Abstract

We calculate theoretically the coupling of a terahertz wave from a dipole into a two-wire waveguide. The field transmission and reflection are obtained using a Single Mode Matching (SMM) technique at the input port of the two-wire waveguide. The results show more than 70 percent coupling efficiency for the waveguide using 500μm radii wires with 2mm center-to-center separation and the exciting field cross section of 1mm × 1mm. The results also show good agreement with the full-wave numerical simulations using the Finite Element Method (FEM).

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  1. M. Y. Frankel, S. Gupta, J. A. Valdmanis, and G. A. Mourou, “Terahertz attenuation and dispersion characteristics of coplanar transmission lines,” IEEE Trans. Microw. Theory Tech. 39(6), 910–916 (1991).
    [CrossRef]
  2. C. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17(5), 851–863 (2000).
    [CrossRef]
  3. S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76(15), 1987–1989 (2000).
    [CrossRef]
  4. R. Mendis and D. Grischkowsky, “Plastic ribbon thz waveguides,” J. Appl. Phys. 88(7), 4449–4451 (2000).
    [CrossRef]
  5. K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 (2004).
    [CrossRef] [PubMed]
  6. J. A. Deibel, K. Wang, M. D. Escarra, and D. M. Mittleman, “Enhanced coupling of terahertz radiation to cylindrical wire waveguides,” Opt. Express 14(1), 279–290 (2006).
    [CrossRef] [PubMed]
  7. M. K. Mbonye, V. Astley, W. L. Chan, J. A. Deibel, and D. M. Mittleman, “A terahertz dual wire waveguide,” in Lasers and Electro-Optics Conference, Optical Society of America, 2007, paper CThLL1.
  8. M. K. Mbonye, R. Mendis, and D. M. Mittleman, “A terahertz two-wire waveguide with low bending loss,” Appl. Phys. Lett. 95(23), 233506 (2009).
    [CrossRef]
  9. H. Pahlevaninezhad, T. E. Darcie, and B. Heshmat, “Two-wire waveguide for terahertz,” Opt. Express 18(7), 7415–7420 (2010).
    [CrossRef] [PubMed]
  10. S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. 70(5), 559‒561 (1997).
    [CrossRef]
  11. D. Dragoman and M. Dragoman, “Terahertz fields and applications,” Elsevier, Progress in Quantum Electronics 28(1), 1–66 (2004), doi:.
    [CrossRef]
  12. S. M. Duffy, S. Verghese, A. McIntosh, A. Jackson, A. C. Gossard, and S. Matsuura, “Accurate modeling of dual dipole and slot elements used with photomixers for coherent terahertz output power,” IEEE Trans. Microw. Theory Tech. 49(6), 1032–1038 (2001).
    [CrossRef]
  13. S. Matsuura and H. Ito, “Generation of CW terahertz radiation with photomixing,” Top. Appl. Phys. 97, 157–202 (2005).
    [CrossRef]
  14. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. Total radiated power,” J. Opt. Soc. Am. 67(12), 1607–1614 (1977).
    [CrossRef]
  15. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67(12), 1615–1619 (1977).
    [CrossRef]
  16. W. Lukosz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. III. Radiation patterns of dipoles with arbitrary orientation,” J. Opt. Soc. Am. 69(11), 1495–1502 (1979).
    [CrossRef]
  17. J. Y. Courtois, J. M. Courty, and J. C. Mertz, “Internal dynamics of multilevel atoms near a vacuum-dielectric interface,” Phys. Rev. A 53(3), 1862–1878 (1996).
    [CrossRef] [PubMed]
  18. L. Luan, P. R. Sievert, and J. B. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half space,” J. Phys. 8, 264 (2006), doi:.
  19. P. U. Jepsen and S. R. Keiding, “Radiation patterns from lens-coupled terahertz antennas,” Opt. Lett. 20(8), 807–809 (1995).
    [CrossRef] [PubMed]
  20. C. Fattinger and D. Grischkowsky, “Terahertz beams,” Appl. Phys. Lett. 54(6), 490 (1989).
    [CrossRef]
  21. P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, “Generation and detection of terahertz pulses from biased semiconductor antennas,” J. Opt. Soc. Am. B 13(11), 2424–2436 (1996).
    [CrossRef]
  22. R. Gordon, “Vectorial method for calculating the Fresnel reflection of surface plasmon polaritons,” Phys. Rev. B 74, 153417 (2006). URL http://link.aps.org/abstract/PRB/v74/e153417 .
  23. R. Gordon, “Light in a subwavelength slit in a metal: Propagation and reflection,” Phys. Rev. B 73, 153405 (2006). URL http://link.aps.org/abstract/PRB/v73/e153405 .
  24. D. M. Pozar, Microwave engineering: 3rd Ed. (John Wiley & Sons, 2005), Chap.4.
  25. A. Yariv, and P. Yeh, Optical waves in crystals: propagation and control of laser radiation (John Wiley & Sons, 1984), Chap.11.
  26. J. D. Jackson, Classical electrodynamics 3rd Ed. (John Wiley & Sons,1999), pp. 390–394.

