Abstract

A dispersive body-of-revolution finite-difference time-domain method is developed to simulate metal-cladded near-field scanning optical microscope (NSOM) probes. Two types of NSOM probe (aperture and plasmon NSOM probes) are analyzed and designed with this fast method. The influence of the metal-cladding thickness and the excitation mode on the performance of the NSOM probes is studied. We introduce a new scheme of illumination-mode NSOM by employing the plasmon NSOM probe with the TM01 mode excitation. Such a NSOM probe is designed, and we demonstrate its advantages over the conventional aperture NSOM probe by scanning across a metallic object.

© 2005 Optical Society of America

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    [CrossRef]
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2003 (2)

F. I. Baida, D. V. Labeke, Y. Pagani, “Body-of-revolution FDTD simulations of improved tip performance for scanning near-field optical microscopes,” Opt. Commun. 225, 241–252 (2003).
[CrossRef]

W. X. Sun, Z. X. Shen, “Optimizing the near field around silver tips,” J. Opt. Soc. Am. A 20, 2254–2259 (2003).
[CrossRef]

2002 (4)

J. T. Krug, E. J. Sánchez, X. S. Xie, “Design of near-field optical probes with optimal field enhancement by finite difference time domain electromagnetic simulation,” J. Chem. Phys. 116, 10895–10901 (2002).
[CrossRef]

A. Kramer, W. Trabesinger, B. Hecht, U. P. Wild, “Optical near-field enhancement at a metal tip probed by a single fluorophore,” Appl. Phys. Lett. 80, 1652–1654 (2002).
[CrossRef]

W. Trabesinger, A. Kramer, M. Kreiter, B. Hecht, U. P. Wild, “Single-molecule near-field optical energy transfer microscopy,” Appl. Phys. Lett. 81, 2118–2120 (2002).
[CrossRef]

Y. C. Martin, H. K. Wickramasinghe, “Resolution test for apertureless near-field optical microscopy,” J. Appl. Phys. 91, 3363–3368 (2002).
[CrossRef]

2001 (3)

J. L. Young, R. O. Nelson, “A summary and systematic analysis of FDTD algorithm for linearly dispersive media,” IEEE Antennas Propag. Mag. 43, 61–77 (2001).
[CrossRef]

Y. C. Martin, H. F. Hamann, H. K. Wickramasinghe, “Strength of the electric field in apertureless near-field optical microscopy,” J. Appl. Phys. 89, 5774–5778 (2001).
[CrossRef]

J. L. Bohn, D. J. Nesbitt, A. Gallagher, “Field enhancement in apertureless near-field scanning optical microscopy,” J. Opt. Soc. Am. A 18, 2998–3006 (2001).
[CrossRef]

2000 (3)

M. S. Mirotznik, D. W. Prather, J. N. Mait, W. A. Beck, S. Y. Shi, X. Gao, “Three-dimensional analysis of subwave-length diffractive optical elements with the finite-difference time-domain method,” Appl. Opt. 39, 2871–2880 (2000).
[CrossRef]

W. Yu, D. Arakaki, R. Mittra, “On the solution of a class of large body problems with full or partial circular symmetry by using the finite-difference time-domain (FDTD) method,” IEEE Trans. Antennas Propag. 48, 1810–1817 (2000).
[CrossRef]

F. L. Teixeira, W. C. Chew, “Finite-difference computation of transient electromagnetic waves for cylindrical geometries in complex media,” IEEE Trans. Geosci. Remote Sens. 38, 1530–1543 (2000).
[CrossRef]

1999 (2)

E. J. Sánchez, L. Novotny, X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[CrossRef]

D. W. Prather, S. Y. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A 16, 1131–1142 (1999).
[CrossRef]

1998 (3)

C. Durkana, I. V. Shvets, “Reflection-mode scanning near-field optical microscopy: influence of sample type, tip shape, and polarization of light,” J. Appl. Phys. 83, 1171–1176 (1998).
[CrossRef]

L. Novotny, E. J. Sanchez, X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams,” Ultramicroscopy 71, 1–4 (1998).
[CrossRef]

M. Ashino, M. Ohtsu, “Fabrication and evaluation of a localized plasmon resonance probe for near-field optical microscopy/spectroscopy,” Appl. Phys. Lett. 72, 1299–1301 (1998).
[CrossRef]

1997 (3)

O. J. F. Martin, C. Girard, “Controlling and tuning strong optical field gradients at a local probe microscope tip apex,” Appl. Phys. Lett. 70, 705–707 (1997).
[CrossRef]

