Abstract

In this paper, a novel broadband 3dB directional coupler with very flat coupling based on bridged parallel plate dielectric waveguide (PPDW) is proposed and demonstrated. In the uniform coupling section, a bridge structure between the two PPDWs is employed to obtain accurate coupling value and achieve a broadband coupling. It is found that this new type of coupling structure exhibits excellent performance at terahertz frequencies. In order to achieve strong isolation between the adjacent ports and reduce the power reflection in all ports, two quarter-circle bend arms are introduced as the curved transition sections to connect the uniform coupling section. For this bridged coupler, it only needs the value of the uniform coupling length as short as 400μm to achieve a broadband 3dB coupling. In this case, the coupler’s average return loss is greater than 28dB, average isolation is better than 27dB and average coupler loss is only 0.9dB, over a percentage bandwidth of 12.5% at 1THz. Compared to the conventional PPDW coupler, the bridged PPDW coupler shows significantly greater bandwidth (about 4.2 times), compact and mechanically stable with a much shorter uniform coupling length (reduced about 61%), which may have potential applications for terahertz integrated circuits and systems.

© 2011 OSA

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References

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  1. P. H. Siegel, “Terahertz technology,” IEEE Trans. Microw. Theory Tech. 50(3), 910–928 (2002).
    [CrossRef]
  2. B. Clough, J. Liu, and X.-C. Zhang, ““All air-plasma” terahertz spectroscopy,” Opt. Lett. 36(13), 2399–2401 (2011).
    [CrossRef] [PubMed]
  3. M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, K. J. Ahn, Q. H. Park, P. C. M. Planken, and D. S. Kim, “Near field imaging of terahertz focusing onto rectangular apertures,” Opt. Express 16(25), 20484–20489 (2008).
    [CrossRef] [PubMed]
  4. V. P. Wallace, E. MacPherson, J. A. Zeitler, and C. Reid, “Three-dimensional imaging of optically opaque materials using nonionizing terahertz radiation,” J. Opt. Soc. Am. A 25, 3120–3133 (2008).
    [CrossRef]
  5. R. Degl’Innocenti, M. Montinaro, J. Xu, V. Piazza, P. Pingue, A. Tredicucci, F. Beltram, H. E. Beere, and D. A. Ritchie, “Differential near-field scanning optical microscopy with THz quantum cascade laser sources,” Opt. Express 17(26), 23785–23792 (2009).
    [CrossRef] [PubMed]
  6. C. Y. Jiang, J. S. Liu, B. Sun, K. J. Wang, S. X. Li, and J. Q. Yao, “Time-dependent theoretical model for terahertz wave detector using a parametric process,” Opt. Express 18(17), 18180–18189 (2010).
    [CrossRef] [PubMed]
  7. K. Nielsen, H. K. Rasmussen, A. J. L. Adam, P. C. M. Planken, O. Bang, and P. U. Jepsen, “Bendable, low-loss Topas fibers for the terahertz frequency range,” Opt. Express 17(10), 8592–8601 (2009).
    [CrossRef] [PubMed]
  8. K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 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(16), 161904 (2005).
    [CrossRef]
  10. R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26(11), 846–848 (2001).
    [CrossRef] [PubMed]
  11. R. Mendis, “Guided-wave THz time-domain spectroscopy of highly doped silicon using parallel-plate waveguides,” Electron. Lett. 42(1), 19–21 (2006).