2010 (1)

2009 (1)

M. K. Mbonye, R. Mendis, and D. M. Mittleman, “A terahertz two-wire waveguide with low bending loss,” Appl. Phys. Lett. 95(23), 233506 (2009).
[CrossRef]

2006 (2)

J. A. Deibel, K. Wang, M. D. Escarra, and D. M. Mittleman, “Enhanced coupling of terahertz radiation to cylindrical wire waveguides,” Opt. Express 14(1), 279–290 (2006).
[CrossRef] [PubMed]

L. Luan, P. R. Sievert, and J. B. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half space,” J. Phys. 8, 264 (2006), doi:.

2005 (1)

S. Matsuura and H. Ito, “Generation of CW terahertz radiation with photomixing,” Top. Appl. Phys. 97, 157–202 (2005).
[CrossRef]

2004 (2)

D. Dragoman and M. Dragoman, “Terahertz fields and applications,” Elsevier, Progress in Quantum Electronics 28(1), 1–66 (2004), doi:.
[CrossRef]

K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 (2004).
[CrossRef] [PubMed]

2001 (1)

S. M. Duffy, S. Verghese, A. McIntosh, A. Jackson, A. C. Gossard, and S. Matsuura, “Accurate modeling of dual dipole and slot elements used with photomixers for coherent terahertz output power,” IEEE Trans. Microw. Theory Tech. 49(6), 1032–1038 (2001).
[CrossRef]

2000 (3)

C. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17(5), 851–863 (2000).
[CrossRef]

S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76(15), 1987–1989 (2000).
[CrossRef]

R. Mendis and D. Grischkowsky, “Plastic ribbon thz waveguides,” J. Appl. Phys. 88(7), 4449–4451 (2000).
[CrossRef]

1997 (1)

S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. 70(5), 559‒561 (1997).
[CrossRef]

1996 (2)

J. Y. Courtois, J. M. Courty, and J. C. Mertz, “Internal dynamics of multilevel atoms near a vacuum-dielectric interface,” Phys. Rev. A 53(3), 1862–1878 (1996).
[CrossRef] [PubMed]

P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, “Generation and detection of terahertz pulses from biased semiconductor antennas,” J. Opt. Soc. Am. B 13(11), 2424–2436 (1996).
[CrossRef]

1995 (1)

1991 (1)

M. Y. Frankel, S. Gupta, J. A. Valdmanis, and G. A. Mourou, “Terahertz attenuation and dispersion characteristics of coplanar transmission lines,” IEEE Trans. Microw. Theory Tech. 39(6), 910–916 (1991).
[CrossRef]

1989 (1)

C. Fattinger and D. Grischkowsky, “Terahertz beams,” Appl. Phys. Lett. 54(6), 490 (1989).
[CrossRef]

1979 (1)

1977 (2)

Courtois, J. Y.

J. Y. Courtois, J. M. Courty, and J. C. Mertz, “Internal dynamics of multilevel atoms near a vacuum-dielectric interface,” Phys. Rev. A 53(3), 1862–1878 (1996).
[CrossRef] [PubMed]

Courty, J. M.

J. Y. Courtois, J. M. Courty, and J. C. Mertz, “Internal dynamics of multilevel atoms near a vacuum-dielectric interface,” Phys. Rev. A 53(3), 1862–1878 (1996).
[CrossRef] [PubMed]

Darcie, T. E.

Deibel, J. A.

Dragoman, D.

D. Dragoman and M. Dragoman, “Terahertz fields and applications,” Elsevier, Progress in Quantum Electronics 28(1), 1–66 (2004), doi:.
[CrossRef]

Dragoman, M.

D. Dragoman and M. Dragoman, “Terahertz fields and applications,” Elsevier, Progress in Quantum Electronics 28(1), 1–66 (2004), doi:.
[CrossRef]

Duffy, S. M.

