
  19. D. P. Biss, K. S. Youngworth, and T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt. 45, 470–479 (2006).
    [CrossRef] [PubMed]
  20. T. G. Brown, “Unconventional polarization states: Beam propagation, focusing, and imaging,” in Progress in Optics, E. Wolf, eds. (Elsevier, 2011), 56, pp. 81–129.
    [CrossRef]
  21. T. D. Visser and J. T. Foley, “On the wavefront spacing of focused, radially polarized beams,” J. Opt. Soc. Am. A 22, 2527–2531 (2005).
    [CrossRef]
  22. H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
    [CrossRef]
  23. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  24. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Seventh (expanded) edition, (Cambridge University Press, 1999).
  25. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).
  26. R.W. Schoonover and T.D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express 14, 5733–5745 (2006).
    [CrossRef] [PubMed]
  27. R. Martínez-Herrero and P.M. Mejías, “Stokes-parameters representation in terms of the radial and azimuthal field components: A proposal,” Opt. Laser Techn. 42, 1099–1102 (2010).
    [CrossRef]
  28. G.J. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge University Press, Cambridge, 2011), pp. 91–93.
  29. D.W. Diehl, R.W. Schoonover, and T.D. Visser, “The structure of focused, radially polarized fields,” Opt. Express 14, 3030–3038 (2006).
    [CrossRef] [PubMed]

2012 (3)

2011 (1)

X. Pang, T. D. Visser, and E. Wolf, “Phase anomaly and phase singularities of the field in the focal region of high-numerical aperture systems,” Opt. Commun. 284, 5517–5522 (2011).
[CrossRef]

2010 (2)

T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
[CrossRef]

R. Martínez-Herrero and P.M. Mejías, “Stokes-parameters representation in terms of the radial and azimuthal field components: A proposal,” Opt. Laser Techn. 42, 1099–1102 (2010).
[CrossRef]

2009 (1)

2008 (1)

2007 (2)

W. Zhu, A. Agrawal, and A. Nahata, “Direct measurement of the Gouy phase shift for surface plasmon-polaritons,” Opt. Express 15, 9995–10001 (2007).
[CrossRef] [PubMed]

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[CrossRef]

2006 (3)

2005 (1)

2004 (4)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

2000 (1)

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

1891 (1)

L. G. Gouy, “Sur la propagation anomale des ondes,” Annales des Chimie et de Physique 6eséries 24, 145–213 (1891).

1890 (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Comptes Rendus hebdomadaires des Séances de l’Académie des Sciences 110, 1251–1253 (1890).

Abramowitz, M.

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Agrawal, A.

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Baumann, S. M.

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Biss, D. P.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Seventh (expanded) edition, (Cambridge University Press, 1999).

Brown, T. G.

D. P. Biss, K. S. Youngworth, and T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt. 45, 470–479 (2006).
[CrossRef] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
[CrossRef] [PubMed]

T. G. Brown, “Unconventional polarization states: Beam propagation, focusing, and imaging,” in Progress in Optics, E. Wolf, eds. (Elsevier, 2011), 56, pp. 81–129.
[CrossRef]

Chen, H.

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[CrossRef]

Choudhury, A.

Diehl, D.W.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Dufour, M. L.

G. Lamouche, M. L. Dufour, B. Gauthier, and J.-P. Monchalin, “Gouy phase anomaly in optical coherence tomography,” Opt. Commun. 239, 297–301 (2004).
[CrossRef]

Fischer, D. G.

Foley, J. T.

Galvez, E. J.

Gauthier, B.

G. Lamouche, M. L. Dufour, B. Gauthier, and J.-P. Monchalin, “Gouy phase anomaly in optical coherence tomography,” Opt. Commun. 239, 297–301 (2004).
[CrossRef]

Gbur, G.J.

G.J. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge University Press, Cambridge, 2011), pp. 91–93.

Gouy, L. G.

