Abstract

Gouy phase anomaly of the coaxial focus array generated by a Dammann zone plate (DZP) was investigated. Based on Debye vectorial diffraction theory, the longitudinal-differential (LD) phase profile was presented in the focal region of a high numerical aperture aplanatic objective with a DZP located before as the pupil filter. The numerical simulations show that there is a Gouy phase shift exactly located at the position of every diffraction order along the axial direction. Besides the well-known Gouy phase, a type of extra irregular phase shift among some diffraction orders is demonstrated. This interesting phenomenon of the extra phase shift, physically, is attributed to transversal confinement inside the pupil aperture of the focusing system, which is induced by the discontinuities in phase retardation between any two adjacent zones of DZPs. These extra irregular phase shifts are very different for different kinds of DZPs, and thus this extra phase shift effect can be seen as a characteristic of certain DZPs. In addition, the 3D differential phase maps are presented for investigation of the subtle phase evolution in the off-axis volume.

© 2014 Optical Society of America

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References

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  1. F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express 18, 13563–13573 (2010).
    [CrossRef]
  2. A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474–5488 (2002).
    [CrossRef]
  3. L. Wilson and R. J. Zhang, “3D Localization of weak scatterers in digital holographic microscopy using Rayleigh-Sommerfeld back-propagation,” Opt. Express 20, 16735–16744 (2012).
    [CrossRef]
  4. Z. L. Horvath and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999).
    [CrossRef]
  5. N. Olivier and E. Beaurepaire, “Third-harmonic generation microscopy with focus-engineered beams: a numerical study,” Opt. Express 16, 14703–14715 (2008).
    [CrossRef]
  6. 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]
  7. S. M. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. 26, 485–487 (2001).
    [CrossRef]
  8. 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]
  9. Q. Zhan, “Second-order tilted wave interpretation of the Gouy phase shift under high numerical aperture uniform illumination,” Opt. Commun. 242, 351–360 (2004).
    [CrossRef]
  10. E. Y. S. Yew and C. J. R. Sheppard, “Fractional Gouy phase,” Opt. Lett. 33, 1363–1365 (2008).
    [CrossRef]
  11. M. S. Kim, A. D. Assafrao, T. Scharf, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Longitudinal-differential phase distribution near the focus of a high numerical aperture lens: study of wavefront spacing and Gouy phase,” J. Mod. Opt. 60, 197–201 (2013).
    [CrossRef]
  12. X. Pang and T. D. Visser, “Manifestation of the Gouy phase in strongly focused, radially polarized beams,” Opt. Express 21, 8331–8341 (2013).
    [CrossRef]
  13. M. S. Kim, T. Scharf, A. D. Assafrao, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Phase anomalies in Bessel-Gauss beams,” Opt. Express 20, 28929–28940 (2012).
    [CrossRef]
  14. P. Martelli, M. Tacca, A. Gatto, G. Moneta, and M. Martinelli, “Gouy phase shift in nondiffracting Bessel beams,” Opt. Express 18, 7108–7120 (2010).
    [CrossRef]
  15. X. Pang, G. Gbur, and T. D. Visser, “The Gouy phase of Airy beams,” Opt. Lett. 36, 2492–2494 (2011).
    [CrossRef]
  16. 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]
  17. H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional period structures,” Opt. Acta 24, 505–515 (1977).
    [CrossRef]
  18. C. Zhou and L. Liu, “Numercial study of Dammann array illuminators,” Appl. Opt. 34, 5961–5969 (1995).
    [CrossRef]
  19. J. Yu, C. Zhou, W. Jia, W. Cao, S. Wang, J. Ma, and H. Cao, “Three-dimensional Dammann array,” Appl. Opt. 51, 1619–1630 (2012).
    [CrossRef]
  20. J. Yu, C. Zhou, W. Jia, A. Hu, W. Cao, J. Wu, and S. Wang, “Three-dimensional Dammann vortex array with tunable topological charge,” Appl. Opt. 51, 2485–2490 (2012).
    [CrossRef]
  21. G. T. Francia, “Super-gain antennas and optical resolving power,” Il Nuovo Cimento 9, 426–438 (1952).
  22. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14, 11277–11291 (2006).
    [CrossRef]
  23. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
    [CrossRef]
  24. E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, and A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
    [CrossRef]