S. M. Duffy, S. Verghese, A. McIntosh, A. Jackson, A. C. Gossard, and S. Matsuura, “Accurate modeling of dual dipole and slot elements used with photomixers for coherent terahertz output power,” IEEE Trans. Microw. Theory Tech. 49(6), 1032–1038 (2001).
[CrossRef]

Escarra, M. D.

Fattinger, C.

C. Fattinger and D. Grischkowsky, “Terahertz beams,” Appl. Phys. Lett. 54(6), 490 (1989).
[CrossRef]

Frankel, M. Y.

M. Y. Frankel, S. Gupta, J. A. Valdmanis, and G. A. Mourou, “Terahertz attenuation and dispersion characteristics of coplanar transmission lines,” IEEE Trans. Microw. Theory Tech. 39(6), 910–916 (1991).
[CrossRef]

Gallot, C. G.

Gossard, A. C.

S. M. Duffy, S. Verghese, A. McIntosh, A. Jackson, A. C. Gossard, and S. Matsuura, “Accurate modeling of dual dipole and slot elements used with photomixers for coherent terahertz output power,” IEEE Trans. Microw. Theory Tech. 49(6), 1032–1038 (2001).
[CrossRef]

Grischkowsky, D.

C. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17(5), 851–863 (2000).
[CrossRef]

S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76(15), 1987–1989 (2000).
[CrossRef]

R. Mendis and D. Grischkowsky, “Plastic ribbon thz waveguides,” J. Appl. Phys. 88(7), 4449–4451 (2000).
[CrossRef]

C. Fattinger and D. Grischkowsky, “Terahertz beams,” Appl. Phys. Lett. 54(6), 490 (1989).
[CrossRef]

Gupta, S.

M. Y. Frankel, S. Gupta, J. A. Valdmanis, and G. A. Mourou, “Terahertz attenuation and dispersion characteristics of coplanar transmission lines,” IEEE Trans. Microw. Theory Tech. 39(6), 910–916 (1991).
[CrossRef]

Heshmat, B.

Ito, H.

S. Matsuura and H. Ito, “Generation of CW terahertz radiation with photomixing,” Top. Appl. Phys. 97, 157–202 (2005).
[CrossRef]

Jackson, A.

S. M. Duffy, S. Verghese, A. McIntosh, A. Jackson, A. C. Gossard, and S. Matsuura, “Accurate modeling of dual dipole and slot elements used with photomixers for coherent terahertz output power,” IEEE Trans. Microw. Theory Tech. 49(6), 1032–1038 (2001).
[CrossRef]

Jacobsen, R. H.

Jamison, S. P.

C. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17(5), 851–863 (2000).
[CrossRef]

S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76(15), 1987–1989 (2000).
[CrossRef]

Jepsen, P. U.

Keiding, S. R.

Ketterson, J. B.

L. Luan, P. R. Sievert, and J. B. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half space,” J. Phys. 8, 264 (2006), doi:.

Kunz, R. E.

Luan, L.

L. Luan, P. R. Sievert, and J. B. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half space,” J. Phys. 8, 264 (2006), doi:.

Lukosz, W.

Matsuura, S.

S. Matsuura and H. Ito, “Generation of CW terahertz radiation with photomixing,” Top. Appl. Phys. 97, 157–202 (2005).
[CrossRef]

S. M. Duffy, S. Verghese, A. McIntosh, A. Jackson, A. C. Gossard, and S. Matsuura, “Accurate modeling of dual dipole and slot elements used with photomixers for coherent terahertz output power,” IEEE Trans. Microw. Theory Tech. 49(6), 1032–1038 (2001).
[CrossRef]

S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. 70(5), 559‒561 (1997).
[CrossRef]

Mbonye, M. K.

M. K. Mbonye, R. Mendis, and D. M. Mittleman, “A terahertz two-wire waveguide with low bending loss,” Appl. Phys. Lett. 95(23), 233506 (2009).
[CrossRef]

McGowan, R. W.

S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76(15), 1987–1989 (2000).
[CrossRef]

C. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17(5), 851–863 (2000).
[CrossRef]

McIntosh, A.

S. M. Duffy, S. Verghese, A. McIntosh, A. Jackson, A. C. Gossard, and S. Matsuura, “Accurate modeling of dual dipole and slot elements used with photomixers for coherent terahertz output power,” IEEE Trans. Microw. Theory Tech. 49(6), 1032–1038 (2001).
[CrossRef]

Mendis, R.

