Abstract

A straightforward procedure is described for accurately creating an incident focused light pulse in the 3-D finite-difference time-domain (FDTD) electromagnetic simulation of the image space of an aplanatic converging lens. In this procedure, the focused light pulse is approximated by a finite sum of plane waves, and each plane wave is introduced into the FDTD simulation grid using the total-field/scattered-field (TF/SF) approach. The accuracy of our results is demonstrated by comparison with exact theoretical formulas.

© 2008 Optical Society of America

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References

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  1. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Boston, 2005).
  2. H. K. Roy, Y. Liu, H. Subramanian, D. Kunte, P. Pradhan, R. K. Wali, J. Koetsier, M. J. Goldberg, Z. Bogojevic, and V. Backman, "Detection of the colorectal cancer (CRC) field effect through partial wave spectroscopic microscopy (PWS)," Gastroenterology 132, A169 (2007).
  3. E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. Roy. Soc. A 253, 349-357 (1959).
    [CrossRef]
  4. G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge University Press, New York, NY, 1997).
  5. R. W. Ziolkowski and J. B. Judkins, "Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response-time," J. Opt. Soc. Am. B 10, 186-198 (1993).
    [CrossRef]
  6. D. B. Davidson and R. W. Ziolkowski, "Body-of-revolution finite-difference time-domain modeling of spacetime focusing by a 3-dimensional lens," J. Opt. Soc. Am. A 11, 1471-1490 (1994).
    [CrossRef]
  7. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. A 253, 358-379 (1959).
    [CrossRef]
  8. W. Sun, S. Pan, and Y. Jiang, "Computation of the optical trapping force on small particles illuminated with a focused light beam using a FDTD method," J. Mod. Opt. pp. 2691-2700 (2006).
    [CrossRef]
  9. L. Cheng, H. Zhiwei, L. Fake, Z. Wei, D. W. Hutmacher, and C. Sheppard, "Near-field effects on coherent anti-Stokes Raman scattering microscopy imaging," Opt. Express 15, 4118-4131 (2007).
    [CrossRef]
  10. W. A. Challener, I. K. Sendur, and C. Peng, "Scattered field formulation of finite difference time domain for a focused light beam in dense media with lossy materials," Opt. Express 11, 3160-3170 (2003).
    [CrossRef] [PubMed]
  11. P. T¨or¨ok, P. R. T. Munro, and E. E. Kriezis, "High numerical aperture vectorial imaging in coherent optical microscopes," Opt. Express 16, 507-523 (2008).
    [CrossRef] [PubMed]
  12. C. Liu and S.-H. Park, "Numerical analysis of an annular-aperture solid immersion lens," Opt. Lett. 29, 1742-1744 (2004).
    [CrossRef] [PubMed]
  13. J. Liu, B. Xu, and T. C. Chong, "Three-dimensional finite-difference time-domain analysis of optical disk storage system," Jpn. J. Appl. Phys. Part 1  39, 687-692 (2000).
    [CrossRef]
  14. K. S¸endur, W. Challener, and C. Peng, "Ridge waveguide as a near field aperture for high density data storage," J. Appl. Phys. 96, 2743-2752 (2004).
    [CrossRef]
  15. S.-Y. Sung and Y.-G. Lee, "Trapping of a micro-bubble by non-paraxial Gaussian beam: Computation using the FDTD method," Opt. Express 16, 3463-3473 (2008).
    [CrossRef] [PubMed]
  16. K. Choi, J. W. M. Chon, M. Gu, and B. Lee, "Characterization of a subwavelength-scale 3D void structure using the FDTD-based confocal laser scanning microscopic image mapping technique," Opt. Express 15, 10,767-10,781 (2007).
    [CrossRef]
  17. M. Wang, J. Wu, J. Xu, D. Ge, H. Li, and J. Feng, "FDTD simulation on the interaction between Gaussian beam and biaxial anisotropic metamaterial slabs," Int. J. Infrared Millim. Waves 29, 167-178 (2008).
    [CrossRef]
  18. Y.-F. Chau and D. P. Tsai, "Near-field optics imaging in silica waveguide using near-field scanning optical microscope," Jpn. J. Appl. Phys. Part 1 46, 238-242 (2007).
    [CrossRef]
  19. J. B. Judkins and R.W. Ziolkowski, "Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings," J. Opt. Soc. Am. A 12, 1974 (1995).
    [CrossRef]
  20. J. B. Judkins, C.W. Haggans, and R.W. Ziolkowski, "Two-dimensional finite-difference time-domain simulation for rewritable optical disk surface structure design," Appl. Opt. 35, 2477 (1996).
    [PubMed]
  21. M. Born and E. Wolf, Principles of Optics : Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, Cambridge, 1999).
    [PubMed]
  22. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University Press, Cambridge, 1992).
  23. S. Bochkanov and V. Bystritsky, "Computation of Gauss-Legendre quadrature rule nodes and weights," Alglib.net- Web Resource. Date Accessed: 08/2008. URL http://www.alglib.net/integral/gq/glegendre.php.