L. G. Gouy, “Sur la propagation anomale des ondes,” Annales des Chimie et de Physique 6eséries 24, 145–213 (1891).

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Comptes Rendus hebdomadaires des Séances de l’Académie des Sciences 110, 1251–1253 (1890).

Heckenberg, N. R.

Hoogeboom, A.

Kalb, D. M.

Klaassen, T.

Kumar, V.

Lamouche, G.

G. Lamouche, M. L. Dufour, B. Gauthier, and J.-P. Monchalin, “Gouy phase anomaly in optical coherence tomography,” Opt. Commun. 239, 297–301 (2004).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Li, Y.-P.

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[CrossRef]

Martínez-Herrero, R.

R. Martínez-Herrero and P. M. Mejías, “Propagation and parametric characterization of the polarization structure of paraxial radially and azimuthally polarized beams,” Opt. Laser Techn. 44, 482–485 (2012).
[CrossRef]

R. Martínez-Herrero and P.M. Mejías, “Stokes-parameters representation in terms of the radial and azimuthal field components: A proposal,” Opt. Laser Techn. 42, 1099–1102 (2010).
[CrossRef]

Mejías, P. M.

R. Martínez-Herrero and P. M. Mejías, “Propagation and parametric characterization of the polarization structure of paraxial radially and azimuthally polarized beams,” Opt. Laser Techn. 44, 482–485 (2012).
[CrossRef]

Mejías, P.M.

R. Martínez-Herrero and P.M. Mejías, “Stokes-parameters representation in terms of the radial and azimuthal field components: A proposal,” Opt. Laser Techn. 42, 1099–1102 (2010).
[CrossRef]

Milione, G.

Monchalin, J.-P.

G. Lamouche, M. L. Dufour, B. Gauthier, and J.-P. Monchalin, “Gouy phase anomaly in optical coherence tomography,” Opt. Commun. 239, 297–301 (2004).
[CrossRef]

Nahata, A.

Nieminen, T. A.

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Pang, X.

X. Pang, D. G. Fischer, and T. D. Visser, “A generalized Gouy phase for focused partially coherent light and its implications for interferometry,” J. Opt. Soc. Am. A 29, 989–993. (2012).
[CrossRef]

X. Pang, T. D. Visser, and E. Wolf, “Phase anomaly and phase singularities of the field in the focal region of high-numerical aperture systems,” Opt. Commun. 284, 5517–5522 (2011).
[CrossRef]

Philip, G. M.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Rubinsztein-Dunlop, H.

Schoonover, R.W.

Sheppard, C. J. R.

Stegun, I.A.

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

van Exter, M. P.

Visser, T. D.

X. Pang, D. G. Fischer, and T. D. Visser, “A generalized Gouy phase for focused partially coherent light and its implications for interferometry,” J. Opt. Soc. Am. A 29, 989–993. (2012).
[CrossRef]

X. Pang, T. D. Visser, and E. Wolf, “Phase anomaly and phase singularities of the field in the focal region of high-numerical aperture systems,” Opt. Commun. 284, 5517–5522 (2011).
[CrossRef]

T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
[CrossRef]

T. D. Visser and J. T. Foley, “On the wavefront spacing of focused, radially polarized beams,” J. Opt. Soc. Am. A 22, 2527–2531 (2005).
[CrossRef]

Visser, T.D.

Viswanathan, N. K.

Woerdman, J. P.

T. Klaassen, A. Hoogeboom, M. P. van Exter, and J. P. Woerdman, “Gouy phase of nonparaxial eigenmodes in a folded resonator,” J. Opt. Soc. Am. A 21, 1689–1693 (2004).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Wolf, E.

X. Pang, T. D. Visser, and E. Wolf, “Phase anomaly and phase singularities of the field in the focal region of high-numerical aperture systems,” Opt. Commun. 284, 5517–5522 (2011).
[CrossRef]

T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
[CrossRef]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Seventh (expanded) edition, (Cambridge University Press, 1999).

