Abstract

We derive solutions for radially polarized Bessel-Gauss beams in free-space by superimposing decentered Gaussian beams with differing polarization states. We numerically show that the analytical result is applicable even for large semi-aperture angles, and we experimentally confirm the analytical expression by employing a fiber-based mode-converter.

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microw. Opt. Acoust.2, 105–112 (1978).
    [CrossRef]
  2. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64, 491–495 (1987).
    [CrossRef]
  3. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43: 6, 1155–1166 (1996).
  4. M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev.6, 607–621 (2012).
    [CrossRef]
  5. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419, 145–147 (2002).
    [CrossRef] [PubMed]
  6. W. P. Putnam, D. N. Schimpf, G. Abram, and F. X. Kärtner, “Bessel-Gauss beam enhancement cavities for high-intensity applications,” Opt. Express20, 24429–24443 (2012)
    [CrossRef] [PubMed]
  7. R. D. Romea and W. D. Kimura, “Modeling of inverse Cerenkov laser acceleration with axicon laser-beam focusing,” Phy. Rev. D42, 1807–1818 (1990).
    [CrossRef]
  8. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179, 1–7 (2000).
    [CrossRef]
  9. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7, 77–87 (2000).
    [CrossRef] [PubMed]
  10. E. Y. S. Yew and C. J. R. Sheppard, “Tight focusing of radially polarized Gaussian and Bessel-Gauss beams,” Opt. Lett.32, 3417–3419 (2007).
    [CrossRef] [PubMed]
  11. R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett.19, 427–429 (1994).
    [CrossRef] [PubMed]
  12. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett.21, 9–11 (1996).
    [CrossRef] [PubMed]
  13. P. L. Greene and D. G. Hall, “Properties and diffraction of vector Bessel-Gauss beams, ” J. Opt. Soc. Am. A15, 3020–3027 (1998).
    [CrossRef]
  14. A. R. Al-Rashed and B. E. A. Saleh, “Decentered Gaussian beams,” Appl. Opt.34, 6819–6825 (1995).
    [CrossRef] [PubMed]
  15. D. N. Schimpf, J. Schulte, W. P. Putnam, and F. X. Kärtner, “Generalizing higher-order Bessel-Gauss beams: analytical description and demonstration,” Opt. Express20, 26852–26867 (2012).
    [CrossRef] [PubMed]
  16. L. W. Davis, “Theory of electromagnetic beams,” Phy. Rev. A19, 1177–1179 (1979).
    [CrossRef]
  17. H. A. Haus, Waves and Fields in Optoelectronics (CBLS, 2004).
  18. L. W. Davis, “TM and TE electromagnetic beams in free space,” Opt. Lett.6, 22–23 (1981).
    [CrossRef] [PubMed]
  19. R. I. Hernandez-Aranda, J. C. Gutiérrez-Vega, M. Guizar-Sicairos, and M. A. Bandres, “Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems,” Opt. Express14, 8974–8988 (2006).
    [CrossRef] [PubMed]
  20. R. S. Kurti, K. Halterman, R. K. Shori, and M. J. Wardlaw, “Discrete cylindrical vector beam generation from an array of optical fibers,” Opt. Express17, 13982–13988 (2009).
    [CrossRef] [PubMed]
  21. Li Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett.23, 409–411 (1998).
    [CrossRef]
  22. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proceedings of the Royal Societyof London. Series A, Mathematical and Physical Sciences253, 358–379 (1959).
    [CrossRef]
  23. S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett.34, 2525–2527 (2009).
    [CrossRef] [PubMed]
  24. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett.21, 1948–1950 (1996).
    [CrossRef] [PubMed]
  25. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett.32, 1468–1470 (2007)
    [CrossRef] [PubMed]
  26. I. K. Hwang, S. H. Yun, and B. Y. Kim, “Long-period fiber gratings based on periodic microbends,” Opt. Lett.24, 1263–1265 (1999).
    [CrossRef]
  27. O. Brzobohaty, T. Cizmar, and P. Zemanek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express16, 12688–12700 (2008).
    [CrossRef] [PubMed]
  28. M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun.132, 1–7 (1996).
    [CrossRef]
  29. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am.60, 1168–1177 (1970).
    [CrossRef]

2012 (3)

2009 (2)

2008 (1)

2007 (2)

2006 (1)

2002 (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419, 145–147 (2002).
[CrossRef] [PubMed]

2000 (2)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179, 1–7 (2000).
[CrossRef]

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7, 77–87 (2000).
[CrossRef] [PubMed]

1999 (1)

1998 (2)

1996 (4)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43: 6, 1155–1166 (1996).