2008 (3)

2007 (4)

L. Cheng, H. Zhiwei, L. Fake, Z. Wei, D. W. Hutmacher, and C. Sheppard, "Near-field effects on coherent anti-Stokes Raman scattering microscopy imaging," Opt. Express 15, 4118-4131 (2007).
[CrossRef]

Y.-F. Chau and D. P. Tsai, "Near-field optics imaging in silica waveguide using near-field scanning optical microscope," Jpn. J. Appl. Phys. Part 1 46, 238-242 (2007).
[CrossRef]

H. K. Roy, Y. Liu, H. Subramanian, D. Kunte, P. Pradhan, R. K. Wali, J. Koetsier, M. J. Goldberg, Z. Bogojevic, and V. Backman, "Detection of the colorectal cancer (CRC) field effect through partial wave spectroscopic microscopy (PWS)," Gastroenterology 132, A169 (2007).

K. Choi, J. W. M. Chon, M. Gu, and B. Lee, "Characterization of a subwavelength-scale 3D void structure using the FDTD-based confocal laser scanning microscopic image mapping technique," Opt. Express 15, 10,767-10,781 (2007).
[CrossRef]

2006 (1)

W. Sun, S. Pan, and Y. Jiang, "Computation of the optical trapping force on small particles illuminated with a focused light beam using a FDTD method," J. Mod. Opt. pp. 2691-2700 (2006).
[CrossRef]

2004 (2)

K. S¸endur, W. Challener, and C. Peng, "Ridge waveguide as a near field aperture for high density data storage," J. Appl. Phys. 96, 2743-2752 (2004).
[CrossRef]

C. Liu and S.-H. Park, "Numerical analysis of an annular-aperture solid immersion lens," Opt. Lett. 29, 1742-1744 (2004).
[CrossRef] [PubMed]

2003 (1)

2000 (1)

J. Liu, B. Xu, and T. C. Chong, "Three-dimensional finite-difference time-domain analysis of optical disk storage system," Jpn. J. Appl. Phys. Part 1  39, 687-692 (2000).
[CrossRef]

1996 (1)

1995 (1)

1994 (1)

1993 (1)

1959 (2)

E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. Roy. Soc. A 253, 349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. A 253, 358-379 (1959).
[CrossRef]

Backman, V.

H. K. Roy, Y. Liu, H. Subramanian, D. Kunte, P. Pradhan, R. K. Wali, J. Koetsier, M. J. Goldberg, Z. Bogojevic, and V. Backman, "Detection of the colorectal cancer (CRC) field effect through partial wave spectroscopic microscopy (PWS)," Gastroenterology 132, A169 (2007).

Bogojevic, Z.

H. K. Roy, Y. Liu, H. Subramanian, D. Kunte, P. Pradhan, R. K. Wali, J. Koetsier, M. J. Goldberg, Z. Bogojevic, and V. Backman, "Detection of the colorectal cancer (CRC) field effect through partial wave spectroscopic microscopy (PWS)," Gastroenterology 132, A169 (2007).

Challener, W.