S. A. Cummer, “An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy,” IEEE Trans. Antennas Propag. 45, 392–400 (1997).
[CrossRef]

R. Bachelot, P. Gleyzes, A. C. Boccara, “Reflection-mode scanning near-field optical microscopy using an apertureless metallic tip,” Appl. Opt. 36, 2160–2170 (1997).
[CrossRef] [PubMed]

1996 (1)

Y. Chen, R. Mittra, “Finite-difference time-domain algorithm for solving Maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. 44, 832–839 (1996).
[CrossRef]

1995 (4)

D. A. Christensen, “Analysis of near field tip patterns including object interaction using finite-difference time-domain calculations,” Ultramicroscopy 57, 189–195 (1995).
[CrossRef]

O. J. F. Martin, C. Girard, A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[CrossRef] [PubMed]

L. Novotny, D. W. Pohl, B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
[CrossRef]

D. Van Labeke, D. Barchiesi, F. Baida, “Optical characterization of nanosources used in scanning near-field optical microscopy,” J. Opt. Soc. Am. A 12, 695–703 (1995).
[CrossRef]

1994 (2)

1993 (1)

1992 (1)

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Al-Bader, S. J.

Arakaki, D.

W. Yu, D. Arakaki, R. Mittra, “On the solution of a class of large body problems with full or partial circular symmetry by using the finite-difference time-domain (FDTD) method,” IEEE Trans. Antennas Propag. 48, 1810–1817 (2000).
[CrossRef]

Ashino, M.

M. Ashino, M. Ohtsu, “Fabrication and evaluation of a localized plasmon resonance probe for near-field optical microscopy/spectroscopy,” Appl. Phys. Lett. 72, 1299–1301 (1998).
[CrossRef]

Bachelot, R.

Baida, F.

Baida, F. I.

F. I. Baida, D. V. Labeke, Y. Pagani, “Body-of-revolution FDTD simulations of improved tip performance for scanning near-field optical microscopes,” Opt. Commun. 225, 241–252 (2003).
[CrossRef]

Barchiesi, D.

Beck, W. A.

Betzig, E.

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Beversluis, M. R.

A. Bouhelier, J. Renger, M. R. Beversluis, L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. (Oxford) 210, 220–224 (2003).
[CrossRef]

Boccara, A. C.

Bohn, J. L.

Bouhelier, A.

A. Bouhelier, J. Renger, M. R. Beversluis, L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. (Oxford) 210, 220–224 (2003).
[CrossRef]

Chen, Y.

Y. Chen, R. Mittra, “Finite-difference time-domain algorithm for solving Maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. 44, 832–839 (1996).
[CrossRef]

Chew, W. C.

F. L. Teixeira, W. C. Chew, “Finite-difference computation of transient electromagnetic waves for cylindrical geometries in complex media,” IEEE Trans. Geosci. Remote Sens. 38, 1530–1543 (2000).
[CrossRef]

Christensen, D. A.

D. A. Christensen, “Analysis of near field tip patterns including object interaction using finite-difference time-domain calculations,” Ultramicroscopy 57, 189–195 (1995).
[CrossRef]

Corrado, B. J.

Cummer, S. A.

S. A. Cummer, “An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy,” IEEE Trans. Antennas Propag. 45, 392–400 (1997).
[CrossRef]

Davidson, D. B.

Dereux, A.

O. J. F. Martin, C. Girard, A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[CrossRef] [PubMed]

Durkana, C.

C. Durkana, I. V. Shvets, “Reflection-mode scanning near-field optical microscopy: influence of sample type, tip shape, and polarization of light,” J. Appl. Phys. 83, 1171–1176 (1998).
[CrossRef]

Fillard, J. P.

J. P. Fillard, Near Field Optics and Nanoscopy (World Scientific, Singapore, 1996), pp. 264–267.

Gallagher, A.

Gao, X.

Girard, C.

O. J. F. Martin, C. Girard, “Controlling and tuning strong optical field gradients at a local probe microscope tip apex,” Appl. Phys. Lett. 70, 705–707 (1997).
[CrossRef]

O. J. F. Martin, C. Girard, A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[CrossRef] [PubMed]

Gleyzes, P.

Hamann, H. F.

Y. C. Martin, H. F. Hamann, H. K. Wickramasinghe, “Strength of the electric field in apertureless near-field optical microscopy,” J. Appl. Phys. 89, 5774–5778 (2001).
[CrossRef]

Hecht, B.