    [CrossRef]
  12. M. Cho, J. Kim, H. Park, Y. Han, K. Moon, E. Jung, and H. Han, “Highly birefringent terahertz polarization maintaining plastic photonic crystal fibers,” Opt. Express 16(1), 7–12 (2008).
    [CrossRef] [PubMed]
  13. M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon Photonic Crystal Fiber as Terahertz Waveguide,” Jpn. J. Appl. Phys. 43(No. 2B), L317–L319 (2004).
    [CrossRef]
  14. O. Mitrofanov and J. A. Harrington, “Dielectric-lined cylindrical metallic THz waveguides: mode structure and dispersion,” Opt. Express 18(3), 1898–1903 (2010).
    [CrossRef] [PubMed]
  15. B. Bowden, J. A. Harrington, and O. Mitrofanov, “Fabrication of terahertz hollow-glass metallic waveguides with inner dielectric coatings,” J. Appl. Phys. 104(9), 093110 (2008).
    [CrossRef]
  16. D. Chen and H. Chen, “A novel low-loss Terahertz waveguide: polymer tube,” Opt. Express 18(4), 3762–3767 (2010).
    [CrossRef] [PubMed]
  17. D. Chen, “Mode Property of Terahertz Polymer Tube,” J. Lightwave Technol. 28(18), 2708–2713 (2010).
    [CrossRef]
  18. A. Dupuis, A. Mazhorova, F. Désévédavy, M. Rozé, and M. Skorobogatiy, “Spectral characterization of porous dielectric subwavelength THz fibers fabricated using a microstructured molding technique,” Opt. Express 18(13), 13813–13828 (2010).
    [CrossRef] [PubMed]
  19. A. Hassani, A. Dupuis, and M. Skorobogatiy, “Porous polymer fibers for low-loss Terahertz guiding,” Opt. Express 16(9), 6340–6351 (2008).
    [CrossRef] [PubMed]
  20. A. Dupuis, K. Stoeffler, B. Ung, C. Dubois, and M. Skorobogatiy, “Transmission measurements of hollow-core THz Bragg fibers,” J. Opt. Soc. Am. B 28(4), 896–907 (2011).
    [CrossRef]
  21. L. Ye, R. Xu, Z. Wang, and W. Lin, “A novel broadband coaxial probe to parallel plate dielectric waveguide transition at THz frequency,” Opt. Express 18(21), 21725–21731 (2010).
    [CrossRef] [PubMed]
  22. T. Yoneyama and S. Nishida, “Nonradiative Dielectric Waveguide for Millimeter-Wave Intergrated Circuits,” IEEE Trans. Microwave Theory Tech., vol. MTT-29, no.11, pp. 1188–1192, Nov. (1981).
  23. T. Yoneyama, N. Tozawa, and S. Nishida, “Coupling Characteristics of Nonradiative Dielectric Waveguide,” IEEE Trans. Microwave Theory Tech., vol. MTT-31, no. 8, pp. 648–654, Aug. (1983).
  24. M. Pu, N. Yao, C. Hu, X. Xin, Z. Zhao, C. Wang, and X. Luo, “Directional coupler and nonlinear Mach-Zehnder interferometer based on metal-insulator-metal plasmonic waveguide,” Opt. Express 18(20), 21030–21037 (2010).
    [CrossRef] [PubMed]
  25. G. K. C. Kwan and N. K. Das, “Excitation of a parallel-plate dielectric waveguide using a coaxial probe-basic characteristics and experiments,” IEEE Trans. Microw. Theory Tech. 50(6), 1609–1620 (2002).
    [CrossRef]
  26. K. Nielsen, H. K. Rasmussen, P. U. Jepsen, and O. Bang, “Broadband terahertz fiber directional coupler,” Opt. Lett. 35(17), 2879–2881 (2010).
    [CrossRef] [PubMed]
  27. K. Solbach and L. Wolff, “The electromagnetic fields and the phase constants of dielectric image lines,” IEEE Trans. Microwave Theory Tech., vol. MTT-26, pp. 266–274, Apr. (1978).
  28. K. Solbach, “The Calculation and the Measurement of the Coupling Properties of Dielectric Image Lines of Rectangular Cross Sections,” IEEE Trans. MTT, vol.27, pp.54–58, Jan. (1979).

