## Efficient terahertz slot-line waveguides |

Optics Express, Vol. 19, Issue 26, pp. B47-B55 (2011)

http://dx.doi.org/10.1364/OE.19.000B47

Acrobat PDF (1571 KB)

### Abstract

We present two solutions to the challenge of radiation loss of slot-lines at terahertz frequencies: using a slot-line in a homogeneous medium, and using a slot-line on a layered substrate. A theoretical analysis of the slot-line in a homogeneous medium as a terahertz transmission line is presented. The absorption coefficient is obtained in terms of the waveguide dimensions using the field distribution of the slot-line. Results show that the slot-line in a homogeneous medium and the slot-line on a layered substrate can be effective transmission lines for terahertz waves with 2 cm^{−1} and 3 cm^{−1} absorption due to conductor loss. Full-wave numerical simulations using the Finite Element Method (FEM) are applied to validate the theory.

© 2011 OSA

## 1. Introduction

^{−1}loss at 1 THz [1

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

3. J. Liu, R. Mendis, and D. M. Mittleman, “The transition from a TEM-like mode to a plasmonic mode in parallel-plate waveguides,” Appl. Phys. Lett. **98**(23), 231113 (2011). [CrossRef]

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

5. H. Pahlevaninezhad, T. E. Darcie, and B. Heshmat, “Two-wire waveguide for terahertz,” Opt. Express **18**(7), 7415–7420 (2010). [CrossRef] [PubMed]

6. H. Pahlevaninezhad and T. E. Darcie, “Coupling of terahertz waves to a two-wire waveguide,” Opt. Express **18**(22), 22614–22624 (2010). [CrossRef] [PubMed]

7. 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]

8. D. R. Grischkowsky, M. B. Ketchen, C.-C. Chi, I. N. Duling, N. J. Halas, J.-M. Halbout, and P. G. May, “Capacitance free generation and detection of subpicosecond electrical pulses on coplanar transmission lines,” IEEE J. Quantum Electron. **24**(2), 221–225 (1988). [CrossRef]

9. D. Grischkowsky, I. I. I. Duling III, J. C. Chen, and C. C. Chi, “Electromagnetic shock waves from transmission lines,” Phys. Rev. Lett. **59**(15), 1663–1666 (1987). [CrossRef] [PubMed]

10. D. Grischkowsky, “Optoelectronic characterization of transmission lines and waveguides by terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. **6**(6), 1122–1135 (2000). [CrossRef]

12. C. Fattinger and D. Grischkowsky, “Observation of electromagnetic shock waves from propagating surface-dipole distributions,” Phys. Rev. Lett. **62**(25), 2961–2964 (1989). [CrossRef] [PubMed]

13. D. K. Kleinman and D. H. Auston, “Theory of Electrooptic shock radiation in nonlinear optical media,” IEEE J. Quantum Electron. **20**(8), 964–970 (1984). [CrossRef]

^{−1}and 3 cm

^{−1}absorption, respectively, due to conductor loss.

## 2. Slot-lines in a homogeneous medium

### 2.1 Theoretical Analysis of the slot-line in a homogeneous medium

*V*is the absolute value of the potential on the plates and

_{0}*s*is the separation. The equipotential curves in the

*uv*-plane are simply the lines parallel to the

*v*-axis corresponding to a family of confocal hyperbolas whose foci are located at distance

*s/2*from origin in the

*xy*-plane:

*(1-V/V*with the

_{0})π/2*x*-axis. The electric field can be derived from the gradient of the potential:that, with some algebraic simplifications, yields where,and

*η*is the characteristic impedance of the surrounding medium. Note that the metal plate thicknesses are assumed to be zero. The effect of the finite thicknesses of the plates will be discussed later.

14. Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express **17**(16), 13502–13515 (2009). [CrossRef] [PubMed]

### 2.2 Loss estimation of the slot-line in a homogeneous medium

*δ*, the skin depth, and relate to the tangential magnetic field just outside the conductor surface by [15] where

*ξ*is the normal coordinate inward into the conductor,

*H*is the tangential magnetic field outside the surface,

_{||}*E*and

_{c}*H*are the electric and magnetic field inside the conductor,

_{c}*μ*is the permeability of the conductor, and

_{c}*σ*is the conductivity. The time-averaged power absorbed per unit area due to ohmic losses in the body of the conductor is then

*x*-axis, like the ones shown in Fig. 4 . This hyperbola corresponds to the lines

*u = ± u*in the

_{0}*uv*-plane when

*u*is very close to

_{0}*π/2*and depends on the thickness of the plates.