K. S¸endur, W. Challener, and C. Peng, "Ridge waveguide as a near field aperture for high density data storage," J. Appl. Phys. 96, 2743-2752 (2004).
[CrossRef]

Challener, W. A.

Chau, Y.-F.

Y.-F. Chau and D. P. Tsai, "Near-field optics imaging in silica waveguide using near-field scanning optical microscope," Jpn. J. Appl. Phys. Part 1 46, 238-242 (2007).
[CrossRef]

Cheng, L.

Choi, K.

K. Choi, J. W. M. Chon, M. Gu, and B. Lee, "Characterization of a subwavelength-scale 3D void structure using the FDTD-based confocal laser scanning microscopic image mapping technique," Opt. Express 15, 10,767-10,781 (2007).
[CrossRef]

Chon, J. W. M.

K. Choi, J. W. M. Chon, M. Gu, and B. Lee, "Characterization of a subwavelength-scale 3D void structure using the FDTD-based confocal laser scanning microscopic image mapping technique," Opt. Express 15, 10,767-10,781 (2007).
[CrossRef]

Chong, T. C.

J. Liu, B. Xu, and T. C. Chong, "Three-dimensional finite-difference time-domain analysis of optical disk storage system," Jpn. J. Appl. Phys. Part 1  39, 687-692 (2000).
[CrossRef]

Davidson, D. B.

Fake, L.

Feng, J.

M. Wang, J. Wu, J. Xu, D. Ge, H. Li, and J. Feng, "FDTD simulation on the interaction between Gaussian beam and biaxial anisotropic metamaterial slabs," Int. J. Infrared Millim. Waves 29, 167-178 (2008).
[CrossRef]

Ge, D.

M. Wang, J. Wu, J. Xu, D. Ge, H. Li, and J. Feng, "FDTD simulation on the interaction between Gaussian beam and biaxial anisotropic metamaterial slabs," Int. J. Infrared Millim. Waves 29, 167-178 (2008).
[CrossRef]

Goldberg, M. J.

H. K. Roy, Y. Liu, H. Subramanian, D. Kunte, P. Pradhan, R. K. Wali, J. Koetsier, M. J. Goldberg, Z. Bogojevic, and V. Backman, "Detection of the colorectal cancer (CRC) field effect through partial wave spectroscopic microscopy (PWS)," Gastroenterology 132, A169 (2007).

Gu, M.

K. Choi, J. W. M. Chon, M. Gu, and B. Lee, "Characterization of a subwavelength-scale 3D void structure using the FDTD-based confocal laser scanning microscopic image mapping technique," Opt. Express 15, 10,767-10,781 (2007).
[CrossRef]

Haggans, C.W.

Hutmacher, D. W.

Jiang, Y.

W. Sun, S. Pan, and Y. Jiang, "Computation of the optical trapping force on small particles illuminated with a focused light beam using a FDTD method," J. Mod. Opt. pp. 2691-2700 (2006).
[CrossRef]

Judkins, J. B.

Koetsier, J.

H. K. Roy, Y. Liu, H. Subramanian, D. Kunte, P. Pradhan, R. K. Wali, J. Koetsier, M. J. Goldberg, Z. Bogojevic, and V. Backman, "Detection of the colorectal cancer (CRC) field effect through partial wave spectroscopic microscopy (PWS)," Gastroenterology 132, A169 (2007).

Kriezis, E. E.

Kunte, D.

H. K. Roy, Y. Liu, H. Subramanian, D. Kunte, P. Pradhan, R. K. Wali, J. Koetsier, M. J. Goldberg, Z. Bogojevic, and V. Backman, "Detection of the colorectal cancer (CRC) field effect through partial wave spectroscopic microscopy (PWS)," Gastroenterology 132, A169 (2007).

Lee, B.

K. Choi, J. W. M. Chon, M. Gu, and B. Lee, "Characterization of a subwavelength-scale 3D void structure using the FDTD-based confocal laser scanning microscopic image mapping technique," Opt. Express 15, 10,767-10,781 (2007).
[CrossRef]

Lee, Y.-G.

Li, H.