2011

2010

O. Mitrofanov and J. A. Harrington, “Dielectric-lined cylindrical metallic THz waveguides: mode structure and dispersion,” Opt. Express 18(3), 1898–1903 (2010).
[CrossRef] [PubMed]

D. Chen and H. Chen, “A novel low-loss Terahertz waveguide: polymer tube,” Opt. Express 18(4), 3762–3767 (2010).
[CrossRef] [PubMed]

A. Dupuis, A. Mazhorova, F. Désévédavy, M. Rozé, and M. Skorobogatiy, “Spectral characterization of porous dielectric subwavelength THz fibers fabricated using a microstructured molding technique,” Opt. Express 18(13), 13813–13828 (2010).
[CrossRef] [PubMed]

C. Y. Jiang, J. S. Liu, B. Sun, K. J. Wang, S. X. Li, and J. Q. Yao, “Time-dependent theoretical model for terahertz wave detector using a parametric process,” Opt. Express 18(17), 18180–18189 (2010).
[CrossRef] [PubMed]

K. Nielsen, H. K. Rasmussen, P. U. Jepsen, and O. Bang, “Broadband terahertz fiber directional coupler,” Opt. Lett. 35(17), 2879–2881 (2010).
[CrossRef] [PubMed]

M. Pu, N. Yao, C. Hu, X. Xin, Z. Zhao, C. Wang, and X. Luo, “Directional coupler and nonlinear Mach-Zehnder interferometer based on metal-insulator-metal plasmonic waveguide,” Opt. Express 18(20), 21030–21037 (2010).
[CrossRef] [PubMed]

L. Ye, R. Xu, Z. Wang, and W. Lin, “A novel broadband coaxial probe to parallel plate dielectric waveguide transition at THz frequency,” Opt. Express 18(21), 21725–21731 (2010).
[CrossRef] [PubMed]

D. Chen, “Mode Property of Terahertz Polymer Tube,” J. Lightwave Technol. 28(18), 2708–2713 (2010).
[CrossRef]

2009

2008

2006

R. Mendis, “Guided-wave THz time-domain spectroscopy of highly doped silicon using parallel-plate waveguides,” Electron. Lett. 42(1), 19–21 (2006).
[CrossRef]

2005

T.-I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 (2005).
[CrossRef]

2004

M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon Photonic Crystal Fiber as Terahertz Waveguide,” Jpn. J. Appl. Phys. 43(No. 2B), L317–L319 (2004).
[CrossRef]

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

2002

P. H. Siegel, “Terahertz technology,” IEEE Trans. Microw. Theory Tech. 50(3), 910–928 (2002).
[CrossRef]

G. K. C. Kwan and N. K. Das, “Excitation of a parallel-plate dielectric waveguide using a coaxial probe-basic characteristics and experiments,” IEEE Trans. Microw. Theory Tech. 50(6), 1609–1620 (2002).
[CrossRef]

2001

Adam, A. J. L.

Ahn, K. J.

Bang, O.

Beere, H. E.

Beltram, F.

Bowden, B.

B. Bowden, J. A. Harrington, and O. Mitrofanov, “Fabrication of terahertz hollow-glass metallic waveguides with inner dielectric coatings,” J. Appl. Phys. 104(9), 093110 (2008).
[CrossRef]

Chen, D.

Chen, H.

Cho, M.

Clough, B.

Das, N. K.

G. K. C. Kwan and N. K. Das, “Excitation of a parallel-plate dielectric waveguide using a coaxial probe-basic characteristics and experiments,” IEEE Trans. Microw. Theory Tech. 50(6), 1609–1620 (2002).
[CrossRef]

Degl’Innocenti, R.

Désévédavy, F.

Dubois, C.

Dupuis, A.

Goto, M.

M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon Photonic Crystal Fiber as Terahertz Waveguide,” Jpn. J. Appl. Phys. 43(No. 2B), L317–L319 (2004).
[CrossRef]

Grischkowsky, D.

