Abstract

We present a comprehensive experimental study comparing the propagation characteristics of the virtually unknown TE1 mode to the well-known TEM mode of the parallel-plate waveguide (PPWG), for THz pulse applications. We demonstrate that it is possible to overcome the undesirable effects caused by the TE1 mode’s inherent low-frequency cutoff, making it a viable THz wave-guiding option, and that for certain applications, the TE1 mode may even be more desirable than the TEM mode. This study presents a whole new dimension to the THz technological capabilities offered by the PPWG, via the possible use of the TE1 mode.

© 2009 OSA

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References

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  1. R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26(11), 846–848 (2001).
    [CrossRef]
  2. R. Mendis and D. Grischkowsky, “THz interconnect with low loss and low group velocity dispersion,” IEEE Microw. Wirel. Compon. Lett. 11(11), 444–446 (2001).
    [CrossRef]
  3. H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. 29(15), 1751–1753 (2004).
    [CrossRef] [PubMed]
  4. S. Coleman and D. Grischkowsky, “Parallel plate THz transmitter,” Appl. Phys. Lett. 84(5), 654–656 (2004).
    [CrossRef]
  5. R. Mendis, “Guided-wave THz time-domain spectroscopy of highly doped silicon using parallel-plate waveguides,” Electron. Lett. 42(1), 19–21 (2006).
    [CrossRef]
  6. J. S. Melinger, N. Laman, S. S. Harsha, and D. Grischkowsky, “Line narrowing of terahertz vibrational modes for organic thin polycrystalline films within a parallel plate waveguide,” Appl. Phys. Lett. 89(25), 252221 (2006).
    [CrossRef]
  7. J. S. Melinger, N. Laman, S. S. Harsha, S. Cheng, and D. Grischkowsky, “High-resolution waveguide terahertz spectroscopy of partially oriented organic polycrystalline films,” J. Phys. Chem. A 111(43), 10977–10987 (2007).
    [CrossRef] [PubMed]
  8. N. Laman, S. S. Harsha, D. Grischkowsky, and J. S. Melinger, “High-resolution waveguide THz spectroscopy of biological molecules,” Biophys. J. 94(3), 1010–1020 (2008).
    [CrossRef]
  9. J. Zhang and D. Grischkowsky, “Waveguide THz time-domain spectroscopy of nm water layers,” Opt. Lett. 29(14), 1617–1619 (2004).
    [CrossRef] [PubMed]
  10. M. Nagel, M. Forst, and H. Kurz, “THz biosensing devices: fundamentals and technology,” J. Phys. Condens. Matter 18(18), S601–S618 (2006).
    [CrossRef]
  11. M. M. Awad and R. A. Cheville, “Transmission terahertz waveguide-based imaging below the diffraction limit,” Appl. Phys. Lett. 86(22), 221107 (2005).
    [CrossRef]
  12. D. G. Cooke and P. U. Jepsen, “Optical modulation of terahertz pulses in a parallel plate waveguide,” Opt. Express 16(19), 15123–15129 (2008).
    [CrossRef] [PubMed]
  13. Z. Jian, J. Pearce, and D. M. Mittleman, “Defect modes in photonic crystal slabs studied using terahertz time-domain spectroscopy,” Opt. Lett. 29(17), 2067–2069 (2004).
    [CrossRef] [PubMed]
  14. Y. Zhao and D. Grischkowsky, “2-D terahertz metallic photonic crystals in parallel-plate waveguides,” IEEE Trans. Microw. Theory Tech. 55(4), 656–663 (2007).
    [CrossRef]
  15. A. L. Bingham and D. Grischkowsky, “High Q, one-dimensional terahertz photonic waveguides,” Appl. Phys. Lett. 90(9), 091105 (2007).
    [CrossRef]
  16. T. Prasad, V. L. Colvin, Z. Jian, and D. M. Mittleman, “Superprism effect in a metal-clad terahertz photonic crystal slab,” Opt. Lett. 32(6), 683–685 (2007).
    [CrossRef] [PubMed]
  17. S. S. Harsha, N. Laman, and D. Grischkowsky, “High-Q terahertz Bragg resonances within a metal parallel plate waveguide,” Appl. Phys. Lett. 94(9), 091118 (2009).
    [CrossRef]
  18. N. Marcuvitz, Waveguide Handbook (Peregrinus, 1993).
  19. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).
  20. R. Mendis and D. M. Mittleman, “An investigation of the lowest-order transverse-electric (TE1) mode of the parallel-plate waveguide for THz pulse propagation,” J. Opt. Soc. Am. B 26(9), A6–A13 (2009).
    [CrossRef]
  21. R. Mendis, “THz transmission characteristics of dielectric-filled parallel-plate waveguides,” J. Appl. Phys. 101(8), 083115 (2007).
    [CrossRef]
  22. R. Mendis and D. M. Mittleman, “Whispering-gallery-mode THz-pulse propagation on a single curved metallic plate,” in Conference on Lasers and Electro-Optics 2009, paper CThQ1.
  23. R. Mendis and D. M. Mittleman, “A beam-scanning THz prism with effective refractive index less than unity,” presented at the International Workshop on Optical Terahertz Science and Technology, California, USA, 2009.