M. Wang, J. Wu, J. Xu, D. Ge, H. Li, and J. Feng, "FDTD simulation on the interaction between Gaussian beam and biaxial anisotropic metamaterial slabs," Int. J. Infrared Millim. Waves 29, 167-178 (2008).
[CrossRef]

Liu, C.

Liu, J.

J. Liu, B. Xu, and T. C. Chong, "Three-dimensional finite-difference time-domain analysis of optical disk storage system," Jpn. J. Appl. Phys. Part 1  39, 687-692 (2000).
[CrossRef]

Liu, Y.

H. K. Roy, Y. Liu, H. Subramanian, D. Kunte, P. Pradhan, R. K. Wali, J. Koetsier, M. J. Goldberg, Z. Bogojevic, and V. Backman, "Detection of the colorectal cancer (CRC) field effect through partial wave spectroscopic microscopy (PWS)," Gastroenterology 132, A169 (2007).

Munro, P. R. T.

Pan, S.

W. Sun, S. Pan, and Y. Jiang, "Computation of the optical trapping force on small particles illuminated with a focused light beam using a FDTD method," J. Mod. Opt. pp. 2691-2700 (2006).
[CrossRef]

Park, S.-H.

Peng, C.

Pradhan, P.

H. K. Roy, Y. Liu, H. Subramanian, D. Kunte, P. Pradhan, R. K. Wali, J. Koetsier, M. J. Goldberg, Z. Bogojevic, and V. Backman, "Detection of the colorectal cancer (CRC) field effect through partial wave spectroscopic microscopy (PWS)," Gastroenterology 132, A169 (2007).

Richards, B.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. A 253, 358-379 (1959).
[CrossRef]

Roy, H. K.

H. K. Roy, Y. Liu, H. Subramanian, D. Kunte, P. Pradhan, R. K. Wali, J. Koetsier, M. J. Goldberg, Z. Bogojevic, and V. Backman, "Detection of the colorectal cancer (CRC) field effect through partial wave spectroscopic microscopy (PWS)," Gastroenterology 132, A169 (2007).

S¸endur, K.

K. S¸endur, W. Challener, and C. Peng, "Ridge waveguide as a near field aperture for high density data storage," J. Appl. Phys. 96, 2743-2752 (2004).
[CrossRef]

Sendur, I. K.

Sheppard, C.

Subramanian, H.

H. K. Roy, Y. Liu, H. Subramanian, D. Kunte, P. Pradhan, R. K. Wali, J. Koetsier, M. J. Goldberg, Z. Bogojevic, and V. Backman, "Detection of the colorectal cancer (CRC) field effect through partial wave spectroscopic microscopy (PWS)," Gastroenterology 132, A169 (2007).

Sun, W.

W. Sun, S. Pan, and Y. Jiang, "Computation of the optical trapping force on small particles illuminated with a focused light beam using a FDTD method," J. Mod. Opt. pp. 2691-2700 (2006).
[CrossRef]

Sung, S.-Y.

T¨or¨ok, P.

Tsai, D. P.

Y.-F. Chau and D. P. Tsai, "Near-field optics imaging in silica waveguide using near-field scanning optical microscope," Jpn. J. Appl. Phys. Part 1 46, 238-242 (2007).
[CrossRef]

Wali, R. K.

H. K. Roy, Y. Liu, H. Subramanian, D. Kunte, P. Pradhan, R. K. Wali, J. Koetsier, M. J. Goldberg, Z. Bogojevic, and V. Backman, "Detection of the colorectal cancer (CRC) field effect through partial wave spectroscopic microscopy (PWS)," Gastroenterology 132, A169 (2007).

Wang, M.

M. Wang, J. Wu, J. Xu, D. Ge, H. Li, and J. Feng, "FDTD simulation on the interaction between Gaussian beam and biaxial anisotropic metamaterial slabs," Int. J. Infrared Millim. Waves 29, 167-178 (2008).
[CrossRef]

Wei, Z.

Wolf, E.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. A 253, 358-379 (1959).
[CrossRef]

E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. Roy. Soc. A 253, 349-357 (1959).
[CrossRef]

Wu, J.