Youngworth, K. S.

Zhan, Q.

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[CrossRef]

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
[CrossRef] [PubMed]

Zhang, Y.

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[CrossRef]

Zhu, W.

Annales des Chimie et de Physique (1)

L. G. Gouy, “Sur la propagation anomale des ondes,” Annales des Chimie et de Physique 6eséries 24, 145–213 (1891).

Appl. Opt. (2)

Comptes Rendus hebdomadaires des Séances de l’Académie des Sciences (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Comptes Rendus hebdomadaires des Séances de l’Académie des Sciences 110, 1251–1253 (1890).

J. Opt. Soc. Am. A (3)

Opt. Commun. (4)

X. Pang, T. D. Visser, and E. Wolf, “Phase anomaly and phase singularities of the field in the focal region of high-numerical aperture systems,” Opt. Commun. 284, 5517–5522 (2011).
[CrossRef]

T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

G. Lamouche, M. L. Dufour, B. Gauthier, and J.-P. Monchalin, “Gouy phase anomaly in optical coherence tomography,” Opt. Commun. 239, 297–301 (2004).
[CrossRef]

Opt. Express (6)

Opt. Laser Techn. (2)

R. Martínez-Herrero and P.M. Mejías, “Stokes-parameters representation in terms of the radial and azimuthal field components: A proposal,” Opt. Laser Techn. 42, 1099–1102 (2010).
[CrossRef]

R. Martínez-Herrero and P. M. Mejías, “Propagation and parametric characterization of the polarization structure of paraxial radially and azimuthally polarized beams,” Opt. Laser Techn. 44, 482–485 (2012).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Other (4)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Seventh (expanded) edition, (Cambridge University Press, 1999).

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

G.J. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge University Press, Cambridge, 2011), pp. 91–93.

T. G. Brown, “Unconventional polarization states: Beam propagation, focusing, and imaging,” in Progress in Optics, E. Wolf, eds. (Elsevier, 2011), 56, pp. 81–129.
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

A high-numerical-aperture focusing system with an incident beam that is radially polarized.

Fig. 2
Fig. 2

The Gouy phase δz along the optical axis (v = 0) of the electric field component ez for selected values of the semi-aperture angle α (blue curve: α = 40°, red curve: α = 50°, olive curve: α = 60°). The beam-size parameter β = 3.

Fig. 3
Fig. 3

The Gouy phase δz along the optical axis (v = 0) of the electric field component ez for selected values of the beam-size parameter β = f/ω0 (green curve: β = 1, blue curve: β = 2, red curve: β = 3, olive curve: β = 4). The semi-aperture angle α = 60°.

Fig. 4
Fig. 4

The Gouy phase of the longitudinal component ez (red curve) and that of the radial component eρ (blue curve) along an oblique ray through focus under an angle θ = 35°. Here α = 40° and β = 1.

Fig. 5
Fig. 5

The Poincaré sphere with Cartesian axes (s1,s2,s3) adapted for focused, radially polarized fields.

Fig. 6
Fig. 6

The normalized moduli of the longitudinal component ez (red curve) and that of the radial component eρ (blue curve) of the electric field along an oblique ray under an angle θ = 35° with the z-axis. Here we have chosen α = 40° and β = 1.

Fig. 7
Fig. 7

The Stokes parameters along an oblique ray through focus which makes an angle θ = 35° with the z-axis (s1: blue curve, s2: red curve, s3: olive curve). Here α = 40° and β = 1.

Fig. 8
Fig. 8

Defining the angles ψ and χ of a polarization ellipse.

Fig. 9
Fig. 9

The orientation angle ψ of the polarization ellipse along an oblique ray through focus under an angle θ = 35°. Here we have chosen α = 40° and β = 1.

Fig. 10
Fig. 10

The ellipticity angle χ of the polarization ellipse along an oblique ray through focus under an angle θ = 35°. Here we have chosen α = 40° and β = 1.