2013 (2)

M. S. Kim, A. D. Assafrao, T. Scharf, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Longitudinal-differential phase distribution near the focus of a high numerical aperture lens: study of wavefront spacing and Gouy phase,” J. Mod. Opt. 60, 197–201 (2013).
[CrossRef]

X. Pang and T. D. Visser, “Manifestation of the Gouy phase in strongly focused, radially polarized beams,” Opt. Express 21, 8331–8341 (2013).
[CrossRef]

2012 (5)

2011 (1)

2010 (4)

P. Martelli, M. Tacca, A. Gatto, G. Moneta, and M. Martinelli, “Gouy phase shift in nondiffracting Bessel beams,” Opt. Express 18, 7108–7120 (2010).
[CrossRef]

F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express 18, 13563–13573 (2010).
[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]

E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, and A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
[CrossRef]

2008 (2)

2006 (1)

2004 (2)

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]

Q. Zhan, “Second-order tilted wave interpretation of the Gouy phase shift under high numerical aperture uniform illumination,” Opt. Commun. 242, 351–360 (2004).
[CrossRef]

2002 (1)

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474–5488 (2002).
[CrossRef]

2001 (1)

1999 (1)

Z. L. Horvath and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999).
[CrossRef]

1995 (1)

1977 (1)

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional period structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

1959 (1)

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

1952 (1)

G. T. Francia, “Super-gain antennas and optical resolving power,” Il Nuovo Cimento 9, 426–438 (1952).

Assafrao, A. D.

M. S. Kim, A. D. Assafrao, T. Scharf, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Longitudinal-differential phase distribution near the focus of a high numerical aperture lens: study of wavefront spacing and Gouy phase,” J. Mod. Opt. 60, 197–201 (2013).
[CrossRef]

M. S. Kim, T. Scharf, A. D. Assafrao, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Phase anomalies in Bessel-Gauss beams,” Opt. Express 20, 28929–28940 (2012).
[CrossRef]

Beaurepaire, E.

Bor, Z.

Z. L. Horvath and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999).
[CrossRef]

Cao, H.

Cao, W.

Chaumet, P. C.

E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, and A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
[CrossRef]

Cheong, F. C.

Dammann, H.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional period structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

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]

Feng, S. M.

Ferrand, P.

E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, and A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
[CrossRef]

Francia, G. T.

G. T. Francia, “Super-gain antennas and optical resolving power,” Il Nuovo Cimento 9, 426–438 (1952).

Gatto, A.

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.

Grier, D. G.

Herzig, H. P.

M. S. Kim, A. D. Assafrao, T. Scharf, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Longitudinal-differential phase distribution near the focus of a high numerical aperture lens: study of wavefront spacing and Gouy phase,” J. Mod. Opt. 60, 197–201 (2013).
[CrossRef]

M. S. Kim, T. Scharf, A. D. Assafrao, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Phase anomalies in Bessel-Gauss beams,” Opt. Express 20, 28929–28940 (2012).
[CrossRef]

Horvath, Z. L.

Z. L. Horvath and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999).
[CrossRef]

Hu, A.

Jia, W.

Kim, M. S.

M. S. Kim, A. D. Assafrao, T. Scharf, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Longitudinal-differential phase distribution near the focus of a high numerical aperture lens: study of wavefront spacing and Gouy phase,” J. Mod. Opt. 60, 197–201 (2013).
[CrossRef]

M. S. Kim, T. Scharf, A. D. Assafrao, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Phase anomalies in Bessel-Gauss beams,” Opt. Express 20, 28929–28940 (2012).
[CrossRef]

Klotz, E.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional period structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Krishnatreya, B. J.

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]

Lasser, T.

Le Moal, E.

E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, and A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
[CrossRef]

Leitgeb, R. A.

Leutenegger, M.

Liu, L.

Ma, J.

Martelli, P.

Martinelli, M.

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]

Moneta, G.

Mudry, E.

E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, and A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
[CrossRef]

Olivier, N.

Pang, X.

Pereira, S. F.