T.-I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 (2005).
[CrossRef]

R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26(11), 846–848 (2001).
[CrossRef] [PubMed]

Han, H.

Han, Y.

Harrington, J. A.

O. Mitrofanov and J. A. Harrington, “Dielectric-lined cylindrical metallic THz waveguides: mode structure and dispersion,” Opt. Express 18(3), 1898–1903 (2010).
[CrossRef] [PubMed]

B. Bowden, J. A. Harrington, and O. Mitrofanov, “Fabrication of terahertz hollow-glass metallic waveguides with inner dielectric coatings,” J. Appl. Phys. 104(9), 093110 (2008).
[CrossRef]

Hassani, A.

Hu, C.

Jeon, T.-I.

T.-I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 (2005).
[CrossRef]

Jepsen, P. U.

Jiang, C. Y.

Jung, E.

Kang, J. H.

Kim, D. S.

Kim, J.

Kwan, G. K. C.

G. K. C. Kwan and N. K. Das, “Excitation of a parallel-plate dielectric waveguide using a coaxial probe-basic characteristics and experiments,” IEEE Trans. Microw. Theory Tech. 50(6), 1609–1620 (2002).
[CrossRef]

Lee, J. W.

Li, S. X.

Lin, W.

Liu, J.

Liu, J. S.

Luo, X.

MacPherson, E.

Mazhorova, A.

Mendis, R.

R. Mendis, “Guided-wave THz time-domain spectroscopy of highly doped silicon using parallel-plate waveguides,” Electron. Lett. 42(1), 19–21 (2006).
[CrossRef]

R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26(11), 846–848 (2001).
[CrossRef] [PubMed]

Mitrofanov, O.

O. Mitrofanov and J. A. Harrington, “Dielectric-lined cylindrical metallic THz waveguides: mode structure and dispersion,” Opt. Express 18(3), 1898–1903 (2010).
[CrossRef] [PubMed]

B. Bowden, J. A. Harrington, and O. Mitrofanov, “Fabrication of terahertz hollow-glass metallic waveguides with inner dielectric coatings,” J. Appl. Phys. 104(9), 093110 (2008).
[CrossRef]

Mittleman, D. M.

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

Montinaro, M.

Moon, K.

Nielsen, K.

Ono, S.

M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon Photonic Crystal Fiber as Terahertz Waveguide,” Jpn. J. Appl. Phys. 43(No. 2B), L317–L319 (2004).
[CrossRef]

Park, H.

Park, Q. H.

Piazza, V.

Pingue, P.

Planken, P. C. M.

Pu, M.

Quema, A.

M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon Photonic Crystal Fiber as Terahertz Waveguide,” Jpn. J. Appl. Phys. 43(No. 2B), L317–L319 (2004).
[CrossRef]

Rasmussen, H. K.

Reid, C.

Ritchie, D. A.

Rozé, M.

Sarukura, N.

M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon Photonic Crystal Fiber as Terahertz Waveguide,” Jpn. J. Appl. Phys. 43(No. 2B), L317–L319 (2004).
[CrossRef]

Seo, M. A.

Siegel, P. H.

P. H. Siegel, “Terahertz technology,” IEEE Trans. Microw. Theory Tech. 50(3), 910–928 (2002).
[CrossRef]

Skorobogatiy, M.

Stoeffler, K.

Sun, B.

Takahashi, H.

M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon Photonic Crystal Fiber as Terahertz Waveguide,” Jpn. J. Appl. Phys. 43(No. 2B), L317–L319 (2004).
[CrossRef]

Tredicucci, A.

Ung, B.

Wallace, V. P.

Wang, C.

Wang, K.

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

Wang, K. J.

Wang, Z.

Xin, X.

Xu, J.

Xu, R.

Yao, J. Q.

Yao, N.

Ye, L.

Zeitler, J. A.

Zhang, J.

T.-I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 (2005).
[CrossRef]

Zhang, X.-C.