A. Kramer, W. Trabesinger, B. Hecht, U. P. Wild, “Optical near-field enhancement at a metal tip probed by a single fluorophore,” Appl. Phys. Lett. 80, 1652–1654 (2002).
[CrossRef]

W. Trabesinger, A. Kramer, M. Kreiter, B. Hecht, U. P. Wild, “Single-molecule near-field optical energy transfer microscopy,” Appl. Phys. Lett. 81, 2118–2120 (2002).
[CrossRef]

L. Novotny, D. W. Pohl, B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
[CrossRef]

Imtaar, M.

Kramer, A.

A. Kramer, W. Trabesinger, B. Hecht, U. P. Wild, “Optical near-field enhancement at a metal tip probed by a single fluorophore,” Appl. Phys. Lett. 80, 1652–1654 (2002).
[CrossRef]

W. Trabesinger, A. Kramer, M. Kreiter, B. Hecht, U. P. Wild, “Single-molecule near-field optical energy transfer microscopy,” Appl. Phys. Lett. 81, 2118–2120 (2002).
[CrossRef]

Kreiter, M.

W. Trabesinger, A. Kramer, M. Kreiter, B. Hecht, U. P. Wild, “Single-molecule near-field optical energy transfer microscopy,” Appl. Phys. Lett. 81, 2118–2120 (2002).
[CrossRef]

Krug, J. T.

J. T. Krug, E. J. Sánchez, X. S. Xie, “Design of near-field optical probes with optimal field enhancement by finite difference time domain electromagnetic simulation,” J. Chem. Phys. 116, 10895–10901 (2002).
[CrossRef]

Labeke, D. V.

F. I. Baida, D. V. Labeke, Y. Pagani, “Body-of-revolution FDTD simulations of improved tip performance for scanning near-field optical microscopes,” Opt. Commun. 225, 241–252 (2003).
[CrossRef]

Labeke, D. Van

Mait, J. N.

Martin, O. J. F.

O. J. F. Martin, C. Girard, “Controlling and tuning strong optical field gradients at a local probe microscope tip apex,” Appl. Phys. Lett. 70, 705–707 (1997).
[CrossRef]

O. J. F. Martin, C. Girard, A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[CrossRef] [PubMed]

Martin, Y. C.

Y. C. Martin, H. K. Wickramasinghe, “Resolution test for apertureless near-field optical microscopy,” J. Appl. Phys. 91, 3363–3368 (2002).
[CrossRef]

Y. C. Martin, H. F. Hamann, H. K. Wickramasinghe, “Strength of the electric field in apertureless near-field optical microscopy,” J. Appl. Phys. 89, 5774–5778 (2001).
[CrossRef]

Mirotznik, M. S.

Mittra, R.

W. Yu, D. Arakaki, R. Mittra, “On the solution of a class of large body problems with full or partial circular symmetry by using the finite-difference time-domain (FDTD) method,” IEEE Trans. Antennas Propag. 48, 1810–1817 (2000).
[CrossRef]

Y. Chen, R. Mittra, “Finite-difference time-domain algorithm for solving Maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. 44, 832–839 (1996).
[CrossRef]

Moyer, P. J.

M. A. Paesler, P. J. Moyer, Near-Field Optics: Theory, Instrumentation, and Applications (Wiley, New York, 1996), Part 3.

Nelson, R. O.

J. L. Young, R. O. Nelson, “A summary and systematic analysis of FDTD algorithm for linearly dispersive media,” IEEE Antennas Propag. Mag. 43, 61–77 (2001).
[CrossRef]

Nesbitt, D. J.

Novotny, L.

E. J. Sánchez, L. Novotny, X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[CrossRef]

L. Novotny, E. J. Sanchez, X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams,” Ultramicroscopy 71, 1–4 (1998).
[CrossRef]

L. Novotny, D. W. Pohl, B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
[CrossRef]

A. Bouhelier, J. Renger, M. R. Beversluis, L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. (Oxford) 210, 220–224 (2003).
[CrossRef]

Ohtsu, M.

M. Ashino, M. Ohtsu, “Fabrication and evaluation of a localized plasmon resonance probe for near-field optical microscopy/spectroscopy,” Appl. Phys. Lett. 72, 1299–1301 (1998).
[CrossRef]

M. Ohtsu, Near-Field Nano/Atom Optics and Technology (Springer, Tokyo, 1998).
[CrossRef]

Paesler, M. A.

M. A. Paesler, P. J. Moyer, Near-Field Optics: Theory, Instrumentation, and Applications (Wiley, New York, 1996), Part 3.

Pagani, Y.

