Abstract

We experimentally and theoretically investigate the lowest-order transverse-electric (TE1) mode of the parallel-plate waveguide (PPWG) for the propagation of broadband THz pulses. We demonstrate undistorted THz pulse propagation via the single TE1 mode, solving the group-velocity-dispersion and spectral-filtering problems caused by the mode’s low-frequency cutoff. We observe a remarkable counterintuitive property of the TE1 mode: its attenuation decreases with increasing frequency for all frequencies above cutoff. This phenomenon has not been observed with any other THz waveguide to date, and it can enable extremely low-loss propagation. We present a physical interpretation of this frequency-dependent behavior using a simple plane-wave description of the TE1 mode propagation. We also find that it is possible to achieve almost 100% coupling to the TE1 mode from a focused free-space Gaussian beam. In addition, using the above plane-wave analysis, we show how to mitigate the diffraction losses inherent to long path-length PPWGs via the use of transverse-concave plates.

© 2009 Optical Society of America

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References

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  1. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17, 851-863 (2000).
    [CrossRef]
  2. 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, 1987-1989 (2000).
    [CrossRef]
  3. R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguide,” J. Appl. Phys. 88, 4449-4451 (2001).
    [CrossRef]
  4. R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26, 846-848 (2001).
    [CrossRef]
  5. R. Mendis and D. Gischkowsky, “THz interconnect with low loss and low group velocity dispersion,” IEEE Microw. Wirel. Compon. Lett. 11, 444-446 (2001).
    [CrossRef]
  6. H. Han, H. Park, M. Cho, and J. Kim, “Terahertz pulse propagation in a plastic photonic crystal fiber,” Appl. Phys. Lett. 80, 2634-2636 (2002).
    [CrossRef]
  7. T. -I. Jeon and D. Grischkowsky, “Direct optoelectronic generation and detection of sub-ps electrical pulses on sub-mm coaxial transmission lines,” Appl. Phys. Lett. 85, 6092-6094 (2004).
    [CrossRef]
  8. K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432, 376-379 (2004).
    [CrossRef] [PubMed]
  9. T. -I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86, 161904 (2005).
    [CrossRef]
  10. K. Wang and D. M. Mittleman, “Guided propagation of terahertz pulses on metal wires,” J. Opt. Soc. Am. B 22, 2001-2008 (2005).
    [CrossRef]
  11. M. Wachter, M. Nagel, and H. Kurz, “Metallic slit waveguide for dispersion-free low-loss terahertz signal transmission,” Appl. Phys. Lett. 90, 061111 (2007).
    [CrossRef]
  12. R. Mendis, “Comment on “Low-loss terahertz ribbon waveguides”,” Appl. Opt. 47, 4231-4234 (2008).
    [CrossRef] [PubMed]
  13. H. Nishihara, T. Inoue, and J. Koyama, “Low-loss parallel-plate waveguide at 10.6 μm,” Appl. Phys. Lett. 25, 391-393 (1974).
    [CrossRef]
  14. E. Garmire, T. McMahon, and M. Bass, “Flexible infrared-transmissive metal waveguides,” Appl. Phys. Lett. 29, 254-256 (1976).
    [CrossRef]
  15. Y. Mizushima, T. Sugeta, T. Urisu, H. Nishihara, and J. Koyama, “Ultralow loss waveguide for long distance transmission,” Appl. Opt. 19, 3259-3260 (1980).
    [CrossRef] [PubMed]
  16. N. Marcuvitz, Waveguide Handbook (Peregrinus, 1993).
  17. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).
  18. Y. Jiang, C. Jing, W. A. Peebles, D. L. Brower, and J. L. Doane, “Improved performance of an optically pumped FIR laser using metallic waveguide,” Rev. Sci. Instrum. 63, 4672-4674 (1992).
    [CrossRef]
  19. T. A. Abele, D. A. Alsberg, and P. T. Hutchison, “A high-capacity digital communication system using TE01 transmission in circular waveguide,” IEEE Trans. Microwave Theory Tech. 23, 326-333 (1975).
    [CrossRef]
  20. J. T. Verdeyen, Laser Electronics (Prentice Hall, 1995).