*C*, the domain of integration, is the hyperbola shown in Fig. 4. The following parameterization of the curve

*C*is used to calculate the integral:that yields,where,

*s << w*). Otherwise the equipotential curves on the

*uv*-plane in Fig. 2(c) cannot be simply the lines parallel to the

*v*-axis. The power flowing on the lossless line, iswhere

*S'*, the surface of integration, is the cross section of the transmission line. Calculation of

*P*includes a complicated surface integral on the cross section of the slot-line. However, the integration can be calculated on the simpler

_{0}*uv*-plane shown in Fig. 2(c) instead that yields

## 3. Slot-lines on a layered substrate

*e*as an eigenfunction [16]. Therefore, the modes must have the formwhere the wave number

^{ikz}*k*is a conserved quantity due to the continuous translational symmetry. In the lower half-space, the periodic substrate also creates a discrete translational symmetry in the

_{z}*y*-direction, resulting in the modes in Bloch state forms:where

*u*is a periodic function of

_{ky}*y*with the same periodicity as the substrate. This periodicity induces a so-called “photonic band gap” in the band structure of the crystal. A mode that has a frequency within the gap cannot propagate through the crystal and have the amplitude that decays exponentially into the crystal since

*k*becomes an imaginary number. The size of the band gap depends on the thickness of the layers, periodicity, dielectric constants of the layers, and frequency. For two materials with refractive indices

_{y}*n*and

_{1}*n*and thicknesses

_{2}*d*and

_{1}*d*=

_{2}*a*–

*d*the gap is maximized when

_{1}*d*=

_{1}n_{1}*d*(

_{2}n_{2}*a*is the periodicity) [17

17. P. Yeh, A. Yariv, and C. Hong, “Electromagnetic propagation in perodic media. I. General theory,” J. Opt. Soc. Am. **67**(4), 423–438 (1977). [CrossRef]

_{m}in this case is [18,19]with the corresponding vacuum wavelength

*λ*that satisfies the quarter-wavelength conditions:

_{m}*λ*/

_{m}*n*= 4

_{1}*d*and

_{1}*λ*/

_{m}*n*= 4

_{2}*d*.

_{2}*x*-direction (Fig. 1c), leaving only the

*z*-direction for waves to propagate. The field distribution of the surface mode supported by the slot-line on a layered substrate is difficult to obtain analytically due to the presence of two metal plates of the slot-line. However, we present the mode profile supported by the waveguide obtained from the numerical simulations using Finite Element Method (FEM) method. Figure 5(b) shows the electric field distribution of the surface mode supported by a slot-line on a quarter-wave stack of Si/SiO

_{2}with 10 μm separation of the metal plates. The field amplitude is exponentially damped on both sides of the substrate as expected from the theory. Similar to the slot-line in a homogeneous medium, the field is highly concentrated at the edges of the plate, but more distributed inside the substrate due to higher average refractive index of the substrate.

## 4. Results and discussion

_{2}layered substrate. The slot-lines are excited by a surface current source at 1THz, emulating a current generated in a terahertz photomixer. The simulations are bounded by a rectangular box with 0.5 mm × 0.5 mm × 2 mm size, and with radiation boundaries assigned to the walls to avoid reflection, as shown in Fig. 6. The absorption coefficient can be calculated from the power measured at two different lengths of the transmission line.

^{−1}loss for a slot-line on a half-space GaAs substrate with 20 μm separation of the plates. However, conductor loss is the only dominant absorption mechanism for the slot-line in GaAs that changes in proportion to the square root of frequency, allowing significantly longer distance wave propagation (Fig. 6b).

*s*, the separation of the plates, obtained both from the theory and the simulations at 1 THz for a slot-line in GaAs. At the lower limit, when the separation goes to zero, the edges of the two plates touch each other and the transmission line cannot support the TEM mode. The conductor loss is a decreasing function of

*s*. There is also a knee in the curve after which the loss changes rather slowly. The results from the simulations, illustrated by green squares in Fig. 7(a), show good agreement with the theoretical expectations (blue solid line). The conductor loss for slot-lines with enough separation (> 5 μm) is in the order of 2 cm

^{−1}, consistent with the conductor loss experimentally measured on a 20-mm-long coplanar transmission line with 15 μm separation of the lines at 1THz in [9

9. D. Grischkowsky, I. I. I. Duling III, J. C. Chen, and C. C. Chi, “Electromagnetic shock waves from transmission lines,” Phys. Rev. Lett. **59**(15), 1663–1666 (1987). [CrossRef] [PubMed]

*α*=

*A*

_{c}f^{1/2}+

*A*

_{rad}f^{3}was presented for the absorption, where

*A*and

_{c}*A*are resistive and radiative loss coefficients, respectively and

_{rad}*f*is the frequency in THz. Curve fitting of experimental data resulted in

*A*= 2 cm

_{c}^{−1},

*A*= 6.5 cm

_{rad}^{−1}.