F. I. Baida, D. V. Labeke, Y. Pagani, “Body-of-revolution FDTD simulations of improved tip performance for scanning near-field optical microscopes,” Opt. Commun. 225, 241–252 (2003).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, London, 1985), pp. 350–357.

Pohl, D. W.

L. Novotny, D. W. Pohl, B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
[CrossRef]

Prather, D. W.

Raether, H.

H. Raether, Surface Plasmons (Springer, Berlin, 1988), pp. 16–18.

Renger, J.

A. Bouhelier, J. Renger, M. R. Beversluis, L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. (Oxford) 210, 220–224 (2003).
[CrossRef]

Sanchez, E. J.

L. Novotny, E. J. Sanchez, X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams,” Ultramicroscopy 71, 1–4 (1998).
[CrossRef]

Sánchez, E. J.

J. T. Krug, E. J. Sánchez, X. S. Xie, “Design of near-field optical probes with optimal field enhancement by finite difference time domain electromagnetic simulation,” J. Chem. Phys. 116, 10895–10901 (2002).
[CrossRef]

E. J. Sánchez, L. Novotny, X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[CrossRef]

Shen, Z. X.

Shi, S. Y.

Shvets, I. V.

C. Durkana, I. V. Shvets, “Reflection-mode scanning near-field optical microscopy: influence of sample type, tip shape, and polarization of light,” J. Appl. Phys. 83, 1171–1176 (1998).
[CrossRef]

Sun, W. X.

Taflove, A.

A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, Mass., 1998).

Teixeira, F. L.

F. L. Teixeira, W. C. Chew, “Finite-difference computation of transient electromagnetic waves for cylindrical geometries in complex media,” IEEE Trans. Geosci. Remote Sens. 38, 1530–1543 (2000).
[CrossRef]

Thornburg, W. Q.

Trabesinger, W.

W. Trabesinger, A. Kramer, M. Kreiter, B. Hecht, U. P. Wild, “Single-molecule near-field optical energy transfer microscopy,” Appl. Phys. Lett. 81, 2118–2120 (2002).
[CrossRef]

A. Kramer, W. Trabesinger, B. Hecht, U. P. Wild, “Optical near-field enhancement at a metal tip probed by a single fluorophore,” Appl. Phys. Lett. 80, 1652–1654 (2002).
[CrossRef]

Trautman, J. K.

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Wickramasinghe, H. K.

Y. C. Martin, H. K. Wickramasinghe, “Resolution test for apertureless near-field optical microscopy,” J. Appl. Phys. 91, 3363–3368 (2002).
[CrossRef]

Y. C. Martin, H. F. Hamann, H. K. Wickramasinghe, “Strength of the electric field in apertureless near-field optical microscopy,” J. Appl. Phys. 89, 5774–5778 (2001).
[CrossRef]

Wild, U. P.

A. Kramer, W. Trabesinger, B. Hecht, U. P. Wild, “Optical near-field enhancement at a metal tip probed by a single fluorophore,” Appl. Phys. Lett. 80, 1652–1654 (2002).
[CrossRef]

W. Trabesinger, A. Kramer, M. Kreiter, B. Hecht, U. P. Wild, “Single-molecule near-field optical energy transfer microscopy,” Appl. Phys. Lett. 81, 2118–2120 (2002).
[CrossRef]

Xie, X. S.

J. T. Krug, E. J. Sánchez, X. S. Xie, “Design of near-field optical probes with optimal field enhancement by finite difference time domain electromagnetic simulation,” J. Chem. Phys. 116, 10895–10901 (2002).
[CrossRef]

E. J. Sánchez, L. Novotny, X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[CrossRef]

L. Novotny, E. J. Sanchez, X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams,” Ultramicroscopy 71, 1–4 (1998).
[CrossRef]

Young, J. L.

J. L. Young, R. O. Nelson, “A summary and systematic analysis of FDTD algorithm for linearly dispersive media,” IEEE Antennas Propag. Mag. 43, 61–77 (2001).
[CrossRef]

Yu, W.

W. Yu, D. Arakaki, R. Mittra, “On the solution of a class of large body problems with full or partial circular symmetry by using the finite-difference time-domain (FDTD) method,” IEEE Trans. Antennas Propag. 48, 1810–1817 (2000).
[CrossRef]

Zhu, X. D.

Ziolkowski, R. W.

Appl. Opt. (2)

Appl. Phys. Lett. (4)

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Figures (12)

Fig. 1
Fig. 1

Positions of the field components in a unit cell of the Yee lattice.