M. S. Kim, A. D. Assafrao, T. Scharf, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Longitudinal-differential phase distribution near the focus of a high numerical aperture lens: study of wavefront spacing and Gouy phase,” J. Mod. Opt. 60, 197–201 (2013).
[CrossRef]

M. S. Kim, T. Scharf, A. D. Assafrao, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Phase anomalies in Bessel-Gauss beams,” Opt. Express 20, 28929–28940 (2012).
[CrossRef]

Philip, G. M.

Rao, R.

Richards, B.

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

Rockstuhl, C.

M. S. Kim, A. D. Assafrao, T. Scharf, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Longitudinal-differential phase distribution near the focus of a high numerical aperture lens: study of wavefront spacing and Gouy phase,” J. Mod. Opt. 60, 197–201 (2013).
[CrossRef]

M. S. Kim, T. Scharf, A. D. Assafrao, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Phase anomalies in Bessel-Gauss beams,” Opt. Express 20, 28929–28940 (2012).
[CrossRef]

Rohrbach, A.

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474–5488 (2002).
[CrossRef]

Scharf, T.

M. S. Kim, A. D. Assafrao, T. Scharf, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Longitudinal-differential phase distribution near the focus of a high numerical aperture lens: study of wavefront spacing and Gouy phase,” J. Mod. Opt. 60, 197–201 (2013).
[CrossRef]

M. S. Kim, T. Scharf, A. D. Assafrao, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Phase anomalies in Bessel-Gauss beams,” Opt. Express 20, 28929–28940 (2012).
[CrossRef]

Sentenac, A.

E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, and A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
[CrossRef]

Sheppard, C. J. R.

Stelzer, E. H. K.

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474–5488 (2002).
[CrossRef]

Tacca, M.

Urbach, H. P.

M. S. Kim, A. D. Assafrao, T. Scharf, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Longitudinal-differential phase distribution near the focus of a high numerical aperture lens: study of wavefront spacing and Gouy phase,” J. Mod. Opt. 60, 197–201 (2013).
[CrossRef]

M. S. Kim, T. Scharf, A. D. Assafrao, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Phase anomalies in Bessel-Gauss beams,” Opt. Express 20, 28929–28940 (2012).
[CrossRef]

Visser, T. D.

Viswanathan, N. K.

Wang, S.

Wilson, L.

Winful, H. G.

Wolf, E.

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 an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Wu, J.

Yew, E. Y. S.

Yu, J.

Zhan, Q.

Q. Zhan, “Second-order tilted wave interpretation of the Gouy phase shift under high numerical aperture uniform illumination,” Opt. Commun. 242, 351–360 (2004).
[CrossRef]

Zhang, R. J.

Zhou, C.

Appl. Opt. (3)

Il Nuovo Cimento (1)

G. T. Francia, “Super-gain antennas and optical resolving power,” Il Nuovo Cimento 9, 426–438 (1952).

J. Appl. Phys. (1)

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474–5488 (2002).
[CrossRef]

J. Mod. Opt. (1)

M. S. Kim, A. D. Assafrao, T. Scharf, C. Rockstuhl, S. F. Pereira, H. P. Urbach, and H. P. Herzig, “Longitudinal-differential phase distribution near the focus of a high numerical aperture lens: study of wavefront spacing and Gouy phase,” J. Mod. Opt. 60, 197–201 (2013).
[CrossRef]

Opt. Acta (1)

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional period structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Opt. Commun. (3)

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]

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]

Q. Zhan, “Second-order tilted wave interpretation of the Gouy phase shift under high numerical aperture uniform illumination,” Opt. Commun. 242, 351–360 (2004).
[CrossRef]

Opt. Express (7)

Opt. Lett. (4)

Phys. Rev. E (1)

Z. L. Horvath and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, and A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
[CrossRef]

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

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

Supplementary Material (1)

» Media 1: MOV (3980 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

Schematic of a focusing system with a DZP for generation of a coaxial focus array. The z axis is along the axial direction, and the position of the geometric focus is chosen as the origin. θ denotes the aperture angle and α represents the maximum aperture angle of the focusing system. s=sinθ/sinα is the normalized radial coordinate in the plane of entrance pupil.

Fig. 2.
Fig. 2.

