Abstract

We present measurements and calculations of the terahertz (THz) electric field measured in the near field of a metal tip used in THz apertureless near-field optical microscopy (THz-ANSOM). An analytical model in which we treat the metal tip as a linear wire antenna allows us to predict almost all of the features observed in the measurements, such as the relatively slow decay of the near-field amplitude when the tip-crystal separation increases. When the tip-crystal separation is modulated, in conjunction with lock-in detection at the modulation frequency, a smaller THz spot size is observed underneath the tip. A comparison with analytical expressions shows that in this case the electric field originates predominantly from the tip apex, with negligible contributions from the tip shaft. In the unmodulated case, the observed signal is the spatial integral of the electro-optic (EO) effect over the interaction length between the THz near field and the probe laser pulse. In the modulated case, to a good approximation, we find that the signal is proportional to the value of the THz near field at the surface of the EO crystal only.

© 2007 Optical Society of America

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  1. H.-T. Chen, R. Kersting, and G. C. Cho, 'Terahertz imaging nanometer resolution,' Appl. Phys. Lett. 83, 3009-3012 (2003).
    [CrossRef]
  2. H. T. Chen, S. Kraatz, G. C. Cho, and R. Kersting, 'Identification of a resonant imaging process in aperturless near-field microscopy,' Phys. Rev. Lett. 93, 267401 (2004).
    [CrossRef]
  3. K. Wang, A. Barkan, and D. M. Mittleman, 'Propagation effects in apertureless near-field optical antennas,' Appl. Phys. Lett. 84, 305-308 (2004).
    [CrossRef]
  4. M. Walther, G. S. Chambers, Z. Liu, M. R. Freeman, and F. A. Hegmann, 'Emission and detection of terahertz pulses from a metal-tip antenna,' J. Opt. Soc. Am. B 22, 2357-2365 (2005).
    [CrossRef]
  5. F. F. Buersgens, H.-T. Chen, and R. Kersting, 'Terahertz microscopy of charge carriers in semiconductors,' Appl. Phys. Lett. 88, 112115 (2006).
    [CrossRef]
  6. N. C. J. van der Valk and P. C. M. Planken, 'Electro-optic detection of subwavelength terahertz spot sizes in the near field of a metal tip,' Appl. Phys. Lett. 81, 1558-1560 (2002).
    [CrossRef]
  7. K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, 'Antenna effects in terahertz apertureless near-field optical microscopy,' Appl. Phys. Lett. 85, 2715-2717 (2004).
    [CrossRef]
  8. P. C. M. Planken and N. C. J. van der Valk, 'Spot-size reduction in terahertz apertureless near-field imaging,' Opt. Lett. 29, 2306-2308 (2004).
    [CrossRef] [PubMed]
  9. J. A. Deibel, N. Berndsen, K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, 'Frequency-dependent radiation patterns emitted by THz plasmons on finite length cylindrical metal wires,' Opt. Express 14, 8772-8778 (2006).
    [CrossRef] [PubMed]
  10. W. Denk and D. W. Pohl, 'Near-field optics: microscopy with nanometer-size fields,' J. Vac. Sci. Technol. B 9, 510-513 (1991).
    [CrossRef]
  11. C. A. Balanis, Antenna Theory, Analysis and Design, 2nd Edition (Wiley, 1997).
  12. B. Knoll and F. Keilmann, 'Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy,' Opt. Commun. 182, 321-328 (2000).
    [CrossRef]
  13. J. Van Bladel, 'Field singularities at the tip of a cone,' Proc. IEEE 71, 901-902 (1983).
    [CrossRef]
  14. H. Cory, A. C. Boccara, J. C. Rivoal, and A. Lahrech, 'Electric field intensity variation in the vicinity of a perfectly conducting conical probe: application to near-field microscopy,' Microwave Opt. Technol. Lett. 18, 120-124 (1998).
    [CrossRef]
  15. G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, 'Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,' Rev. Sci. Instrum. 73, 1715-1720 (2002).
    [CrossRef]

2006 (2)

2005 (1)

2004 (4)