Fig. 11
Fig. 11

Illustration of the symmetry properties of the polarization ellipse. The electric field ellipse is shown at selected points along an oblique ray through focus. The ray is under an angle θ = 35°. Also, α = 40° and β = 1.

Fig. 12
Fig. 12

Polarization ellipse of the field at selected points along an oblique ray through focus. The ray is under an angle θ = 10°. Also, α = 40° and β = 1.

Fig. 13
Fig. 13

Polarization ellipse of the field at selected points along an oblique ray through focus. The ray is under an angle θ = 20°. Also, α = 40° and β = 1.

Fig. 14
Fig. 14

Polarization ellipse of the field at selected points along an oblique ray through focus. The ray is under an angle θ = 30°. Also, α = 40° and β = 1.

Equations (35)

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

E ( r , t ) = Re [ e ( r ) exp ( i ω t ) ] ,
H ( r , t ) = Re [ h ( r ) exp ( i ω t ) ] ,
e z ( ρ , z ) = i k f 0 α l ( θ ) sin 2 θ cos 1 / 2 θ e i k z cos θ J 0 ( k ρ sin θ ) d θ ,
e ρ ( ρ , z ) = k f 0 α l ( θ ) sin θ cos 3 / 2 θ e i k z cos θ J 1 ( k ρ sin θ ) d θ ,
l ( θ ) = f sin θ exp [ f 2 sin 2 θ / ω 0 2 ] ,
u = k z sin 2 α ,
v = k ρ sin α .
e z ( u , v ) = i k f 2 0 α sin 3 θ cos 1 / 2 θ e β 2 sin 2 θ e i u cos θ / sin 2 α J 0 ( v sin θ sin α ) d θ ,
e ρ ( u , v ) = k f 2 0 α sin 2 θ cos 3 / 2 θ e β 2 sin 2 θ e i u cos θ / sin 2 α J 1 ( v sin θ sin α ) d θ ,
β = f / ω 0 ,
e z ( u , v ) = e z * ( u , v ) ,
e ρ ( u , v ) = e ρ * ( u , v ) .
δ z ( u , v ) = arg [ e z ( u , v ) ] sign ( u ) k R ,
δ ρ ( u , v ) = arg [ e ρ ( u , v ) ] sign ( u ) k R ,
k R = k z 2 + ρ 2 = 1 sin α u 2 sin 2 α + v 2 ,
sign ( x ) = { 1 if x < 0 , 1 if x > 0 .
δ z ( u , v ) + δ z ( u , v ) = π ( mod 2 π ) .
δ z ( 0 , 0 ) = π / 2 ( mod 2 π ) .
δ ρ ( u , v ) + δ ρ ( u , v ) = 0 ( mod 2 π ) .
lim u 0 δ ρ ( u , v ) = lim u 0 δ ρ ( u , v ) = π ( mod 2 π ) .
S 0 = | e z | 2 + | e ρ | 2 ,
S 1 = | e z | 2 | e ρ | 2 ,
S 2 = 2 | e z | | e ρ | cos δ ,
S 3 = 2 | e z | | e ρ | sin δ ,
v = | u | tan θ / sin α .
| e z ( u , v ) | = | e z ( u , v ) | ,
| e ρ ( u , v ) | = | e ρ ( u , v ) | .
[ δ z ( u , v ) δ ρ ( u , v ) ] + [ δ z ( u , v ) δ ρ ( u , v ) ] = π ( mod 2 π ) ,
s 1 ( u , v ) = s 1 ( u , v ) ,
s 2 ( u , v ) = s 2 ( u , v ) ,
s 3 ( u , v ) = s 3 ( u , v ) .
ψ = 1 2 arctan ( s 2 s 1 ) ,
χ = 1 2 arcsin ( s 3 ) .
ψ ( u , v ) = π ψ ( u , v ) ,
χ ( u , v ) = χ ( u , v ) .

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