Zhao, Z.

Appl. Phys. Lett.

T.-I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 (2005).
[CrossRef]

Electron. Lett.

R. Mendis, “Guided-wave THz time-domain spectroscopy of highly doped silicon using parallel-plate waveguides,” Electron. Lett. 42(1), 19–21 (2006).
[CrossRef]

IEEE Trans. Microw. Theory Tech.

P. H. Siegel, “Terahertz technology,” IEEE Trans. Microw. Theory Tech. 50(3), 910–928 (2002).
[CrossRef]

G. K. C. Kwan and N. K. Das, “Excitation of a parallel-plate dielectric waveguide using a coaxial probe-basic characteristics and experiments,” IEEE Trans. Microw. Theory Tech. 50(6), 1609–1620 (2002).
[CrossRef]

J. Appl. Phys.

B. Bowden, J. A. Harrington, and O. Mitrofanov, “Fabrication of terahertz hollow-glass metallic waveguides with inner dielectric coatings,” J. Appl. Phys. 104(9), 093110 (2008).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys.

M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon Photonic Crystal Fiber as Terahertz Waveguide,” Jpn. J. Appl. Phys. 43(No. 2B), L317–L319 (2004).
[CrossRef]

Nature

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

Opt. Express

K. Nielsen, H. K. Rasmussen, A. J. L. Adam, P. C. M. Planken, O. Bang, and P. U. Jepsen, “Bendable, low-loss Topas fibers for the terahertz frequency range,” Opt. Express 17(10), 8592–8601 (2009).
[CrossRef] [PubMed]

R. Degl’Innocenti, M. Montinaro, J. Xu, V. Piazza, P. Pingue, A. Tredicucci, F. Beltram, H. E. Beere, and D. A. Ritchie, “Differential near-field scanning optical microscopy with THz quantum cascade laser sources,” Opt. Express 17(26), 23785–23792 (2009).
[CrossRef] [PubMed]

O. Mitrofanov and J. A. Harrington, “Dielectric-lined cylindrical metallic THz waveguides: mode structure and dispersion,” Opt. Express 18(3), 1898–1903 (2010).
[CrossRef] [PubMed]

D. Chen and H. Chen, “A novel low-loss Terahertz waveguide: polymer tube,” Opt. Express 18(4), 3762–3767 (2010).
[CrossRef] [PubMed]

A. Dupuis, A. Mazhorova, F. Désévédavy, M. Rozé, and M. Skorobogatiy, “Spectral characterization of porous dielectric subwavelength THz fibers fabricated using a microstructured molding technique,” Opt. Express 18(13), 13813–13828 (2010).
[CrossRef] [PubMed]

C. Y. Jiang, J. S. Liu, B. Sun, K. J. Wang, S. X. Li, and J. Q. Yao, “Time-dependent theoretical model for terahertz wave detector using a parametric process,” Opt. Express 18(17), 18180–18189 (2010).
[CrossRef] [PubMed]

M. Cho, J. Kim, H. Park, Y. Han, K. Moon, E. Jung, and H. Han, “Highly birefringent terahertz polarization maintaining plastic photonic crystal fibers,” Opt. Express 16(1), 7–12 (2008).
[CrossRef] [PubMed]

A. Hassani, A. Dupuis, and M. Skorobogatiy, “Porous polymer fibers for low-loss Terahertz guiding,” Opt. Express 16(9), 6340–6351 (2008).
[CrossRef] [PubMed]

M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, K. J. Ahn, Q. H. Park, P. C. M. Planken, and D. S. Kim, “Near field imaging of terahertz focusing onto rectangular apertures,” Opt. Express 16(25), 20484–20489 (2008).
[CrossRef] [PubMed]

M. Pu, N. Yao, C. Hu, X. Xin, Z. Zhao, C. Wang, and X. Luo, “Directional coupler and nonlinear Mach-Zehnder interferometer based on metal-insulator-metal plasmonic waveguide,” Opt. Express 18(20), 21030–21037 (2010).
[CrossRef] [PubMed]

L. Ye, R. Xu, Z. Wang, and W. Lin, “A novel broadband coaxial probe to parallel plate dielectric waveguide transition at THz frequency,” Opt. Express 18(21), 21725–21731 (2010).
[CrossRef] [PubMed]

Opt. Lett.