P. C. M. Planken and N. C. J. van der Valk, 'Spot-size reduction in terahertz apertureless near-field imaging,' Opt. Lett. 29, 2306-2308 (2004).
[CrossRef] [PubMed]

H. T. Chen, S. Kraatz, G. C. Cho, and R. Kersting, 'Identification of a resonant imaging process in aperturless near-field microscopy,' Phys. Rev. Lett. 93, 267401 (2004).
[CrossRef]

K. Wang, A. Barkan, and D. M. Mittleman, 'Propagation effects in apertureless near-field optical antennas,' Appl. Phys. Lett. 84, 305-308 (2004).
[CrossRef]

K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, 'Antenna effects in terahertz apertureless near-field optical microscopy,' Appl. Phys. Lett. 85, 2715-2717 (2004).
[CrossRef]

2003 (1)

H.-T. Chen, R. Kersting, and G. C. Cho, 'Terahertz imaging nanometer resolution,' Appl. Phys. Lett. 83, 3009-3012 (2003).
[CrossRef]

2002 (2)

G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, 'Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,' Rev. Sci. Instrum. 73, 1715-1720 (2002).
[CrossRef]

N. C. J. van der Valk and P. C. M. Planken, 'Electro-optic detection of subwavelength terahertz spot sizes in the near field of a metal tip,' Appl. Phys. Lett. 81, 1558-1560 (2002).
[CrossRef]

2000 (1)

B. Knoll and F. Keilmann, 'Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy,' Opt. Commun. 182, 321-328 (2000).
[CrossRef]

1998 (1)

H. Cory, A. C. Boccara, J. C. Rivoal, and A. Lahrech, 'Electric field intensity variation in the vicinity of a perfectly conducting conical probe: application to near-field microscopy,' Microwave Opt. Technol. Lett. 18, 120-124 (1998).
[CrossRef]

1991 (1)

W. Denk and D. W. Pohl, 'Near-field optics: microscopy with nanometer-size fields,' J. Vac. Sci. Technol. B 9, 510-513 (1991).
[CrossRef]

1983 (1)

J. Van Bladel, 'Field singularities at the tip of a cone,' Proc. IEEE 71, 901-902 (1983).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Antenna Theory, Analysis and Design, 2nd Edition (Wiley, 1997).

Barkan, A.

K. Wang, A. Barkan, and D. M. Mittleman, 'Propagation effects in apertureless near-field optical antennas,' Appl. Phys. Lett. 84, 305-308 (2004).
[CrossRef]

Berndsen, N.

Boccara, A. C.

H. Cory, A. C. Boccara, J. C. Rivoal, and A. Lahrech, 'Electric field intensity variation in the vicinity of a perfectly conducting conical probe: application to near-field microscopy,' Microwave Opt. Technol. Lett. 18, 120-124 (1998).
[CrossRef]

Buersgens, F. F.

F. F. Buersgens, H.-T. Chen, and R. Kersting, 'Terahertz microscopy of charge carriers in semiconductors,' Appl. Phys. Lett. 88, 112115 (2006).
[CrossRef]

Chambers, G. S.

Chen, H. T.

H. T. Chen, S. Kraatz, G. C. Cho, and R. Kersting, 'Identification of a resonant imaging process in aperturless near-field microscopy,' Phys. Rev. Lett. 93, 267401 (2004).
[CrossRef]

Chen, H.-T.

F. F. Buersgens, H.-T. Chen, and R. Kersting, 'Terahertz microscopy of charge carriers in semiconductors,' Appl. Phys. Lett. 88, 112115 (2006).
[CrossRef]

H.-T. Chen, R. Kersting, and G. C. Cho, 'Terahertz imaging nanometer resolution,' Appl. Phys. Lett. 83, 3009-3012 (2003).
[CrossRef]

Cho, G. C.

H. T. Chen, S. Kraatz, G. C. Cho, and R. Kersting, 'Identification of a resonant imaging process in aperturless near-field microscopy,' Phys. Rev. Lett. 93, 267401 (2004).
[CrossRef]

H.-T. Chen, R. Kersting, and G. C. Cho, 'Terahertz imaging nanometer resolution,' Appl. Phys. Lett. 83, 3009-3012 (2003).
[CrossRef]

Cory, H.