2008 (1)

2007 (1)

M. Wachter, M. Nagel, and H. Kurz, “Metallic slit waveguide for dispersion-free low-loss terahertz signal transmission,” Appl. Phys. Lett. 90, 061111 (2007).
[CrossRef]

2005 (2)

T. -I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86, 161904 (2005).
[CrossRef]

K. Wang and D. M. Mittleman, “Guided propagation of terahertz pulses on metal wires,” J. Opt. Soc. Am. B 22, 2001-2008 (2005).
[CrossRef]

2004 (2)

T. -I. Jeon and D. Grischkowsky, “Direct optoelectronic generation and detection of sub-ps electrical pulses on sub-mm coaxial transmission lines,” Appl. Phys. Lett. 85, 6092-6094 (2004).
[CrossRef]

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

2002 (1)

H. Han, H. Park, M. Cho, and J. Kim, “Terahertz pulse propagation in a plastic photonic crystal fiber,” Appl. Phys. Lett. 80, 2634-2636 (2002).
[CrossRef]

2001 (3)

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguide,” J. Appl. Phys. 88, 4449-4451 (2001).
[CrossRef]

R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26, 846-848 (2001).
[CrossRef]

R. Mendis and D. Gischkowsky, “THz interconnect with low loss and low group velocity dispersion,” IEEE Microw. Wirel. Compon. Lett. 11, 444-446 (2001).
[CrossRef]

2000 (2)

G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17, 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, 1987-1989 (2000).
[CrossRef]

1992 (1)

Y. Jiang, C. Jing, W. A. Peebles, D. L. Brower, and J. L. Doane, “Improved performance of an optically pumped FIR laser using metallic waveguide,” Rev. Sci. Instrum. 63, 4672-4674 (1992).
[CrossRef]

1980 (1)

1976 (1)

E. Garmire, T. McMahon, and M. Bass, “Flexible infrared-transmissive metal waveguides,” Appl. Phys. Lett. 29, 254-256 (1976).
[CrossRef]

1975 (1)

T. A. Abele, D. A. Alsberg, and P. T. Hutchison, “A high-capacity digital communication system using TE01 transmission in circular waveguide,” IEEE Trans. Microwave Theory Tech. 23, 326-333 (1975).
[CrossRef]

1974 (1)

H. Nishihara, T. Inoue, and J. Koyama, “Low-loss parallel-plate waveguide at 10.6 μm,” Appl. Phys. Lett. 25, 391-393 (1974).
[CrossRef]

Abele, T. A.

T. A. Abele, D. A. Alsberg, and P. T. Hutchison, “A high-capacity digital communication system using TE01 transmission in circular waveguide,” IEEE Trans. Microwave Theory Tech. 23, 326-333 (1975).
[CrossRef]

Alsberg, D. A.

T. A. Abele, D. A. Alsberg, and P. T. Hutchison, “A high-capacity digital communication system using TE01 transmission in circular waveguide,” IEEE Trans. Microwave Theory Tech. 23, 326-333 (1975).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

Bass, M.

E. Garmire, T. McMahon, and M. Bass, “Flexible infrared-transmissive metal waveguides,” Appl. Phys. Lett. 29, 254-256 (1976).
[CrossRef]

Brower, D. L.

Y. Jiang, C. Jing, W. A. Peebles, D. L. Brower, and J. L. Doane, “Improved performance of an optically pumped FIR laser using metallic waveguide,” Rev. Sci. Instrum. 63, 4672-4674 (1992).
[CrossRef]

Cho, M.

H. Han, H. Park, M. Cho, and J. Kim, “Terahertz pulse propagation in a plastic photonic crystal fiber,” Appl. Phys. Lett. 80, 2634-2636 (2002).
[CrossRef]

Doane, J. L.

Y. Jiang, C. Jing, W. A. Peebles, D. L. Brower, and J. L. Doane, “Improved performance of an optically pumped FIR laser using metallic waveguide,” Rev. Sci. Instrum. 63, 4672-4674 (1992).
[CrossRef]

Gallot, G.

Garmire, E.

E. Garmire, T. McMahon, and M. Bass, “Flexible infrared-transmissive metal waveguides,” Appl. Phys. Lett. 29, 254-256 (1976).
[CrossRef]

Gischkowsky, D.