2009 (2)

S. S. Harsha, N. Laman, and D. Grischkowsky, “High-Q terahertz Bragg resonances within a metal parallel plate waveguide,” Appl. Phys. Lett. 94(9), 091118 (2009).
[CrossRef]

R. Mendis and D. M. Mittleman, “An investigation of the lowest-order transverse-electric (TE1) mode of the parallel-plate waveguide for THz pulse propagation,” J. Opt. Soc. Am. B 26(9), A6–A13 (2009).
[CrossRef]

2008 (2)

D. G. Cooke and P. U. Jepsen, “Optical modulation of terahertz pulses in a parallel plate waveguide,” Opt. Express 16(19), 15123–15129 (2008).
[CrossRef] [PubMed]

N. Laman, S. S. Harsha, D. Grischkowsky, and J. S. Melinger, “High-resolution waveguide THz spectroscopy of biological molecules,” Biophys. J. 94(3), 1010–1020 (2008).
[CrossRef]

2007 (5)

J. S. Melinger, N. Laman, S. S. Harsha, S. Cheng, and D. Grischkowsky, “High-resolution waveguide terahertz spectroscopy of partially oriented organic polycrystalline films,” J. Phys. Chem. A 111(43), 10977–10987 (2007).
[CrossRef] [PubMed]

T. Prasad, V. L. Colvin, Z. Jian, and D. M. Mittleman, “Superprism effect in a metal-clad terahertz photonic crystal slab,” Opt. Lett. 32(6), 683–685 (2007).
[CrossRef] [PubMed]

R. Mendis, “THz transmission characteristics of dielectric-filled parallel-plate waveguides,” J. Appl. Phys. 101(8), 083115 (2007).
[CrossRef]

Y. Zhao and D. Grischkowsky, “2-D terahertz metallic photonic crystals in parallel-plate waveguides,” IEEE Trans. Microw. Theory Tech. 55(4), 656–663 (2007).
[CrossRef]

A. L. Bingham and D. Grischkowsky, “High Q, one-dimensional terahertz photonic waveguides,” Appl. Phys. Lett. 90(9), 091105 (2007).
[CrossRef]

2006 (3)

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

J. S. Melinger, N. Laman, S. S. Harsha, and D. Grischkowsky, “Line narrowing of terahertz vibrational modes for organic thin polycrystalline films within a parallel plate waveguide,” Appl. Phys. Lett. 89(25), 252221 (2006).
[CrossRef]

M. Nagel, M. Forst, and H. Kurz, “THz biosensing devices: fundamentals and technology,” J. Phys. Condens. Matter 18(18), S601–S618 (2006).
[CrossRef]

2005 (1)

M. M. Awad and R. A. Cheville, “Transmission terahertz waveguide-based imaging below the diffraction limit,” Appl. Phys. Lett. 86(22), 221107 (2005).
[CrossRef]

2004 (4)

2001 (2)

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

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

Awad, M. M.

M. M. Awad and R. A. Cheville, “Transmission terahertz waveguide-based imaging below the diffraction limit,” Appl. Phys. Lett. 86(22), 221107 (2005).
[CrossRef]

Bingham, A. L.

A. L. Bingham and D. Grischkowsky, “High Q, one-dimensional terahertz photonic waveguides,” Appl. Phys. Lett. 90(9), 091105 (2007).
[CrossRef]

Cao, H.

Cheng, S.

J. S. Melinger, N. Laman, S. S. Harsha, S. Cheng, and D. Grischkowsky, “High-resolution waveguide terahertz spectroscopy of partially oriented organic polycrystalline films,” J. Phys. Chem. A 111(43), 10977–10987 (2007).
[CrossRef] [PubMed]

Cheville, R. A.

M. M. Awad and R. A. Cheville, “Transmission terahertz waveguide-based imaging below the diffraction limit,” Appl. Phys. Lett. 86(22), 221107 (2005).
[CrossRef]

Coleman, S.

S. Coleman and D. Grischkowsky, “Parallel plate THz transmitter,” Appl. Phys. Lett. 84(5), 654–656 (2004).
[CrossRef]

Colvin, V. L.