H. Cory, A. C. Boccara, J. C. Rivoal, and A. Lahrech, 'Electric field intensity variation in the vicinity of a perfectly conducting conical probe: application to near-field microscopy,' Microwave Opt. Technol. Lett. 18, 120-124 (1998).
[CrossRef]

Deibel, J. A.

Denk, W.

W. Denk and D. W. Pohl, 'Near-field optics: microscopy with nanometer-size fields,' J. Vac. Sci. Technol. B 9, 510-513 (1991).
[CrossRef]

Freeman, M. R.

Hegmann, F. A.

Keilmann, F.

B. Knoll and F. Keilmann, 'Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy,' Opt. Commun. 182, 321-328 (2000).
[CrossRef]

Kersting, R.

F. F. Buersgens, H.-T. Chen, and R. Kersting, 'Terahertz microscopy of charge carriers in semiconductors,' Appl. Phys. Lett. 88, 112115 (2006).
[CrossRef]

H. T. Chen, S. Kraatz, G. C. Cho, and R. Kersting, 'Identification of a resonant imaging process in aperturless near-field microscopy,' Phys. Rev. Lett. 93, 267401 (2004).
[CrossRef]

H.-T. Chen, R. Kersting, and G. C. Cho, 'Terahertz imaging nanometer resolution,' Appl. Phys. Lett. 83, 3009-3012 (2003).
[CrossRef]

Knoll, B.

B. Knoll and F. Keilmann, 'Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy,' Opt. Commun. 182, 321-328 (2000).
[CrossRef]

Kraatz, S.

H. T. Chen, S. Kraatz, G. C. Cho, and R. Kersting, 'Identification of a resonant imaging process in aperturless near-field microscopy,' Phys. Rev. Lett. 93, 267401 (2004).
[CrossRef]

Lahrech, A.

H. Cory, A. C. Boccara, J. C. Rivoal, and A. Lahrech, 'Electric field intensity variation in the vicinity of a perfectly conducting conical probe: application to near-field microscopy,' Microwave Opt. Technol. Lett. 18, 120-124 (1998).
[CrossRef]

Liu, Z.

Mittleman, D. M.

J. A. Deibel, N. Berndsen, K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, 'Frequency-dependent radiation patterns emitted by THz plasmons on finite length cylindrical metal wires,' Opt. Express 14, 8772-8778 (2006).
[CrossRef] [PubMed]

K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, 'Antenna effects in terahertz apertureless near-field optical microscopy,' Appl. Phys. Lett. 85, 2715-2717 (2004).
[CrossRef]

K. Wang, A. Barkan, and D. M. Mittleman, 'Propagation effects in apertureless near-field optical antennas,' Appl. Phys. Lett. 84, 305-308 (2004).
[CrossRef]

Planken, P. C. M.

J. A. Deibel, N. Berndsen, K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, 'Frequency-dependent radiation patterns emitted by THz plasmons on finite length cylindrical metal wires,' Opt. Express 14, 8772-8778 (2006).
[CrossRef] [PubMed]

K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, 'Antenna effects in terahertz apertureless near-field optical microscopy,' Appl. Phys. Lett. 85, 2715-2717 (2004).
[CrossRef]

P. C. M. Planken and N. C. J. van der Valk, 'Spot-size reduction in terahertz apertureless near-field imaging,' Opt. Lett. 29, 2306-2308 (2004).
[CrossRef] [PubMed]

G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, 'Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,' Rev. Sci. Instrum. 73, 1715-1720 (2002).
[CrossRef]

N. C. J. van der Valk and P. C. M. Planken, 'Electro-optic detection of subwavelength terahertz spot sizes in the near field of a metal tip,' Appl. Phys. Lett. 81, 1558-1560 (2002).
[CrossRef]

Pohl, D. W.

W. Denk and D. W. Pohl, 'Near-field optics: microscopy with nanometer-size fields,' J. Vac. Sci. Technol. B 9, 510-513 (1991).
[CrossRef]

Rivoal, J. C.