*u*for a slot-line with

_{0}*s*= 10 μm,

*w*= 500 μm made out of gold in GaAs. At

*u*= π/2 the plate thickness is zero, making infinitely high conductor loss. For the finite thickness, however, the loss is relatively independent of the thickness, for the current density is confined to such a small thickness (

_{0}*δ*= 78.6 nm for gold at 1THz) just below the surface of the conductor.

^{−1}at 1THz [21

21. D. Grischkowsky, S. Keiding, M. Exter, and C. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B **7**(10), 2006–2015 (1990). [CrossRef]

^{−1}, an order of magnitude lower than the slot-line on a dielectric substrate (23.9 cm

^{−1}) at 1THz. This shows that the radiation loss is the dominant loss for the asymmetric slot-line, consistent with the experimental results in [9

9. D. Grischkowsky, I. I. I. Duling III, J. C. Chen, and C. C. Chi, “Electromagnetic shock waves from transmission lines,” Phys. Rev. Lett. **59**(15), 1663–1666 (1987). [CrossRef] [PubMed]

10. D. Grischkowsky, “Optoelectronic characterization of transmission lines and waveguides by terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. **6**(6), 1122–1135 (2000). [CrossRef]

22. M. Y. Frankel, R. H. Voelker, and J. N. Hilfiker, “Coplanar transmission lines on thin substrates for high-speed low-loss propagation,” IEEE Trans. Microw. Theory Tech. **42**(3), 396–402 (1994). [CrossRef]

_{2}(4d

_{1}= λ / n

_{1}, 4d

_{2}= λ / n

_{2}) is chosen to maximize the photonic band gap size [16,17

17. P. Yeh, A. Yariv, and C. Hong, “Electromagnetic propagation in perodic media. I. General theory,” J. Opt. Soc. Am. **67**(4), 423–438 (1977). [CrossRef]

*s*, the separation of the plates, obtained from numerical simulations at 1THz. Similar to the slot-line in a homogeneous medium, the conductor loss is a decreasing function of

*s*. The results show the conductor loss for this slot-line with enough separation (> 5 μm) is in the order of 3 cm

^{−1}.

## 5. Summary

## Acknowledgement

## References and links

1. | R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. |

2. | H. Zhan, R. Mendis, and D. M. Mittleman, “Characterization of the terahertz near-field output of parallel-plate waveguides,” J. Opt. Soc. Am. B |

3. | J. Liu, R. Mendis, and D. M. Mittleman, “The transition from a TEM-like mode to a plasmonic mode in parallel-plate waveguides,” Appl. Phys. Lett. |

4. | K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature |

5. | H. Pahlevaninezhad, T. E. Darcie, and B. Heshmat, “Two-wire waveguide for terahertz,” Opt. Express |

6. | H. Pahlevaninezhad and T. E. Darcie, “Coupling of terahertz waves to a two-wire waveguide,” Opt. Express |

7. | 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. |

8. | D. R. Grischkowsky, M. B. Ketchen, C.-C. Chi, I. N. Duling, N. J. Halas, J.-M. Halbout, and P. G. May, “Capacitance free generation and detection of subpicosecond electrical pulses on coplanar transmission lines,” IEEE J. Quantum Electron. |

9. | D. Grischkowsky, I. I. I. Duling III, J. C. Chen, and C. C. Chi, “Electromagnetic shock waves from transmission lines,” Phys. Rev. Lett. |

10. | D. Grischkowsky, “Optoelectronic characterization of transmission lines and waveguides by terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. |

11. | J. V. Jelley, |

12. | C. Fattinger and D. Grischkowsky, “Observation of electromagnetic shock waves from propagating surface-dipole distributions,” Phys. Rev. Lett. |

13. | D. K. Kleinman and D. H. Auston, “Theory of Electrooptic shock radiation in nonlinear optical media,” IEEE J. Quantum Electron. |

14. | Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express |

15. | J. D. Jackson, |

16. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, |

17. | P. Yeh, A. Yariv, and C. Hong, “Electromagnetic propagation in perodic media. I. General theory,” J. Opt. Soc. Am. |