D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett.21, 9–11 (1996).
[CrossRef] [PubMed]

M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett.21, 1948–1950 (1996).
[CrossRef] [PubMed]

M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun.132, 1–7 (1996).
[CrossRef]

1995 (1)

1994 (1)

1990 (1)

R. D. Romea and W. D. Kimura, “Modeling of inverse Cerenkov laser acceleration with axicon laser-beam focusing,” Phy. Rev. D42, 1807–1818 (1990).
[CrossRef]

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64, 491–495 (1987).
[CrossRef]

1981 (1)

1979 (1)

L. W. Davis, “Theory of electromagnetic beams,” Phy. Rev. A19, 1177–1179 (1979).
[CrossRef]

1978 (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microw. Opt. Acoust.2, 105–112 (1978).
[CrossRef]

1970 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proceedings of the Royal Societyof London. Series A, Mathematical and Physical Sciences253, 358–379 (1959).
[CrossRef]

Abram, G.

Al-Rashed, A. R.

Arnold, C. B.

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev.6, 607–621 (2012).
[CrossRef]

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43: 6, 1155–1166 (1996).

Bandres, M. A.

Brown, T.

Brzobohaty, O.

Chen, M.

Chen, W.

Cizmar, T.

Collins, S. A.

Davis, L. W.

L. W. Davis, “TM and TE electromagnetic beams in free space,” Opt. Lett.6, 22–23 (1981).
[CrossRef] [PubMed]

L. W. Davis, “Theory of electromagnetic beams,” Phy. Rev. A19, 1177–1179 (1979).
[CrossRef]

Dholakia, K.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419, 145–147 (2002).
[CrossRef] [PubMed]

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179, 1–7 (2000).
[CrossRef]

Duocastella, M.

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev.6, 607–621 (2012).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179, 1–7 (2000).
[CrossRef]

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43: 6, 1155–1166 (1996).

Garcés-Chávez, V.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419, 145–147 (2002).
[CrossRef] [PubMed]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179, 1–7 (2000).
[CrossRef]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64, 491–495 (1987).
[CrossRef]

Greene, P. L.

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64, 491–495 (1987).
[CrossRef]

Guizar-Sicairos, M.

Gutiérrez-Vega, J. C.

Hall, D. G.

Halterman, K.

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (CBLS, 2004).

Hernandez-Aranda, R. I.

Huang, M.

Huang, W.

Hwang, I. K.

Jackel, S.

Jordan, R. H.

Kärtner, F. X.

Kim, B. Y.

Kimura, W. D.

R. D. Romea and W. D. Kimura, “Modeling of inverse Cerenkov laser acceleration with axicon laser-beam focusing,” Phy. Rev. D42, 1807–1818 (1990).
[CrossRef]

Kristensen, P.

Kurti, R. S.

Leuchs, G.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179, 1–7 (2000).
[CrossRef]

Lumer, Y.

Machavariani, G.

McGloin, D.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419, 145–147 (2002).
[CrossRef] [PubMed]

Meir, A.

Melville, H.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419, 145–147 (2002).
[CrossRef] [PubMed]

Moshe, I.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64, 491–495 (1987).
[CrossRef]

Putnam, W. P.

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179, 1–7 (2000).
[CrossRef]

Ramachandran, S.

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proceedings of the Royal Societyof London. Series A, Mathematical and Physical Sciences253, 358–379 (1959).
[CrossRef]

Romea, R. D.

R. D. Romea and W. D. Kimura, “Modeling of inverse Cerenkov laser acceleration with axicon laser-beam focusing,” Phy. Rev. D42, 1807–1818 (1990).
[CrossRef]

Saleh, B. E. A.

Santarsiero, M.

M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun.132, 1–7 (1996).
[CrossRef]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43: 6, 1155–1166 (1996).

Schadt, M.

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43: 6, 1155–1166 (1996).

Schimpf, D. N.

Schulte, J.

Sheppard, C. J. R.