H. Cory, A. C. Boccara, J. C. Rivoal, and A. Lahrech, 'Electric field intensity variation in the vicinity of a perfectly conducting conical probe: application to near-field microscopy,' Microwave Opt. Technol. Lett. 18, 120-124 (1998).
[CrossRef]

Schouten, R. N.

G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, 'Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,' Rev. Sci. Instrum. 73, 1715-1720 (2002).
[CrossRef]

Van Bladel, J.

J. Van Bladel, 'Field singularities at the tip of a cone,' Proc. IEEE 71, 901-902 (1983).
[CrossRef]

van der Valk, N.

G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, 'Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,' Rev. Sci. Instrum. 73, 1715-1720 (2002).
[CrossRef]

van der Valk, N. C. J.

J. A. Deibel, N. Berndsen, K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, 'Frequency-dependent radiation patterns emitted by THz plasmons on finite length cylindrical metal wires,' Opt. Express 14, 8772-8778 (2006).
[CrossRef] [PubMed]

K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, 'Antenna effects in terahertz apertureless near-field optical microscopy,' Appl. Phys. Lett. 85, 2715-2717 (2004).
[CrossRef]

P. C. M. Planken and N. C. J. van der Valk, 'Spot-size reduction in terahertz apertureless near-field imaging,' Opt. Lett. 29, 2306-2308 (2004).
[CrossRef] [PubMed]

N. C. J. van der Valk and P. C. M. Planken, 'Electro-optic detection of subwavelength terahertz spot sizes in the near field of a metal tip,' Appl. Phys. Lett. 81, 1558-1560 (2002).
[CrossRef]

Walther, M.

Wang, K.

J. A. Deibel, N. Berndsen, K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, 'Frequency-dependent radiation patterns emitted by THz plasmons on finite length cylindrical metal wires,' Opt. Express 14, 8772-8778 (2006).
[CrossRef] [PubMed]

K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, 'Antenna effects in terahertz apertureless near-field optical microscopy,' Appl. Phys. Lett. 85, 2715-2717 (2004).
[CrossRef]

K. Wang, A. Barkan, and D. M. Mittleman, 'Propagation effects in apertureless near-field optical antennas,' Appl. Phys. Lett. 84, 305-308 (2004).
[CrossRef]

Wenckebach, W. Th.

G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, 'Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,' Rev. Sci. Instrum. 73, 1715-1720 (2002).
[CrossRef]

Zhao, G.

G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, 'Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,' Rev. Sci. Instrum. 73, 1715-1720 (2002).
[CrossRef]

Appl. Phys. Lett. (5)

H.-T. Chen, R. Kersting, and G. C. Cho, 'Terahertz imaging nanometer resolution,' Appl. Phys. Lett. 83, 3009-3012 (2003).
[CrossRef]

K. Wang, A. Barkan, and D. M. Mittleman, 'Propagation effects in apertureless near-field optical antennas,' Appl. Phys. Lett. 84, 305-308 (2004).
[CrossRef]

F. F. Buersgens, H.-T. Chen, and R. Kersting, 'Terahertz microscopy of charge carriers in semiconductors,' Appl. Phys. Lett. 88, 112115 (2006).
[CrossRef]

N. C. J. van der Valk and P. C. M. Planken, 'Electro-optic detection of subwavelength terahertz spot sizes in the near field of a metal tip,' Appl. Phys. Lett. 81, 1558-1560 (2002).
[CrossRef]

K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, 'Antenna effects in terahertz apertureless near-field optical microscopy,' Appl. Phys. Lett. 85, 2715-2717 (2004).
[CrossRef]

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

J. Vac. Sci. Technol. B (1)

W. Denk and D. W. Pohl, 'Near-field optics: microscopy with nanometer-size fields,' J. Vac. Sci. Technol. B 9, 510-513 (1991).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

H. Cory, A. C. Boccara, J. C. Rivoal, and A. Lahrech, 'Electric field intensity variation in the vicinity of a perfectly conducting conical probe: application to near-field microscopy,' Microwave Opt. Technol. Lett. 18, 120-124 (1998).
[CrossRef]

Opt. Commun. (1)

B. Knoll and F. Keilmann, 'Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy,' Opt. Commun. 182, 321-328 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