M. Wang, J. Wu, J. Xu, D. Ge, H. Li, and J. Feng, "FDTD simulation on the interaction between Gaussian beam and biaxial anisotropic metamaterial slabs," Int. J. Infrared Millim. Waves 29, 167-178 (2008).
[CrossRef]

Xu, B.

J. Liu, B. Xu, and T. C. Chong, "Three-dimensional finite-difference time-domain analysis of optical disk storage system," Jpn. J. Appl. Phys. Part 1  39, 687-692 (2000).
[CrossRef]

Xu, J.

M. Wang, J. Wu, J. Xu, D. Ge, H. Li, and J. Feng, "FDTD simulation on the interaction between Gaussian beam and biaxial anisotropic metamaterial slabs," Int. J. Infrared Millim. Waves 29, 167-178 (2008).
[CrossRef]

Zhiwei, H.

Ziolkowski, R. W.

Ziolkowski, R.W.

Appl. Opt. (1)

Gastroenterology (1)

H. K. Roy, Y. Liu, H. Subramanian, D. Kunte, P. Pradhan, R. K. Wali, J. Koetsier, M. J. Goldberg, Z. Bogojevic, and V. Backman, "Detection of the colorectal cancer (CRC) field effect through partial wave spectroscopic microscopy (PWS)," Gastroenterology 132, A169 (2007).

Int. J. Infrared Millim. Waves (1)

M. Wang, J. Wu, J. Xu, D. Ge, H. Li, and J. Feng, "FDTD simulation on the interaction between Gaussian beam and biaxial anisotropic metamaterial slabs," Int. J. Infrared Millim. Waves 29, 167-178 (2008).
[CrossRef]

J. Appl. Phys. (1)

K. S¸endur, W. Challener, and C. Peng, "Ridge waveguide as a near field aperture for high density data storage," J. Appl. Phys. 96, 2743-2752 (2004).
[CrossRef]

J. Mod. Opt (1)

W. Sun, S. Pan, and Y. Jiang, "Computation of the optical trapping force on small particles illuminated with a focused light beam using a FDTD method," J. Mod. Opt. pp. 2691-2700 (2006).
[CrossRef]

J. Opt. Soc. Am. A (2)

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

Jpn. J. Appl. Phys (2)

Y.-F. Chau and D. P. Tsai, "Near-field optics imaging in silica waveguide using near-field scanning optical microscope," Jpn. J. Appl. Phys. Part 1 46, 238-242 (2007).
[CrossRef]

J. Liu, B. Xu, and T. C. Chong, "Three-dimensional finite-difference time-domain analysis of optical disk storage system," Jpn. J. Appl. Phys. Part 1  39, 687-692 (2000).
[CrossRef]

Opt. Express (5)

Opt. Lett. (1)

Proc. Roy. Soc. A (2)

E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. Roy. Soc. A 253, 349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. A 253, 358-379 (1959).
[CrossRef]

Other (5)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Boston, 2005).

G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge University Press, New York, NY, 1997).

M. Born and E. Wolf, Principles of Optics : Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, Cambridge, 1999).
[PubMed]

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University Press, Cambridge, 1992).

S. Bochkanov and V. Bystritsky, "Computation of Gauss-Legendre quadrature rule nodes and weights," Alglib.net- Web Resource. Date Accessed: 08/2008. URL http://www.alglib.net/integral/gq/glegendre.php.

Supplementary Material (1)

» Media 1: MPEG (2146 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

The geometry of the problem. (a) The optical system and the simulation space. (b) The polarization of the electric field on an incident ray (AP) and the corresponding ray in the image space after refraction through the lens (PF).

Fig. 2.
Fig. 2.

Quadrature positions for the two-dimensional numerical integral in (7), for N=5, M=7. One-dimensional quadrature rules are used for both θ and ϕ , regarding the other as constant.

Fig. 3.
Fig. 3.

Grayscale plots of the x̂ component of the electric field (in dB) belonging to the focused light pulse evaluated in the x=0 plane at different time instants. (a)–(f) t=7τ to t=17τ in 2τ intervals. (Media 1)

Fig. 4.
Fig. 4.