R. Mendis and D. Gischkowsky, “THz interconnect with low loss and low group velocity dispersion,” IEEE Microw. Wirel. Compon. Lett. 11, 444-446 (2001).
[CrossRef]

Grischkowsky, D.

T. -I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86, 161904 (2005).
[CrossRef]

T. -I. Jeon and D. Grischkowsky, “Direct optoelectronic generation and detection of sub-ps electrical pulses on sub-mm coaxial transmission lines,” Appl. Phys. Lett. 85, 6092-6094 (2004).
[CrossRef]

R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26, 846-848 (2001).
[CrossRef]

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguide,” J. Appl. Phys. 88, 4449-4451 (2001).
[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, 1987-1989 (2000).
[CrossRef]

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

Han, H.

H. Han, H. Park, M. Cho, and J. Kim, “Terahertz pulse propagation in a plastic photonic crystal fiber,” Appl. Phys. Lett. 80, 2634-2636 (2002).
[CrossRef]

Hutchison, P. T.

T. A. Abele, D. A. Alsberg, and P. T. Hutchison, “A high-capacity digital communication system using TE01 transmission in circular waveguide,” IEEE Trans. Microwave Theory Tech. 23, 326-333 (1975).
[CrossRef]

Inoue, T.

H. Nishihara, T. Inoue, and J. Koyama, “Low-loss parallel-plate waveguide at 10.6 μm,” Appl. Phys. Lett. 25, 391-393 (1974).
[CrossRef]

Jamison, S. P.

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, 1987-1989 (2000).
[CrossRef]

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

Jeon, T. -I.

T. -I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86, 161904 (2005).
[CrossRef]

T. -I. Jeon and D. Grischkowsky, “Direct optoelectronic generation and detection of sub-ps electrical pulses on sub-mm coaxial transmission lines,” Appl. Phys. Lett. 85, 6092-6094 (2004).
[CrossRef]

Jiang, Y.

Y. Jiang, C. Jing, W. A. Peebles, D. L. Brower, and J. L. Doane, “Improved performance of an optically pumped FIR laser using metallic waveguide,” Rev. Sci. Instrum. 63, 4672-4674 (1992).
[CrossRef]

Jing, C.

Y. Jiang, C. Jing, W. A. Peebles, D. L. Brower, and J. L. Doane, “Improved performance of an optically pumped FIR laser using metallic waveguide,” Rev. Sci. Instrum. 63, 4672-4674 (1992).
[CrossRef]

Kim, J.

H. Han, H. Park, M. Cho, and J. Kim, “Terahertz pulse propagation in a plastic photonic crystal fiber,” Appl. Phys. Lett. 80, 2634-2636 (2002).
[CrossRef]

Koyama, J.

Y. Mizushima, T. Sugeta, T. Urisu, H. Nishihara, and J. Koyama, “Ultralow loss waveguide for long distance transmission,” Appl. Opt. 19, 3259-3260 (1980).
[CrossRef] [PubMed]

H. Nishihara, T. Inoue, and J. Koyama, “Low-loss parallel-plate waveguide at 10.6 μm,” Appl. Phys. Lett. 25, 391-393 (1974).
[CrossRef]

Kurz, H.

M. Wachter, M. Nagel, and H. Kurz, “Metallic slit waveguide for dispersion-free low-loss terahertz signal transmission,” Appl. Phys. Lett. 90, 061111 (2007).
[CrossRef]

Marcuvitz, N.

N. Marcuvitz, Waveguide Handbook (Peregrinus, 1993).

McGowan, R. W.

G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17, 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, 1987-1989 (2000).
[CrossRef]

McMahon, T.

E. Garmire, T. McMahon, and M. Bass, “Flexible infrared-transmissive metal waveguides,” Appl. Phys. Lett. 29, 254-256 (1976).
[CrossRef]

Mendis, R.