Cooke, D. G.

Forst, M.

M. Nagel, M. Forst, and H. Kurz, “THz biosensing devices: fundamentals and technology,” J. Phys. Condens. Matter 18(18), S601–S618 (2006).
[CrossRef]

Grischkowsky, D.

S. S. Harsha, N. Laman, and D. Grischkowsky, “High-Q terahertz Bragg resonances within a metal parallel plate waveguide,” Appl. Phys. Lett. 94(9), 091118 (2009).
[CrossRef]

N. Laman, S. S. Harsha, D. Grischkowsky, and J. S. Melinger, “High-resolution waveguide THz spectroscopy of biological molecules,” Biophys. J. 94(3), 1010–1020 (2008).
[CrossRef]

J. S. Melinger, N. Laman, S. S. Harsha, S. Cheng, and D. Grischkowsky, “High-resolution waveguide terahertz spectroscopy of partially oriented organic polycrystalline films,” J. Phys. Chem. A 111(43), 10977–10987 (2007).
[CrossRef] [PubMed]

Y. Zhao and D. Grischkowsky, “2-D terahertz metallic photonic crystals in parallel-plate waveguides,” IEEE Trans. Microw. Theory Tech. 55(4), 656–663 (2007).
[CrossRef]

A. L. Bingham and D. Grischkowsky, “High Q, one-dimensional terahertz photonic waveguides,” Appl. Phys. Lett. 90(9), 091105 (2007).
[CrossRef]

J. S. Melinger, N. Laman, S. S. Harsha, and D. Grischkowsky, “Line narrowing of terahertz vibrational modes for organic thin polycrystalline films within a parallel plate waveguide,” Appl. Phys. Lett. 89(25), 252221 (2006).
[CrossRef]

S. Coleman and D. Grischkowsky, “Parallel plate THz transmitter,” Appl. Phys. Lett. 84(5), 654–656 (2004).
[CrossRef]

J. Zhang and D. Grischkowsky, “Waveguide THz time-domain spectroscopy of nm water layers,” Opt. Lett. 29(14), 1617–1619 (2004).
[CrossRef] [PubMed]

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

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

Harsha, S. S.

S. S. Harsha, N. Laman, and D. Grischkowsky, “High-Q terahertz Bragg resonances within a metal parallel plate waveguide,” Appl. Phys. Lett. 94(9), 091118 (2009).
[CrossRef]

N. Laman, S. S. Harsha, D. Grischkowsky, and J. S. Melinger, “High-resolution waveguide THz spectroscopy of biological molecules,” Biophys. J. 94(3), 1010–1020 (2008).
[CrossRef]

J. S. Melinger, N. Laman, S. S. Harsha, S. Cheng, and D. Grischkowsky, “High-resolution waveguide terahertz spectroscopy of partially oriented organic polycrystalline films,” J. Phys. Chem. A 111(43), 10977–10987 (2007).
[CrossRef] [PubMed]

J. S. Melinger, N. Laman, S. S. Harsha, and D. Grischkowsky, “Line narrowing of terahertz vibrational modes for organic thin polycrystalline films within a parallel plate waveguide,” Appl. Phys. Lett. 89(25), 252221 (2006).
[CrossRef]

Jepsen, P. U.

Jian, Z.

Kurz, H.

M. Nagel, M. Forst, and H. Kurz, “THz biosensing devices: fundamentals and technology,” J. Phys. Condens. Matter 18(18), S601–S618 (2006).
[CrossRef]

Laman, N.

S. S. Harsha, N. Laman, and D. Grischkowsky, “High-Q terahertz Bragg resonances within a metal parallel plate waveguide,” Appl. Phys. Lett. 94(9), 091118 (2009).
[CrossRef]

N. Laman, S. S. Harsha, D. Grischkowsky, and J. S. Melinger, “High-resolution waveguide THz spectroscopy of biological molecules,” Biophys. J. 94(3), 1010–1020 (2008).
[CrossRef]

J. S. Melinger, N. Laman, S. S. Harsha, S. Cheng, and D. Grischkowsky, “High-resolution waveguide terahertz spectroscopy of partially oriented organic polycrystalline films,” J. Phys. Chem. A 111(43), 10977–10987 (2007).
[CrossRef] [PubMed]

J. S. Melinger, N. Laman, S. S. Harsha, and D. Grischkowsky, “Line narrowing of terahertz vibrational modes for organic thin polycrystalline films within a parallel plate waveguide,” Appl. Phys. Lett. 89(25), 252221 (2006).
[CrossRef]

Linke, R. A.