M. K. Mbonye, R. Mendis, and D. M. Mittleman, “A terahertz two-wire waveguide with low bending loss,” Appl. Phys. Lett. 95(23), 233506 (2009).
[CrossRef]

R. Mendis and D. Grischkowsky, “Plastic ribbon thz waveguides,” J. Appl. Phys. 88(7), 4449–4451 (2000).
[CrossRef]

Mertz, J. C.

J. Y. Courtois, J. M. Courty, and J. C. Mertz, “Internal dynamics of multilevel atoms near a vacuum-dielectric interface,” Phys. Rev. A 53(3), 1862–1878 (1996).
[CrossRef] [PubMed]

Mittleman, D. M.

M. K. Mbonye, R. Mendis, and D. M. Mittleman, “A terahertz two-wire waveguide with low bending loss,” Appl. Phys. Lett. 95(23), 233506 (2009).
[CrossRef]

J. A. Deibel, K. Wang, M. D. Escarra, and D. M. Mittleman, “Enhanced coupling of terahertz radiation to cylindrical wire waveguides,” Opt. Express 14(1), 279–290 (2006).
[CrossRef] [PubMed]

K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 (2004).
[CrossRef] [PubMed]

Mourou, G. A.

M. Y. Frankel, S. Gupta, J. A. Valdmanis, and G. A. Mourou, “Terahertz attenuation and dispersion characteristics of coplanar transmission lines,” IEEE Trans. Microw. Theory Tech. 39(6), 910–916 (1991).
[CrossRef]

Pahlevaninezhad, H.

Sakai, K.

S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. 70(5), 559‒561 (1997).
[CrossRef]

Sievert, P. R.

L. Luan, P. R. Sievert, and J. B. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half space,” J. Phys. 8, 264 (2006), doi:.

Tani, M.

S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. 70(5), 559‒561 (1997).
[CrossRef]

Valdmanis, J. A.

M. Y. Frankel, S. Gupta, J. A. Valdmanis, and G. A. Mourou, “Terahertz attenuation and dispersion characteristics of coplanar transmission lines,” IEEE Trans. Microw. Theory Tech. 39(6), 910–916 (1991).
[CrossRef]

Verghese, S.

S. M. Duffy, S. Verghese, A. McIntosh, A. Jackson, A. C. Gossard, and S. Matsuura, “Accurate modeling of dual dipole and slot elements used with photomixers for coherent terahertz output power,” IEEE Trans. Microw. Theory Tech. 49(6), 1032–1038 (2001).
[CrossRef]

Wang, K.

Appl. Phys. Lett. (4)

S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76(15), 1987–1989 (2000).
[CrossRef]

M. K. Mbonye, R. Mendis, and D. M. Mittleman, “A terahertz two-wire waveguide with low bending loss,” Appl. Phys. Lett. 95(23), 233506 (2009).
[CrossRef]

S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. 70(5), 559‒561 (1997).
[CrossRef]

C. Fattinger and D. Grischkowsky, “Terahertz beams,” Appl. Phys. Lett. 54(6), 490 (1989).
[CrossRef]

Elsevier, Progress in Quantum Electronics (1)

D. Dragoman and M. Dragoman, “Terahertz fields and applications,” Elsevier, Progress in Quantum Electronics 28(1), 1–66 (2004), doi:.
[CrossRef]

IEEE Trans. Microw. Theory Tech. (2)

S. M. Duffy, S. Verghese, A. McIntosh, A. Jackson, A. C. Gossard, and S. Matsuura, “Accurate modeling of dual dipole and slot elements used with photomixers for coherent terahertz output power,” IEEE Trans. Microw. Theory Tech. 49(6), 1032–1038 (2001).
[CrossRef]

M. Y. Frankel, S. Gupta, J. A. Valdmanis, and G. A. Mourou, “Terahertz attenuation and dispersion characteristics of coplanar transmission lines,” IEEE Trans. Microw. Theory Tech. 39(6), 910–916 (1991).
[CrossRef]

J. Appl. Phys. (1)

R. Mendis and D. Grischkowsky, “Plastic ribbon thz waveguides,” J. Appl. Phys. 88(7), 4449–4451 (2000).
[CrossRef]

J. Opt. Soc. Am. (3)

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

J. Phys. (1)

L. Luan, P. R. Sievert, and J. B. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half space,” J. Phys. 8, 264 (2006), doi:.