R. Mendis, “Comment on “Low-loss terahertz ribbon waveguides”,” Appl. Opt. 47, 4231-4234 (2008).
[CrossRef] [PubMed]

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguide,” J. Appl. Phys. 88, 4449-4451 (2001).
[CrossRef]

R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26, 846-848 (2001).
[CrossRef]

R. Mendis and D. Gischkowsky, “THz interconnect with low loss and low group velocity dispersion,” IEEE Microw. Wirel. Compon. Lett. 11, 444-446 (2001).
[CrossRef]

Mittleman, D. M.

Mizushima, Y.

Nagel, M.

M. Wachter, M. Nagel, and H. Kurz, “Metallic slit waveguide for dispersion-free low-loss terahertz signal transmission,” Appl. Phys. Lett. 90, 061111 (2007).
[CrossRef]

Nishihara, H.

Y. Mizushima, T. Sugeta, T. Urisu, H. Nishihara, and J. Koyama, “Ultralow loss waveguide for long distance transmission,” Appl. Opt. 19, 3259-3260 (1980).
[CrossRef] [PubMed]

H. Nishihara, T. Inoue, and J. Koyama, “Low-loss parallel-plate waveguide at 10.6 μm,” Appl. Phys. Lett. 25, 391-393 (1974).
[CrossRef]

Park, H.

H. Han, H. Park, M. Cho, and J. Kim, “Terahertz pulse propagation in a plastic photonic crystal fiber,” Appl. Phys. Lett. 80, 2634-2636 (2002).
[CrossRef]

Peebles, W. A.

Y. Jiang, C. Jing, W. A. Peebles, D. L. Brower, and J. L. Doane, “Improved performance of an optically pumped FIR laser using metallic waveguide,” Rev. Sci. Instrum. 63, 4672-4674 (1992).
[CrossRef]

Sugeta, T.

Urisu, T.

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics (Prentice Hall, 1995).

Wachter, M.

M. Wachter, M. Nagel, and H. Kurz, “Metallic slit waveguide for dispersion-free low-loss terahertz signal transmission,” Appl. Phys. Lett. 90, 061111 (2007).
[CrossRef]

Wang, K.

Zhang, J.

T. -I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86, 161904 (2005).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (7)

H. Nishihara, T. Inoue, and J. Koyama, “Low-loss parallel-plate waveguide at 10.6 μm,” Appl. Phys. Lett. 25, 391-393 (1974).
[CrossRef]

E. Garmire, T. McMahon, and M. Bass, “Flexible infrared-transmissive metal waveguides,” Appl. Phys. Lett. 29, 254-256 (1976).
[CrossRef]

M. Wachter, M. Nagel, and H. Kurz, “Metallic slit waveguide for dispersion-free low-loss terahertz signal transmission,” Appl. Phys. Lett. 90, 061111 (2007).
[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, 1987-1989 (2000).
[CrossRef]

H. Han, H. Park, M. Cho, and J. Kim, “Terahertz pulse propagation in a plastic photonic crystal fiber,” Appl. Phys. Lett. 80, 2634-2636 (2002).
[CrossRef]

T. -I. Jeon and D. Grischkowsky, “Direct optoelectronic generation and detection of sub-ps electrical pulses on sub-mm coaxial transmission lines,” Appl. Phys. Lett. 85, 6092-6094 (2004).
[CrossRef]

T. -I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86, 161904 (2005).
[CrossRef]

IEEE Microw. Wirel. Compon. Lett. (1)

R. Mendis and D. Gischkowsky, “THz interconnect with low loss and low group velocity dispersion,” IEEE Microw. Wirel. Compon. Lett. 11, 444-446 (2001).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

T. A. Abele, D. A. Alsberg, and P. T. Hutchison, “A high-capacity digital communication system using TE01 transmission in circular waveguide,” IEEE Trans. Microwave Theory Tech. 23, 326-333 (1975).
[CrossRef]

J. Appl. Phys. (1)

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguide,” J. Appl. Phys. 88, 4449-4451 (2001).
[CrossRef]

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

Nature (1)

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

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

Y. Jiang, C. Jing, W. A. Peebles, D. L. Brower, and J. L. Doane, “Improved performance of an optically pumped FIR laser using metallic waveguide,” Rev. Sci. Instrum. 63, 4672-4674 (1992).
[CrossRef]

Other (3)

J. T. Verdeyen, Laser Electronics (Prentice Hall, 1995).

N. Marcuvitz, Waveguide Handbook (Peregrinus, 1993).

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

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Figures (11)

Fig. 1
Fig. 1

Time-domain THz waveforms corresponding to (a) input reference, (b) TE 1 -mode propagation in 25.4 mm long PPWG with b = 0.5 mm , (c) TE 1 -mode propagation in 6.4 mm long PPWG with b = 0.5 mm , and (d) TE 1 -mode propagation in 25.4 mm long PPWG with b = 5 mm . The inset (circled) shows the excitation polarization axis with respect to the plate surfaces.