Melinger, J. S.

N. Laman, S. S. Harsha, D. Grischkowsky, and J. S. Melinger, “High-resolution waveguide THz spectroscopy of biological molecules,” Biophys. J. 94(3), 1010–1020 (2008).
[CrossRef]

J. S. Melinger, N. Laman, S. S. Harsha, S. Cheng, and D. Grischkowsky, “High-resolution waveguide terahertz spectroscopy of partially oriented organic polycrystalline films,” J. Phys. Chem. A 111(43), 10977–10987 (2007).
[CrossRef] [PubMed]

J. S. Melinger, N. Laman, S. S. Harsha, and D. Grischkowsky, “Line narrowing of terahertz vibrational modes for organic thin polycrystalline films within a parallel plate waveguide,” Appl. Phys. Lett. 89(25), 252221 (2006).
[CrossRef]

Mendis, R.

R. Mendis and D. M. Mittleman, “An investigation of the lowest-order transverse-electric (TE1) mode of the parallel-plate waveguide for THz pulse propagation,” J. Opt. Soc. Am. B 26(9), A6–A13 (2009).
[CrossRef]

R. Mendis, “THz transmission characteristics of dielectric-filled parallel-plate waveguides,” J. Appl. Phys. 101(8), 083115 (2007).
[CrossRef]

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

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

Mittleman, D. M.

Nagel, M.

M. Nagel, M. Forst, and H. Kurz, “THz biosensing devices: fundamentals and technology,” J. Phys. Condens. Matter 18(18), S601–S618 (2006).
[CrossRef]

Nahata, A.

Pearce, J.

Prasad, T.

Zhang, J.

Zhao, Y.

Y. Zhao and D. Grischkowsky, “2-D terahertz metallic photonic crystals in parallel-plate waveguides,” IEEE Trans. Microw. Theory Tech. 55(4), 656–663 (2007).
[CrossRef]

Appl. Phys. Lett. (5)

S. Coleman and D. Grischkowsky, “Parallel plate THz transmitter,” Appl. Phys. Lett. 84(5), 654–656 (2004).
[CrossRef]

J. S. Melinger, N. Laman, S. S. Harsha, and D. Grischkowsky, “Line narrowing of terahertz vibrational modes for organic thin polycrystalline films within a parallel plate waveguide,” Appl. Phys. Lett. 89(25), 252221 (2006).
[CrossRef]

M. M. Awad and R. A. Cheville, “Transmission terahertz waveguide-based imaging below the diffraction limit,” Appl. Phys. Lett. 86(22), 221107 (2005).
[CrossRef]

A. L. Bingham and D. Grischkowsky, “High Q, one-dimensional terahertz photonic waveguides,” Appl. Phys. Lett. 90(9), 091105 (2007).
[CrossRef]

S. S. Harsha, N. Laman, and D. Grischkowsky, “High-Q terahertz Bragg resonances within a metal parallel plate waveguide,” Appl. Phys. Lett. 94(9), 091118 (2009).
[CrossRef]

Biophys. J. (1)

N. Laman, S. S. Harsha, D. Grischkowsky, and J. S. Melinger, “High-resolution waveguide THz spectroscopy of biological molecules,” Biophys. J. 94(3), 1010–1020 (2008).
[CrossRef]

Electron. Lett. (1)

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 Microw. Wirel. Compon. Lett. (1)

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

IEEE Trans. Microw. Theory Tech. (1)

Y. Zhao and D. Grischkowsky, “2-D terahertz metallic photonic crystals in parallel-plate waveguides,” IEEE Trans. Microw. Theory Tech. 55(4), 656–663 (2007).
[CrossRef]

J. Appl. Phys. (1)

R. Mendis, “THz transmission characteristics of dielectric-filled parallel-plate waveguides,” J. Appl. Phys. 101(8), 083115 (2007).
[CrossRef]

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

J. Phys. Chem. A (1)

J. S. Melinger, N. Laman, S. S. Harsha, S. Cheng, and D. Grischkowsky, “High-resolution waveguide terahertz spectroscopy of partially oriented organic polycrystalline films,” J. Phys. Chem. A 111(43), 10977–10987 (2007).
[CrossRef] [PubMed]

J. Phys. Condens. Matter (1)

M. Nagel, M. Forst, and H. Kurz, “THz biosensing devices: fundamentals and technology,” J. Phys. Condens. Matter 18(18), S601–S618 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Other (4)

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

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

R. Mendis and D. M. Mittleman, “Whispering-gallery-mode THz-pulse propagation on a single curved metallic plate,” in Conference on Lasers and Electro-Optics 2009, paper CThQ1.