H. T. Chen, S. Kraatz, G. C. Cho, and R. Kersting, 'Identification of a resonant imaging process in aperturless near-field microscopy,' Phys. Rev. Lett. 93, 267401 (2004).
[CrossRef]

Proc. IEEE (1)

J. Van Bladel, 'Field singularities at the tip of a cone,' Proc. IEEE 71, 901-902 (1983).
[CrossRef]

Rev. Sci. Instrum. (1)

G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, 'Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,' Rev. Sci. Instrum. 73, 1715-1720 (2002).
[CrossRef]

Other (1)

C. A. Balanis, Antenna Theory, Analysis and Design, 2nd Edition (Wiley, 1997).

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Figures (8)

Fig. 1
Fig. 1

Drawing of a vertical point dipole above a dielectric half-space.

Fig. 2
Fig. 2

EO signal calculated for the sphere model at different lengths of integration p for q = 0 . Note that the curves are not normalized. The inset is a close up with a linear scale.

Fig. 3
Fig. 3

(a) Drawing of a vertical point dipole above a dielectric half-space. (b) Vertical wire antenna above a dielectric half-space. (c) Vertical wire antenna above a dielectric half-space showing the path inside the crystal along which the EO effect is integrated. (d) Schematic of the vertical wire antenna with a triangular current distribution.

Fig. 4
Fig. 4

(a) Drawing of a horizontal point dipole above a dielectric half-space. (b) Horizontal wire antenna above a dielectric half-space. (c) Horizontal wire antenna above a dielectric half-space showing the path inside the crystal along which the EO effect is integrated. (d) Schematic of the horizontal wire antenna with a triangular current distribution.

Fig. 5
Fig. 5

Measured EO signal (squares) as a function of the tip-crystal distance. The solid curve is calculated using the antenna model and the dotted curve is calculated using the sphere model. The inset is a signal recorded by the differential detector as a function of time for a tip-crystal distance of 1 μ m .

Fig. 6
Fig. 6

Measured EO signal as a function of the tip-crystal distance when the tip-crystal separation is modulated. The signal is detected at the tip-crystal modulation frequency. The points represent the measured data, the solid curve is calculated using the antenna model.

Fig. 7
Fig. 7

EO signal calculated for the sphere model and for the antenna model for different lengths d of the antenna with q = 0 μ m and p = 100 μ m .

Fig. 8
Fig. 8

THz near-field intensity distribution measured in a plane directly underneath (a) a 16 μ m diameter gold-coated polystyrene sphere, (b) the metal tip, and (c) a 500 μ m diameter copper sphere.

Equations (53)