Numerical simulation of a 1×5 DZP with 10 periods inside the objective aperture for NA=0.85. (a) and (b) are the radial angular spectrums of TDZP(θ), Gx(θ), and TDZP(θ)Gx(θ) versus t and s2, respectively. (c) The normalized intensity profile (green broken line) and LD phase profile (blue solid line) along the optical axis.

Fig. 3.
Fig. 3.

LD phase profiles of three 1×5 DZPs with different period number Nt inside the objective aperture. (a) Nt=10; (b) Nt=50; (c) Nt=200; (d) and (e) are subsets of the enlarged rectangular areas in (b) and (c), respectively.

Fig. 4.
Fig. 4.

Phase profiles of the focused field of the focusing system with the 1×5 DZP. Absolute phase (pink dot line), LD phase (blue solid line), and ideal spherical wave phase (blue broken line). (b)–(e) are subsets of the enlarged rectangular areas in (a), respectively.

Fig. 5.
Fig. 5.

Numerical results of three binary pure-phase DZPs with different phase retardation. (a) LD phase profile; (b) axial intensity profile. Among these graphs, Red dotted lines, green broken lines, and blue dashed–dotted lines denote the binary (0,π), (0,π/2), and (0,π/4) DZPs, respectively.

Fig. 6.
Fig. 6.

LD phase profile of different kinds of DZPs under condition of NA=0.85. There are 10 periods inside objective aperture for all four kinds of DZPs. (a) 1×5 DZP; (b) 1×7 DZP; (c) 1×2 DZP; (d) 1×4 DZP.

Fig. 7.
Fig. 7.

(a) to (d) are radial angular spectrums of TDZP(θ) (red dot line), Gx(θ) (green broken line), and TDZP(θ)Gx(θ) (blue solid line) versus s2 for NA=0.02, 0.45, 0.85, and 1, respectively. (e) to (g) are normalized axial intensity profiles on the optical axis (green dot line) and LD phase profiles (blue solid line) for NA=0.02, 0.45, 0.85, and 1, respectively.

Fig. 8.
Fig. 8.

3D distribution of the full focused field for a focusing system of NA=0.85 with the 1×5 DZP under x polarization incidence. (a) Axial 2D total intensity distribution on the xz plane, and five transverse 2D total intensity distributions on those five coaxial planes located at z=42.26λ, 21.13λ, 0, 21.13λ, and 42.26λ, respectively. (b) Axial 2D absolute phase distribution on the xz plane, and five transverse 2D absolute phase maps of the x component on five coaxial planes located at z=42.26λ, 21.13λ, 0, 21.13λ, and 42.26λ, respectively. (c) Axial 2D LD phase distribution on the xz plane, and five transverse 2D LD phase maps of the x component on five coaxial planes located at z=42.26λ, 21.13λ, 0, 21.13λ, and 42.26λ, respectively. More detailed results of the phase evolution along the z direction can be found in Media 1.

Equations (14)

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

Eo(x,y,z)=00{TDZP(kx,ky)Et(kx,ky)eikzz/cosθ}ei(kxx+kyy)dkxdky=I{TDZP(ξ)}Ev(x,y,z),
Ev(x,y,z)=0α02πEt(θ,φ)×exp[ik(xsinθcosφ+ysinθsinφzcosθ)]sinθdφdθ,
Et(θ,φ)=tp(Ei·ep)er+ts(Ei·es)es,
Eo(0,0,z)=00TDZP(kx,ky)Et(kx,ky)eikzz/cosθdkxdky.
Eo(0,0,z)=0αTDZP(θ)G(θ)eikzcosθsinθdθ,
G(θ)=02πL(θ){[V(θ,φ)·ep]er+[V(θ,φ)·es]ep}dφ.
Eo(z)=(Eo,x(z)Eo,y(z)Eo,z(z))=cosα1TDZP(t)G(t)eikztdt.
G(t)=(t1/2(1+t)00),
Eo,x(z)=TDZP(t)Gx(t)eikztdt,
TDZP(t)={ei[1+(1)n]π/2tnt<tn+1,n=0,1,2,0t<cosα,ort>1.
Eo,x(z)=mCmδ(zmΔz)I{Gx(s)}.
Φ(z)=arg{Eo,x(z)}kz.
Φ(x,y,z)=arg{Eo(x,y,z)}sign{z}·kx2+y2+z2.
sign(z)={1z>00z=01z<0.

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