E. Y. S. Yew and C. J. R. Sheppard, “Tight focusing of radially polarized Gaussian and Bessel-Gauss beams,” Opt. Lett.32, 3417–3419 (2007).
[CrossRef] [PubMed]

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microw. Opt. Acoust.2, 105–112 (1978).
[CrossRef]

Shori, R. K.

Sibbett, W.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419, 145–147 (2002).
[CrossRef] [PubMed]

Spagnolo Schirripa, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43: 6, 1155–1166 (1996).

Stalder, M.

Wardlaw, M. J.

Wilson, T.

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microw. Opt. Acoust.2, 105–112 (1978).
[CrossRef]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proceedings of the Royal Societyof London. Series A, Mathematical and Physical Sciences253, 358–379 (1959).
[CrossRef]

Yan, M. F.

Yew, E. Y. S.

Youngworth, K.

Yu, Li

Yun, S. H.

Zemanek, P.

Zhu, Z.

Appl. Opt. (1)

IEE J. Microw. Opt. Acoust. (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microw. Opt. Acoust.2, 105–112 (1978).
[CrossRef]

J. Mod. Opt. (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43: 6, 1155–1166 (1996).

J. Opt. Soc. Am. (1)

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

Laser Photonics Rev. (1)

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev.6, 607–621 (2012).
[CrossRef]

Nature (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419, 145–147 (2002).
[CrossRef] [PubMed]

Opt. Commun. (3)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64, 491–495 (1987).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179, 1–7 (2000).
[CrossRef]

M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun.132, 1–7 (1996).
[CrossRef]

Opt. Express (6)

Opt. Lett. (9)

Phy. Rev. A (1)

L. W. Davis, “Theory of electromagnetic beams,” Phy. Rev. A19, 1177–1179 (1979).
[CrossRef]

Phy. Rev. D (1)

R. D. Romea and W. D. Kimura, “Modeling of inverse Cerenkov laser acceleration with axicon laser-beam focusing,” Phy. Rev. D42, 1807–1818 (1990).
[CrossRef]

Proceedings of the Royal Societyof London. Series A, Mathematical and Physical Sciences (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proceedings of the Royal Societyof London. Series A, Mathematical and Physical Sciences253, 358–379 (1959).
[CrossRef]

Other (1)

H. A. Haus, Waves and Fields in Optoelectronics (CBLS, 2004).

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 (8)

Fig. 1
Fig. 1

(a) Geometry for constructing cylindrical Bessel-Gauss beams by superposing decentered Gaussian beams, (b) superposing decentered Gaussian beams for a radially or azimuthally oriented vector potential, (c) for a y-oriented or z-oriented vector potential

Fig. 2
Fig. 2

Vector potential amplitude of Bessel-Gauss beam with azimuthally or radially oriented vector potential: (a) square modulus of the amplitude at the focus (z = 0); and (b) in the far-field (z = 10 mm); and (c) in the focal region. Vector potential amplitude oriented along a Cartesian unit vector potential: (d) square modulus of the amplitude at the focus (z = 0); (e) in the far-field (z = 10 mm); and (f) amplitude in the focal region.

Fig. 3
Fig. 3

Results of numerical simulation. (a) Initial state vector potential amplitude. (b) Vector potential amplitude of propagated initial state; (c) and (d), radial and z-component of the resulting electric field (both on the same scale), respectively.

Fig. 4
Fig. 4

Comparison of the electric fields derived from numerical and analytical vector potential solution for (a) the on-axis z-component of the electric field and (b) the radial field component (at radius of max. field strength); and (c) Ratio of the maximum amplitude of the z-component of the electric field to the maximum radial component of the electric field for different semi-aperture angles φ.

Fig. 5
Fig. 5

(a) Schematic of the experimental setup for fiber-based radially polarized Bessel-Gauss beam generation; (b) refractive index profile, i.e. difference to the cladding index, of the specialty optical fiber; (c) independently measured LPG loss spectrum.

Fig. 6
Fig. 6

(a) Image of the TM01 mode (plane P1); (b) images of the mode after passage through a polarizer at different angles.

Fig. 7
Fig. 7

(a) Partial beam intensity profile at the axicon position (plane P2) and fit. (b) Relative optical path length of the axicon measured with a wavefront sensor.

Fig. 8
Fig. 8

(a) Near-field intensity image of the radially polarized Bessel-Gauss beam (plane 3); (b) comparison between experiment and theoretical expression.