Fig. 2
Fig. 2

Amplitude spectra corresponding to the scans in Figs. 1a, 1b. The latter is given by the dots.

Fig. 3
Fig. 3

(a) Phase and group velocity for the TE 1 mode. The theoretical thick and thin solid curves are for b = 0.5 mm and 5 mm , respectively. The open circles and dots are experimental. (b) Close-up of the phase velocity for b = 5 mm . The solid curve is theoretical and the dots are experimental.

Fig. 4
Fig. 4

(a) The attenuation constant for the TE 1 mode. The theoretical thick and thin solid curves are for b = 0.5 mm and 5 mm , respectively. The dots are experimental. (b) Close-up of the baseline of the theoretical curve for b = 5 mm .

Fig. 5
Fig. 5

The power-coupling efficiency from an input Gaussian beam to several even-symmetric TE modes of the PPWG.

Fig. 6
Fig. 6

Longitudinal cross section of the PPWG depicting the bouncing plane wave.

Fig. 7
Fig. 7

Frequency dependence of the incidence angle θ [left vertical axis] and the number of bounces N per meter length of the waveguide [right vertical axis], for the plane wave corresponding to the TE 1 mode with b = 0.5 mm . The cutoff frequency is indicated by the dashed line.

Fig. 8
Fig. 8

Magnitude of the reflection coefficient for the plane-wave incident at the air–metal boundary for the two orthogonal polarizations s and p, which correspond to the TE 1 and TM 1 modes, respectively. The conductivity of the metal is taken to be 3.96 × 10 7 S m , the DC conductivity of aluminum.

Fig. 9
Fig. 9

Comparison of the attenuation constant derived using the simple plane-wave description and the expressions given in [16]. The derived curves are represented by the dots and squares for the TE 1 and TM 1 modes, respectively, whereas the solid curves are from [16]. The agreement is so good that the corresponding curves are indistinguishable at this scale. The unique frequency dependence of decreasing attenuation for all frequencies above cutoff for the TE 1 mode, in contrast to that of the TM 1 mode, is clearly evident.

Fig. 10
Fig. 10

(a) Transverse cross section of the PPWG showing the concave plate geometry. (b) Longitudinal cross section of the PPWG, indicating the path of the associated plane wave for the case of three bounces within the length L. (c) The equivalent unfolded lateral beam profile of the bouncing wave, where the thin lenses simulate the focusing effect caused by the curvature.

Fig. 11
Fig. 11

Lateral beam size at the output of the PPWG as a function of frequency for varying radii of curvature of the plates.

Equations (11)

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

α = 4 R s ( f c / f ) 2 Z o b 1 ( f c / f ) 2 .
E x = A n β y ε o sin ( β y y ) e j β z z ,
H y = A n β y β z ω μ o ε o sin ( β y y ) e j β z z ,
H z = j A n β y 2 ω μ o ε o cos ( β y y ) e j β z z ,
E x , H y , H z ( e j β y y e + j β y y ) e j β z z .
θ = cos 1 ( β y β o ) = cos 1 ( n λ o 2 b ) .
N = 1 b cot θ ,
r s = Z 2 cos θ i Z 1 cos θ t Z 2 cos θ i + Z 1 cos θ t ,
Z 1 Z 2 σ j ω ε o ,
α = N ( 1 r s 2 ) .
[ A B C D ] = [ 1 d 0 1 ] [ [ 1 0 2 R 1 ] [ 1 2 d 0 1 ] ] ( N 1 ) [ 1 0 2 R 1 ] [ 1 d 0 1 ] ,

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