Nature (1)

K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 (2004).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (1)

J. Y. Courtois, J. M. Courty, and J. C. Mertz, “Internal dynamics of multilevel atoms near a vacuum-dielectric interface,” Phys. Rev. A 53(3), 1862–1878 (1996).
[CrossRef] [PubMed]

Top. Appl. Phys. (1)

S. Matsuura and H. Ito, “Generation of CW terahertz radiation with photomixing,” Top. Appl. Phys. 97, 157–202 (2005).
[CrossRef]

Other (6)

M. K. Mbonye, V. Astley, W. L. Chan, J. A. Deibel, and D. M. Mittleman, “A terahertz dual wire waveguide,” in Lasers and Electro-Optics Conference, Optical Society of America, 2007, paper CThLL1.

R. Gordon, “Vectorial method for calculating the Fresnel reflection of surface plasmon polaritons,” Phys. Rev. B 74, 153417 (2006). URL http://link.aps.org/abstract/PRB/v74/e153417 .

R. Gordon, “Light in a subwavelength slit in a metal: Propagation and reflection,” Phys. Rev. B 73, 153405 (2006). URL http://link.aps.org/abstract/PRB/v73/e153405 .

D. M. Pozar, Microwave engineering: 3rd Ed. (John Wiley & Sons, 2005), Chap.4.

A. Yariv, and P. Yeh, Optical waves in crystals: propagation and control of laser radiation (John Wiley & Sons, 1984), Chap.11.

J. D. Jackson, Classical electrodynamics 3rd Ed. (John Wiley & Sons,1999), pp. 390–394.

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Figures (8)

Fig. 1
Fig. 1

Intersection of the parallel-plate waveguide and the two-wire waveguide.

Fig. 2
Fig. 2

(a) Cross section of the two-wire waveguides with same edge-to-edge wires' distance but different radii, (b) The transmission and reflection coefficients of the power.

Fig. 3
Fig. 3

The amplitude of the electric field obtained from a 3D full-wave simulations with FEM using the Ansoft HFSS excited by (a) a 0.5mm-long parallel-plate waveguide with 1mm × 0.4mm cross section, (b) a dipole, 200μm away from the input port of the two-wire waveguide, at 1THz .

Fig. 4
Fig. 4

Coupling obtained from the theory (solid line), and from full-wave simulations using FEM (dark squares), for a two-wire waveguide with D = 400μm at 1THz (a) the parallel-plate excitation for simulations and w × d = 1mm × 0.4mm for theory, (b) the dipole excitation for simulations and w × d = 1mm × 1mm for theory.

Fig. 5
Fig. 5

Coupling vs. D, for R = 500μm and w × d = 1mm × 1mm.

Fig. 6
Fig. 6

(a) overlap region of the plane wave (black square) and the waveguide field for R = 500μm and D = 2mm, (b) overlap when R = 500μm and D = 3mm

Fig. 7
Fig. 7

Integration contour for coupling when d > D + 2R.

Fig. 8
Fig. 8

(a) Integration contour when D-2R < d < D + 2R, (b) when d < D-2R.

Equations (33)