Fig. 2
Fig. 2

Total-field and scattered-field regions in the present dispersive BOR FDTD method. PML, perfectly matched layer.

Fig. 3
Fig. 3

Three different structures of probes considered in the present paper: (a) a metal-cladded cylindrical waveguide, (b) an aperture NSOM probe (design example 1), (c) a plasmon NSOM probe (design example 2). The probe is excited with a field at the source plane, and the propagation is along the +z direction; n denotes the refractive index of the corresponding materials.

Fig. 4
Fig. 4

Time snapshot of the distribution of the field component Eφ in a metal-cladded cylindrical waveguide along the z direction.

Fig. 5
Fig. 5

Distribution of the normalized field intensity I for an aperture NSOM probe on (a) the x–z plane passing through the axis ρ = 0, (b) the y–z plane passing through the axis ρ = 0, (c) an x–y plane 5 nm away from the probe end. The axis unit is 10−7 m. The probe is excited with the x-polarized HE11 mode. The structural parameters are listed in Table 1. The metal cladding of the probe is indicated by the dashed lines.

Fig. 6
Fig. 6

Maximal value Im of the normalized field intensity I on an x–y plane 5 nm away from the probe as the metal-cladding thickness of the aperture NSOM probe increases.

Fig. 7
Fig. 7

Distribution of the normalized field intensity I at three planes (same as Fig. 5) when the metal-cladding thickness of the aperture NSOM probe is reduced to t = 30 nm. The other configuration is the same as Fig. 5. The axis unit is 10−7 m.

Fig. 8
Fig. 8

Distribution of the normalized field intensity I for a plasmon NSOM probe with the x-polarized HE11 mode excitation: (a) on the x–z plane passing through ρ = 0, (b) on the y–z plane passing through ρ = 0, (c) on the x–y plane 5 nm away from the probe end. The axis unit is 10−7 m. The structural parameters are listed in Table 1. The metal cladding of the probe is indicated by the dashed lines.

Fig. 9
Fig. 9

Distribution of the normalized field intensity I for a plasmon NSOM probe with the TM01 mode excitation: (a) on the x–z plane passing through ρ = 0, (b) on the x–y plane 5 nm away from the probe end. The axis unit is 10−7 m. The structural parameters are the same as those used for Fig. 8.

Fig. 10
Fig. 10

(a) Maximal value Im of the normalized field intensity I and (b) the beam-spot size on an x–y plane 5 nm away from the probe end as the metal-cladding thickness of the plasmon NSOM probe increases.

Fig. 11
Fig. 11

Scanning configuration of a NSOM probe near a metal cuboid.

Fig. 12
Fig. 12

Detected response (i.e., the Pr curve) as the probe moves along the x axis for different types of NSOM probe. Pr is normalized with its value in the absence of the metal cuboid.

Tables (1)

Tables Icon

Table 1 Structural Parameters of the Three Different Types of Probe

Equations (12)

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ɛ 0 E t = × H J p ,
μ 0 H t = × E ,
d J p d t = ɛ 0 ω p 2 E γ J p ,
ɛ ( ω ) ɛ 0 = ɛ r + j ɛ i = ( n r + j n i ) 2 = 1 ω p 2 ω 2 + j γ ω .
A = { A s sin ( ν φ ) A c cos ( ν φ ) ,
{ [ E ρ , c , E φ , s , E z , c , H ρ , s , H φ , c , H z , s , J ρ , c , J φ , s , J z , c ] [ E ρ , s , E φ , c , E z , s , H ρ , c , H φ , s , H z , c , J ρ , s , J φ , c , J z , s ] ,
E k + 1 = E k + Δ t ɛ 0 · ( × H k + 1 / 2 J p k + 1 / 2 ) ,
H k + 1 / 2 = H k 1 / 2 Δ t μ 0 · × E k ,
J p k + 1 / 2 = 1 0.5 γ Δ t 1 + 0.5 γ Δ t · J p k + 1 / 2 = ɛ 0 ω p 2 Δ t 1 + 0.5 γ Δ t · E k ,
Δ t min [ δ s ν max , ( 4 δ s 2 4 c 2 + δ x 2 ω p 2 ) 1 / 2 ] ,
δ s = [ ( ν + 1 ) 2 + 2.8 4 Δ ρ 2 + 1 Δ z 2 ] 1 / 2 .
H z k + 1 / 2 | r = 0 = { H z k 1 / 2 | r = 0 4 Δ t μ 0 Δ r E φ k | r = Δ r / 2 ν = 0 0 otherwise .

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