18. | P. Yeh, |

19. | A. Yariv and P. Yeh, |

20. | D. M. Pozar, “ |

21. | D. Grischkowsky, S. Keiding, M. Exter, and C. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B |

22. | M. Y. Frankel, R. H. Voelker, and J. N. Hilfiker, “Coplanar transmission lines on thin substrates for high-speed low-loss propagation,” IEEE Trans. Microw. Theory Tech. |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(300.6495) Spectroscopy : Spectroscopy, teraherz

**ToC Category:**

Waveguide and Opto-Electronic Devices

**History**

Original Manuscript: September 16, 2011

Revised Manuscript: October 11, 2011

Manuscript Accepted: October 12, 2011

Published: November 16, 2011

**Virtual Issues**

European Conference on Optical Communication 2011 (2011) *Optics Express*

**Citation**

Hamid Pahlevaninezhad, Barmak Heshmat, and Thomas Edward Darcie, "Efficient terahertz slot-line waveguides," Opt. Express **19**, B47-B55 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-26-B47

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### References

- R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett.26(11), 846–848 (2001). [CrossRef] [PubMed]
- H. Zhan, R. Mendis, and D. M. Mittleman, “Characterization of the terahertz near-field output of parallel-plate waveguides,” J. Opt. Soc. Am. B28(3), 558–566 (2011). [CrossRef]
- J. Liu, R. Mendis, and D. M. Mittleman, “The transition from a TEM-like mode to a plasmonic mode in parallel-plate waveguides,” Appl. Phys. Lett.98(23), 231113 (2011). [CrossRef]
- K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature432(7015), 376–379 (2004). [CrossRef] [PubMed]
- H. Pahlevaninezhad, T. E. Darcie, and B. Heshmat, “Two-wire waveguide for terahertz,” Opt. Express18(7), 7415–7420 (2010). [CrossRef] [PubMed]
- H. Pahlevaninezhad and T. E. Darcie, “Coupling of terahertz waves to a two-wire waveguide,” Opt. Express18(22), 22614–22624 (2010). [CrossRef] [PubMed]
- 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]
- D. R. Grischkowsky, M. B. Ketchen, C.-C. Chi, I. N. Duling, N. J. Halas, J.-M. Halbout, and P. G. May, “Capacitance free generation and detection of subpicosecond electrical pulses on coplanar transmission lines,” IEEE J. Quantum Electron.24(2), 221–225 (1988). [CrossRef]
- D. Grischkowsky, I. I. I. Duling, J. C. Chen, and C. C. Chi, “Electromagnetic shock waves from transmission lines,” Phys. Rev. Lett.59(15), 1663–1666 (1987). [CrossRef] [PubMed]
- D. Grischkowsky, “Optoelectronic characterization of transmission lines and waveguides by terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron.6(6), 1122–1135 (2000). [CrossRef]
- J. V. Jelley, Cherenlov radiation and its applications (Pergamon, New York, 1958).
- C. Fattinger and D. Grischkowsky, “Observation of electromagnetic shock waves from propagating surface-dipole distributions,” Phys. Rev. Lett.62(25), 2961–2964 (1989). [CrossRef] [PubMed]
- D. K. Kleinman and D. H. Auston, “Theory of Electrooptic shock radiation in nonlinear optical media,” IEEE J. Quantum Electron.20(8), 964–970 (1984). [CrossRef]
- Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express17(16), 13502–13515 (2009). [CrossRef] [PubMed]
- J. D. Jackson, Classical electrodynamics (John Wiley & Sons, 1999), pp. 352–356.
- J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: Modeling the flow of light, 2nd Edition (Princeton Univ. Press,2008), Chap. 4.
- P. Yeh, A. Yariv, and C. Hong, “Electromagnetic propagation in perodic media. I. General theory,” J. Opt. Soc. Am.67(4), 423–438 (1977). [CrossRef]
- P. Yeh, Optical waves in layered media (Wiley,1988), Chap.6.
- A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley,1984), Chap. 6.
- D. M. Pozar, “Microwave engineering (John Wiley & Sons, 2005), pp. 97-98.
- D. Grischkowsky, S. Keiding, M. Exter, and C. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B7(10), 2006–2015 (1990). [CrossRef]
- M. Y. Frankel, R. H. Voelker, and J. N. Hilfiker, “Coplanar transmission lines on thin substrates for high-speed low-loss propagation,” IEEE Trans. Microw. Theory Tech.42(3), 396–402 (1994). [CrossRef]

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