R. Mendis and D. M. Mittleman, “A beam-scanning THz prism with effective refractive index less than unity,” presented at the International Workshop on Optical Terahertz Science and Technology, California, USA, 2009.

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Figures (8)

Fig. 1
Fig. 1

Time scans corresponding to (a) input reference, (b) TEM-mode propagation, (c) TE1-mode propagation in a 2.5 cm long PPWG with b = 0.5 mm, (d) TE1-mode propagation in a 2.5 cm long PPWG with b = 5 mm. The two insets (circled) show the excitation polarization axes with respect to the transverse cross-section of the PPWG. Although some of the TE1-mode data has been previously published [20], they are presented again here to emphasize the comparison with the TEM results.

Fig. 2
Fig. 2

(a) Amplitude spectra corresponding to the scans in Figs. 1(b) and 1(c), obtained by Fourier-transforming the truncated time-domain waveforms. The latter is given by the dots. (b) Phase and group velocity for the TE1 mode. The thick and thin red curves give the theoretical values for b = 0.5 mm and 5 mm, respectively. The dots and open circles are experimental. (c) Close-up of the phase velocity for b = 5 mm. The red curve is theoretical and the dots are experimental.

Fig. 3
Fig. 3

(a) Comparison of the experimental and theoretical attenuation constant for b = 0.5 mm. The dots and open circles give the experimental values, while the red and blue theoretical curves correspond to the TE1 and TEM modes, respectively. The inset shows the power coupling efficiency to the TE1 and TEM modes from an input Gaussian beam. (b) Theoretical attenuation constant for b = 5 mm, where the thick and thin lines correspond to air-filled and Si-filled PPWGs, respectively. Same color association as above. (c) Close-up of the baselines of the TE1 curves.

Fig. 4
Fig. 4

Time scans corresponding to TE1-mode propagation in (a) 2.5 cm long PPWG, (b) 25.0 cm long straight PPWG, and (c) 26.6 cm long bent PPWG that is shown in (d), whose dimensions are given in cm. For all three PPWGs, b = 10 mm, and the width in the unconfined direction was chosen to be sufficiently larger (10 cm for the longer ones) to allow diffractive spreading.

Fig. 5
Fig. 5

(a) Amplitude spectra associated with the scans in Figs. 4(a) and 4(b). The latter is given by the dots. (b) Spectrum of the 2.5 cm long PPWG adjusted to account for the diffraction losses. (c) Lateral output beam size for TEM-mode propagation (blue curve), and TE1-mode propagation with b = 0.5 mm (thick red curve) and b = 10 mm (thin red curve, overlapping with the blue curve), in a 25 cm long PPWG with an input beam size of 2 cm. (d) One-dimensional coupling coefficient at the output of 2.5 cm and 25.0 cm long PPWGs, with b = 10 mm, an input beam size of 2 cm, and a collecting aperture size of 6 mm.

Fig. 6
Fig. 6

Time scans corresponding to TE1-mode propagation in (a) 6.4 mm long PPWG with b = 1 mm, (b) same PPWG with an integrated resonant cavity, formed by incorporating a square groove in the top plate. Longitudinal cross-sections (along the direction of propagation) are shown inset. (c) and (d) give the respective amplitude spectra, where the spectrum of the cavity-integrated-PPWG shows a strong and narrow resonance dip (red arrow) in addition to the water-vapor absorption lines (green arrows).

Fig. 7
Fig. 7

(a) Power transmission (dots) in the vicinity of the resonance dip, fit to a Lorentzian line-shape indicating a resonance frequency of 0.280 THz, a linewidth of 5 GHz, and an extinction coefficient of 30 dB. (b) Photograph of the aluminum plate containing the square groove. The blue dashed lines demarcate the lateral extent of the propagating THz beam inside the assembled PPWG.

Fig. 8
Fig. 8

Time scans corresponding to TEM-mode propagation in (a) 6.4 mm long PPWG with b = 1 mm, (b) same PPWG with an integrated resonant cavity, formed by incorporating a square groove in the top plate. Longitudinal cross-sections (along the direction of propagation) are shown inset. (c) and (d) give the respective amplitude spectra. The spectrum of the cavity-integrated-PPWG does not show a strong resonance dip as in the TE1 case. Green arrows indicate water-vapor absorption lines.

Equations (5)

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

Eout=ErefTCy2Cxej(βzβo)LeαL/2,
αTE=4nRs(fc/f)2Zob1(fc/f)2,
αTEM=2nRsZob.
|EoutlEouts|=[CxlCxs]eα(LlLs)/2,
fr=c2(m1d1)2+(m2d2)2+(m3d3)2,

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