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V z > 0 = p d R A S 4 π ϵ 0 R A S 3 + ϵ ϵ 0 ϵ + ϵ 0 p d R B S 4 π ϵ R B S 3 , ( z > 0 ) ,
V z 0 = 2 ϵ ϵ 0 + ϵ p d R A S 4 π ϵ R A S 3 ( z 0 ) ,
V ( S ) = 1 4 2 ϵ ϵ 0 + ϵ p d x π ϵ ( x 2 + y 2 + ( z z 0 ) 2 ) 3 2 .
E z = 3 8 2 ϵ ϵ 0 + ϵ p d x ( 2 z 2 z 0 ) π ϵ ( x 2 + y 2 + ( z z 0 ) 2 ) 5 2 .
V ( S ) = 1 4 2 ϵ ϵ 0 + ϵ p d ( z z 0 ) π ϵ ( x 2 + y 2 + ( z z o ) 2 ) 3 2 .
E z = 1 4 2 ϵ ϵ 0 + ϵ p d π ϵ ( x 2 + y 2 + ( z z 0 ) 2 ) 3 2 + 3 8 ( z z o ) p d ( 2 z 2 z o ) π ϵ ( x 2 + y 2 + ( z z o ) 2 ) 5 2 .
E z = 2 ( 1 + ϵ r ) p r 2 4 π ϵ 0 ( r 2 + ( z z 0 ) 2 ) 5 2 + 2 ( 1 + ϵ r ) 2 p ( z z 0 ) 2 4 π ϵ 0 ( r 2 + ( z z 0 ) 2 ) 5 2 .
E z ( 0 , 0 , z ) = 1 2 2 ϵ ϵ 0 + ϵ p d π ϵ ( z z o ) 3 .
S EO p q E z d z ,
S EO p d 4 π ϵ 0 ( 1 ( q + z 0 ) 2 1 ( p + z 0 ) 2 ) ,
p ( z 0 , t ) = d z 0 t I ( z 0 , t ) d t = t I 2 ( t ) d t I 1 ( z 0 ) d z 0 G ( t ) I 1 ( z 0 ) d z 0 .
d E z = 2 ( 1 + ϵ r ) G ( t ) I 1 ( z 0 ) r 2 d z 0 4 π ϵ 0 ( r 2 + ( z z 0 ) 2 ) 5 2 + 2 ( 1 + ϵ r ) 2 G ( t ) I 1 ( z 0 ) ( z z 0 ) 2 d z 0 4 π ϵ 0 ( r 2 + ( z z 0 ) 2 ) 5 2 .
E z = 2 G ( t ) 4 π ϵ 0 ( 1 + ϵ r ) a b ( I 1 ( z 0 ) r 2 ( r 2 + ( z z 0 ) 2 ) 5 2 2 I 1 ( z 0 ) ( z z 0 ) 2 ( r 2 + ( z z 0 ) 2 ) 5 2 ) d z 0 .
I 1 ( z 0 ) = C 0 ( z 0 a ) ( a < z 0 < ( a + b a 2 ) ) ,
I 1 ( z 0 ) = C 0 ( b z 0 ) ( ( a + b a 2 ) < z 0 < b ) ,
E z = 2 C 0 G ( t ) 4 π ϵ 0 ( 1 + ϵ r ) a ( a + b ) 2 ( ( z 0 a ) r 2 ( r 2 + ( z z 0 ) 2 ) 5 2 2 ( z 0 a ) ( z z 0 ) 2 ( r 2 + ( z z 0 ) 2 ) 5 2 ) d z 0 2 C 0 G ( t ) 4 π ϵ 0 ( 1 + ϵ r ) ( a + b ) 2 b ( ( b z 0 ) r 2 ( r 2 + ( z z 0 ) 2 ) 5 2 2 ( b z 0 ) ( z z 0 ) 2 ( r 2 + ( z z 0 ) 2 ) 5 2 ) d z 0 .
R 2 = r 2 + ( z a + b 2 ) 2 ,
R a 2 = r 2 + ( z a ) 2 ,
R b 2 = r 2 + ( z b ) 2 ,
E z = 2 C 0 G ( t ) 4 π ϵ 0 ( 1 + ϵ r ) ( 1 R a + 1 R b 2 R ) ( z 0 ) .
E z = 6 p ( x x 0 ) ( z z 0 ) ( 1 + ϵ r ) 4 π ϵ 0 ( ( x x 0 ) 2 + y 2 + ( z z 0 ) 2 ) 5 2 .
E z = 2 C 0 G ( t ) 4 ϵ 0 π ( 1 + ϵ r ) ( z z 0 ) ( ( z z 0 ) 2 + y 2 ) ( x + a R a ¯ + x a R a 2 x R )
( z 0 ) ,
R 2 = x 2 + y 2 + ( z z 0 ) 2 ,
R a 2 = ( x a ) 2 + y 2 + ( z z 0 ) 2 ,
R a ¯ 2 = ( x + a ) 2 + y 2 + ( z z 0 ) 2 .
S EO p q E z d z ,