Other

T. Yoneyama and S. Nishida, “Nonradiative Dielectric Waveguide for Millimeter-Wave Intergrated Circuits,” IEEE Trans. Microwave Theory Tech., vol. MTT-29, no.11, pp. 1188–1192, Nov. (1981).

T. Yoneyama, N. Tozawa, and S. Nishida, “Coupling Characteristics of Nonradiative Dielectric Waveguide,” IEEE Trans. Microwave Theory Tech., vol. MTT-31, no. 8, pp. 648–654, Aug. (1983).

K. Solbach and L. Wolff, “The electromagnetic fields and the phase constants of dielectric image lines,” IEEE Trans. Microwave Theory Tech., vol. MTT-26, pp. 266–274, Apr. (1978).

K. Solbach, “The Calculation and the Measurement of the Coupling Properties of Dielectric Image Lines of Rectangular Cross Sections,” IEEE Trans. MTT, vol.27, pp.54–58, Jan. (1979).

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Figures (6)

Fig. 1
Fig. 1

(a) Cross-sectional view of the PPDW; (b) 3D view of the PPDW

Fig. 2
Fig. 2

(a) Normalized electric field distribution for the TE10 mode of PPDW at 1THz; (b) Reflection and transmission parameters of PPDW in 0.9~1.2THz

Fig. 3
Fig. 3

(a) Cross-sectional view of the uniform conventional PPDW coupling section; (b) Cross-sectional view of the uniform bridged PPDW coupling section

Fig. 4
Fig. 4

(a) Scattering parameters versus coupling length of the conventional PPDW 3dB coupler at 1THz (a = 100μm, b = 100μm, d = 50μm, ε r = 11.9, ε s = 1, σ = 6.1 × 107 S/m); (b) Scattering parameters versus coupling length of the bridged PPDW coupler at 1THz (a = 100μm, b = 100μm, c = 50μm, d = 50μm, ε r = 11.9, ε s = 1, σ = 6.1 × 107 S/m)

Fig. 5
Fig. 5

(a) The bridged PPDW 3dB directional coupler structures used here with two quarter-circle bend connecting arms; (b) Normalized magnitude of Poynting power vector distribution for this bridged PPDW coupler at 1THz (a = 80μm, b = 40μm, c = 50μm, d = 50μm, r = 50μm, l = 400μm,ε r = 11.9, ε s = 1, σ = 6.1 × 107 S/m)

Fig. 6
Fig. 6

(a) Frequency characteristics for the conventional PPDW coupler with two quarter-circle bend connecting arms with the uniform coupling length l = 1025μm; (b) Frequency characteristics for the bridged PPDW coupler with two quarter-circle bend connecting arms with the uniform coupling length l = 400μm

Equations (6)

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

E y ( x ) = { A e γ s ( x + b / 2 )                                                             < x b / 2 A [ cos [ γ r ( x + b / 2 ) ] + γ s γ r sin [ γ r ( x + b / 2 ) ] ]     b / 2 x b / 2 A [ cos ( γ r b ) + γ s γ r sin ( γ r b ) ] e γ s ( x b / 2 )                      b / 2 x <
γ r = k 0 2 ε r β 2 , γ s = β 2 k 0 2 ε s and k 0 = ω ε 0 μ 0
L = π β e β o
L 3 dB = π 2 ( β e β o ) = L 2
| S 21 | = | cos β e β o 2 l |
| S 41 | = | sin β e β o 2 l |

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