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

E 1 , 2 = m ( a 1 , 2 m + b 1 , 2 m ) e 1 , 2 m ,
H 1 , 2 = m ( a 1 , 2 m b 1 , 2 m ) h 1 , 2 m ,
1 2 ( e l × h m ) z d s = δ l m ,
E 1 t ( z = 0 ) = E 2 t ( z = 0 + ) ,
H 1 t ( z = 0 ) = H 2 t ( z = 0 + ) ,
( 1 + r ) ( e 1 ) T E M + m = 2 b 1 m e 1 m = t ( e 2 ) T E M + m = 2 a 2 m e 2 m ,
( 1 + r ) = ( t ) κ + 1 2 m = 2 a 2 m ( e 2 m × h 1 T E M ) d S ,
κ = 1 2 ( e 2 T E M × h 1 T E M ) d S ,
( 1 r ) ( h 1 ) T E M + m = 2 ( b 1 m ) h 1 m = t ( h 2 ) T E M + m = 2 a 2 m h 2 m .
( 1 r ) κ + 1 2 m = 2 ( b 1 m ) ( e 2 m × h 1 T E M ) d S = t .
| r | 2 = ( κ 2 1 κ 2 + 1 ) 2 ,
| t | 2 = ( 2 κ κ 2 + 1 ) 2 ,
( e 1 ) T E M = { A 1 x b e t w e e n t h e p l a t e s , 0 o t h e r w i s e
( h 1 ) T E M = | e 1 | η 0 y ,
A 1 2 = 2 η 0 w d ,
( e 2 ) T E M = e 2 x x + e 2 y y ,
( h T E M ) T E M = 1 η 0 z × ( e 2 ) T E M = 1 η 0 ( e 2 x y e 2 y x ) ,
e 2 x = { A 2 [ ( x + R / C 2 ) ( x + R / C 2 ) 2 + y 2 ( x + R / C 1 ) ( x + R / C 1 ) 2 + y 2 ] o u t s i d e t h e w i r e s 0 i n s i d e t h e w i r e s ,
e 2 y = { A 2 [ y ( x + R / C 2 ) 2 + y 2 y ( x + R / C 1 ) 2 + y 2 ] 0 i n s i d e t h e w i r e s o u t s i d e t h e w i r e s .
C 1 , 2 = D 2 R D 2 R 1 .
1 2 η 0 ( | e 2 x | 2 + | e 2 y | 2 ) d x d y = 1 ,
A 2 = η 0 π ln ( ( 1 C 1 ) [ C 2 ( D / R 1 ) 1 ] ( C 2 1 ) [ 1 C 1 ( D / R 1 ) ] ) .
κ = A 1 A 2 2 η 0 S 1 [ ( x + R / C 2 ) ( x + R / C 2 ) 2 + y 2 ( x + R / C 1 ) ( x + R / C 1 ) 2 + y 2 ] d s ,
κ = A 1 A 2 2 η 0 { w ln [ ( d D 2 + R C 1 ) 2 + ( w 2 ) 2 ( d D 2 + R C 2 ) 2 + ( w 2 ) 2 ] + 4 ( d D 2 + R C 1 ) tan 1 ( w / 2 d D 2 + R C 1 ) 4 ( d D 2 + R C 2 ) tan 1 ( w / 2 d D 2 + R C 2 ) } ,
κ = A 1 A 2 2 η 0 [ w ln [ ( d D 2 + R C 1 ) 2 + ( w 2 ) 2 ( d D 2 + R C 2 ) 2 + ( w 2 ) 2 ] 2 R 2 ( d D 2 ) 2 ln ( ( d D 2 + R C 1 ) 2 + R 2 ( d D 2 ) 2 ( d D 2 + R C 2 ) 2 + R 2 ( d D 2 ) 2 )
+ 4 ( d D 2 + R C 1 ) tan 1 ( w / 2 d D 2 + R C 1 ) 4 ( d D 2 + R C 2 ) tan 1 ( w / 2 d D 2 + R C 2 ) ] ,
Γ ( F 1 d x + F 2 d y ) = S ( F 2 x F 1 y ) d x d y .
e 2 = A 2 t ( ln ( | z 2 | ) ) = A 2 ( x ( ln ( | z 2 | ) ) x + y ( ln ( | z 2 | ) ) y ) ,
| z 2 | = [ R C 1 ( x + D ) ] 2 + ( C 1 y ) 2 [ R C 2 ( x + D ) ] 2 + ( C 2 y ) 2 .
κ = A 1 A 2 2 η 0 S 1 x [ ln ( | z 3 | ) ] d x d y = A 1 A 2 2 η 0 Γ ln ( | z 3 | ) d y ,
κ = A 1 A 2 2 η 0 { w ln [ ( d D 2 + R C 1 ) 2 + ( w 2 ) 2 ( d D 2 + R C 2 ) 2 + ( w 2 ) 2 ] + 4 ( d D 2 + R C 1 ) tan 1 ( w / 2 d D 2 + R C 1 ) 4 ( d D 2 + R C 2 ) tan 1 ( w / 2 d D 2 + R C 2 ) } .
κ = A 1 A 2 2 η 0 [ w ln [ ( d D 2 + R C 1 ) 2 + ( w 2 ) 2 ( d D 2 + R C 2 ) 2 + ( w 2 ) 2 ] 2 R 2 ( d D 2 ) 2 ln ( ( d D 2 + R C 1 ) 2 + R 2 ( d D 2 ) 2 ( d D 2 + R C 2 ) 2 + R 2 ( d D 2 ) 2 )
+ 4 ( d D 2 + R C 1 ) tan 1 ( w / 2 d D 2 + R C 1 ) 4 ( d D 2 + R C 2 ) tan 1 ( w / 2 d D 2 + R C 2 ) ] .

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