S EO 2 C 0 G ( t ) 4 π ϵ 0 ( 1 + ϵ r ) [ ln ( q a + R q a p a + R p a ) + ln ( q b + R p b p b + R p b ) 2 ln ( b a 2 q + 2 R q b a 2 p + 2 R p ) ] .
R p a 2 = r 2 + ( a + p ) 2 ,
R p b 2 = r 2 + ( b + p ) 2 ,
R q a 2 = r 2 + ( a + q ) 2 ,
R q b 2 = r 2 + ( b + q ) 2 ,
R p 2 = r 2 + ( p + a + b 2 ) 2 ,
R q 2 = r 2 + ( q + a + b 2 ) 2 .
S EO C 0 G ( t ) 4 π ϵ 0 ( 1 + ϵ r ) × [ ln ( [ ( p + z 0 ) 2 + y 2 ] [ ( ( x a ) 2 + y 2 + ( x a ) R q a ) 2 + y 2 ( q + z 0 ) 2 ] [ ( q + z 0 ) 2 + y 2 ] [ ( ( x a ) 2 + y 2 + ( x a ) R p a ) 2 + y 2 ( q + z 0 ) 2 ] ) + ln ( [ ( p + z 0 ) 2 + y 2 ] [ ( ( x + a ) 2 + y 2 + ( x + a ) R q a ¯ ) 2 + y 2 ( q + z 0 ) 2 ] [ ( q + z 0 ) 2 + y 2 ] [ ( ( x + a ) 2 + y 2 + ( x + a ) R p a ¯ ) 2 + y 2 ( q + z 0 ) 2 ] ) 2 ln ( [ ( p + z 0 ) 2 + y 2 ] [ ( x 2 + y 2 + x R q ) 2 + y 2 ( q + z 0 ) 2 ] [ ( q + z 0 ) 2 + y 2 ] [ ( x 2 + y 2 + x R p ) 2 + y 2 ( q + z 0 ) 2 ] ) ] ,
R p a 2 = ( x a ) 2 + y 2 + ( p + z 0 ) 2 ,
R p a ¯ 2 = ( x + a ) 2 + y 2 + ( p + z 0 ) 2 ,
R q a 2 = ( x a ) 2 + y 2 + ( q + z 0 ) 2 ,
R q a ¯ 2 = ( x + a ) 2 + y 2 + ( q + z 0 ) 2 ,
R p 2 = x 2 + y 2 + ( p + z 0 ) 2 ,
R q 2 = x 2 + y 2 + ( q + z 0 ) 2 .
ln ( q a + ( r 2 + ( a + q ) 2 ) 1 2 p a + ( r 2 + ( a + p ) 2 ) 1 2 ) .
( r 2 + ( a + q ) 2 ) 1 2 ( a + q ) + r 2 2 ( a + q ) r 4 8 ( a + q ) 3 + O ( r 6 ) ,
( r 2 + ( a + p ) 2 ) 1 2 ( a + p ) + r 2 2 ( a + p ) r 4 8 ( a + p ) 3 + O ( r 6 ) .
ln ( a + p a + q ) .
S EO 2 C 0 G ( t ) 4 π ϵ 0 ( 1 + ϵ r ) [ ln ( a + p a + q ) + ln ( b + p b + q ) 2 ln ( a + b 2 + p a + b 2 + q ) ] ( r = 0 ) .
S EO 2 C 0 G ( t ) 4 π ϵ 0 ( 1 + ϵ r ) ln ( a + p a + q ) .
E 1 R cos ( θ 2 ) u r 1 R sin ( θ 2 ) u θ ,
E 1 R z ̂ ,
S EO [ S EO a ] a = a 0 2 C 0 G ( t ) 4 π ϵ 0 ( 1 + ϵ r ) ( 1 R p b + 1 R p a 0 1 R q a 0 1 R q b + 2 R q 2 R p ) ,
S EO 2 C 0 G ( t ) 4 π ϵ 0 ( 1 + ϵ r ) ( 1 R q a 0 + 1 R q b 2 R q ) .
S EO [ S EO z 0 ] z 0 = z 1 C 0 G ( t ) 4 π ϵ 0 ( 1 + ϵ r ) ( q + z 1 R q a ( x a + R q a ) p + z 1 R p a ( x a + R p a ) + q + z 1 R q a ¯ ( x + a + R q a ¯ ) p + z 1 R p a ¯ ( x + a + R p a ¯ ) 2 ( q + z 1 ) R q ( x + R q ) + 2 ( p + z 1 ) R p ( x + R p ) ) ,
S EO C 0 G ( t ) 4 π ϵ 0 ( 1 + ϵ r ) 1 ( q + z 1 ) ( x a R q a + x + a R q a ¯ 2 x R q ) ,

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