Tables (2)

Tables Icon

Table 1 The nonzero electromagnetic components of the transverse electric beam.

Tables Icon

Table 2 The nonzero electromagnetic components of the transverse magnetic beams.

Equations (29)

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

A ( x , y , z , t ) = n ^ d k x d k y a ^ ( k x , k y ) e i ( k x x + k y y + k z ) z e i ω t .
k x = K x + α ,
k y = K y + β .
a ( α , β ) = w 0 4 π exp ( ( α 2 + β 2 ) ( w 0 2 4 ) ) exp ( i ( α x d 0 + β y d 0 ) ) .
A d G ( x , y , z = 0 , t ) = n ^ exp ( i ω 0 t ) × 1 w 0 exp ( i K z 2 q 0 [ ( x x d 0 ) 2 + ( y y d 0 ) 2 ] ) exp ( i K z [ ε x x + ε y y ] ) ,
k z = k 0 2 k x 2 k y 2 = k 0 2 ( K x + α ) 2 ( K y + β ) 2
= K z 1 [ 2 ( K x K z ) α K z + 2 ( K y K z ) β K z + α 2 + β 2 K z 2 ]
K z α ε x β ε y 1 2 K z ( α 2 + β 2 ) .
A d G ( x , y , z , t ) = n ^ exp ( i ( K z z ω 0 t φ ) ) × 1 w exp ( i K z 2 q [ ( x x d ) 2 + ( y y d ) 2 ] ) exp ( i K z [ ε x x + ε y y ] ) ,
A d G , γ ( r , θ , z , t ) = n ^ ( γ ) exp ( i ( K z z ω 0 t φ ) ) × 1 w exp ( i K z 2 q ( r 2 + r d 2 2 r r d cos ( θ γ ) ) ) exp ( i K z ε r cos ( θ γ ) ) .
A B G ( r , θ , z , t ) = exp ( i ( K z z ω 0 t ϕ ) ) 1 w exp ( i K z 2 q ( r 2 + r d 2 ) ) × 0 2 π d γ n ^ ( γ ) exp ( i K z r ( ε r d q ) cos ( θ γ ) ) .
J k ( τ ) = 1 2 π i k 0 2 π d η exp { ( τ cos η k η ) } .
A B G ( r , θ , z , t ) = n ^ ψ ( r ) ( r , z ) ,
ψ ( r ) ( r , z ) = exp ( i ( K z z ω 0 t ϕ ) ) 1 w exp ( i K z 2 q ( r 2 + r d 2 ) ) 2 π i J 1 ( K z r ( ε r d q ) ) .
A B G ( r , θ , z , t ) = n ^ ψ ( z ) ( r , z )
ψ ( z ) ( r , z ) = exp ( i ( K z z ω 0 t ϕ ) ) 1 w exp ( i K z 2 q ( r 2 + r d 2 ) ) 2 π J 0 ( K z r ( ε r d q ) ) .
E = A t φ and B = × A .
A + 1 c 2 φ t = 0 ,
E = i ω 0 ( A + c 2 ω 0 2 ( A ) ) and B = × A .
A ( x , y , z , t ) = n ^ ψ ( x , y , z , t ) ,
2 A 1 c 2 2 t 2 A = 0 .
2 ψ ( z ) 1 c 2 2 t 2 ψ ( z ) = 0 .
2 ψ ( r ) 1 r 2 ψ ( r ) 1 c 2 2 t 2 ψ ( r ) = 0 .
ψ ( z ) ( x , y , z , t ) = e i ω 0 t I F T [ F T [ ψ ( z ) ( x , y , z = 0 , t = 0 ) ] e i z k 0 2 k x 2 k y 2 ) ] .
A BG ( r , θ , L , t ) = n ^ ψ ( r ) ,
ψ ( r ) ( r , L ) = exp ( i ( k 0 L ω 0 t i ϕ + Σ ) ) × 1 w exp ( i k 0 2 q ( r 2 + r d 2 ) ) 2 π i J 1 ( k 0 r ( ε ˜ r d q ) ) .
q = A q 0 + B C q 0 + D ; w = 2 / k 0 / Im [ 1 / q ] ; ϕ = arg ( A + B / q 0 ) ;
( r d ε ˜ ) = ( A B C D ) ( r d 0 ε ) ;
Σ = k 0 2 [ C r d 0 r d + B ε ε ˜ ] .

Metrics