Normalized maximum dB-error for the numerical integral (7) describing the electric field belonging to the focused light pulse. The Gauss-Legendre quadrature rule is used for the numerical integral, and the error (15) is normalized globally by E max x . (a) The geometry of the error analysis. The elliptical focal spot [see Fig. 3(d)] is denoted by the gray ellipse. The error is calculated along line segments lx, ly, lz , and lxyz extending from the focus F in different directions. (b) Normalized maximum dB-error ε (r̄) (15) in the x̂ component of the electric field on the line segments lx, ly, lz , and lxyz .

Fig. 5.
Fig. 5.

The effects of quadrature schemes and auxiliary FDTD techniques on the error performance of the the numerical integral (7). The normalized maximum dB-error ε(r̄) in (15) is calculated on the line segment lxyz . (a) Using the extended-midpoint (E-M) and Gauss-Legendre (G-L) quadrature rules for the θ integral, with both local and global error normalization in (15). (b) With and without grid-velocity correction for the 4th dataset in (a).

Equations (22)

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

E ̄ w ( r ̄ , t ) = e ̂ w ( θ , ϕ ) a ( θ , ϕ , t + r c ) r ,
a ( θ , ϕ , t ) = f cos 1 2 ( θ ) E i ( t ) ,
ψ ( θ , ϕ ) = ϕ .
E ̄ ( r ̄ , t ) = 1 2 π c ϕ = 0 2 π θ = 0 θ max e ̂ w ( θ , ϕ ) a ˙ ( θ , ϕ , t ) sin ( θ ) d θ d ϕ
= f 2 π c ϕ = 0 2 π θ = 0 θ max e ̂ w ( θ , ϕ ) E ˙ i ( t ) cos 1 2 ( θ ) sin ( θ ) d θ d ϕ ,
d E ( 0 , t ) = sin ( θ ) d θ d ϕ 2 π c a ˙ ( θ , ϕ , t ) = f cos 1 2 ( θ ) sin ( θ ) d θ d ϕ 2 π c E ˙ i ( t ) .
E ̄ d ( r ̄ , t ) = f 2 π c Σ n , m α n m e ̂ w ( θ n , ϕ m ) E ˙ i ( t n m ) cos 1 2 ( θ n ) sin ( θ n ) , n = 1 N , m = 1 M ,
E ̄ d ( r ̄ , t ) = f 2 π c Σ m b m Σ n a n e ̂ w ( θ n , ϕ m ) E ˙ i ( t n m ) cos 1 2 ( θ n ) sin ( θ n ) , n = 1 N , m = 1 M ,
θ n = ( n 1 2 ) θ m a x N , n = 1 N
ϕ m = ( m 1 2 ) 2 π M , m = 1 M
a n = θ max N ,
b m = 2 π M .
θ n = ( x n + 1 ) θ max 2 , n = 1 N
ϕ m = ( x m + 1 ) π , m = 1 M
a n = w n θ max 2 , n = 1 N
b m = w m π , m = 1 M
E i ( t ) = exp ( ( ( t t 0 ) τ ) 2 2 ) sin ( 2 π f 0 ( t t 0 ) ) ,
E x max = ( 2 π f 0 ) f 2 c [ 2 3 ( 1 cos θ max 3 2 ) + 2 5 ( 1 cos θ max 5 2 ) ] .
E ˜ x ( r ̄ ) = i ω f 2 c ( I 0 + I 2 cos 2 ϕ ) E ˜ i ,
I 0 ( k r , θ ) = 0 θ max cos 1 2 θ sin θ ( 1 + cos θ ) J 0 ( k r sin θ sin θ ) exp ( i k r cos θ cos θ ) d θ ,
I 2 ( k r , θ ) = 0 θ max cos 1 2 θ sin θ ( 1 cos θ ) J 2 ( k r sin θ sin θ ) exp ( i k r cos θ cos θ ) d θ ,
ε ( r ̄ ) = 20 log 10 { max E x FDTD ( r ̄ , t ) E x th ( r ̄ , t ) E x ref } ,

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