## 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

Driven by several emerging scientific and industrial applications, the terahertz (THz) frequency range has attracted intense research attention. Unique properties of THz systems have been exploited in areas like security, inspection, and spectroscopy. However, difficulties in generating, manipulating, and detecting THz waves still restrict most applications to exploratory and scientific investigations.

In terahertz time-domain spectroscopy (THz TDS), a femtosecond optical pulse is divided into two beams and focused on the emitter and detector. The generated THz wave from the emitter is then focused on the detector. Generally, this would require precise alignment of THz lenses, with associated loss. Confining THz radiation within waveguide structures offers tremendous potential advantages in size, performance, and versatility, driving research on many types of THz waveguides. Using transmission lines and waveguides can help the integration of THz systems, avoiding the difficulties associated with THz beam-shaping and beam-steering optics.

There has been strong interest in using metallic waveguides for terahertz frequencies. Supporting a TEM mode, parallel-plate waveguides can be efficient guides with 0.1 cm^{−1} loss at 1 THz [1–3]. Pushing for lower loss, single [4] and two-wire waveguides [5] have been explored. The two-wire waveguide offers both low loss and good coupling to most photoconductive antennas [6]. Planar transmission lines have also been proposed to carry terahertz waves [7]. They are compact, low in cost, and capable of being integrated easily with other components of systems. In the terahertz range, however, loss and dispersion prove to be two major obstacles, limiting the applicability of conventional planar transmission lines.

Of many planar alternatives, the slot-line structure with a thin slot in a conductive coating on one side of a dielectric substrate (Fig. 1a ), is quite compatible with terahertz photoconductive switches. The electrostatic field resulting from the applied bias voltage in the emitter is exactly the same as the field distribution of propagating mode. This allows essentially perfect coupling efficiency of waves into the transmission lines [8]. However, at higher frequencies where the substrate thickness is larger than the wavelength, the slot-line, like other asymmetrical transmission lines, becomes extremely lossy due to electromagnetic shock-wave radiation [9,10]. This radiation occurs when charges move faster than the phase velocity of electromagnetic radiation in a material [11]. The geometry of conventional slot-lines results in some of the field lines in the dielectric region, with the rest in the air region above the substrate. Hence, in asymmetrical transmission lines, the group velocity of the propagating mode is higher than the phase velocity in the dielectric, yielding emission of electromagnetic shock-wave radiation [12,13]. Also, as a result of the phase mismatch at the dielectric-air interface, the slot-line cannot support a pure TEM mode. Therefore, the basic electrical parameters of the slot-line, like the characteristic impedance and the phase velocity vary with frequency, making the slot-line a dispersive line.

In this paper, we propose two solutions to the problem of shock-wave radiation loss for slot-lines: using a slot-line in a homogeneous medium (Fig. 1b), and using a slot-line on a layered substrate (Fig. 1c). The former is suitable for broadband applications for it supports a pure TEM mode that is free from cutoff frequency and group velocity dispersion. The latter, on the other hand, is more appropriate for narrowband applications. Theoretical and numerical analysis are used to verify that the slot-line in a homogeneous medium and a slot-line on a layered substrate can be used for guiding terahertz waves with 2 cm^{−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

Analysis of the TEM mode supported by a slot-line in a homogenous medium entails solving the 2D Laplace equation for the electric potential function on the cross-section of the slot-line with fixed potential values on the two metal plates. Conformal mapping can be exploited to solve the problem. We found that the following complex analytic function maps the cross section of the slot-line to two parallel lines like the cross section of a parallel-plate waveguide as shown in Fig. 2(b,c) :

The potential function in the region between the two parallel-plate is

*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:

The asymptotes of these hyperbolas make the angle *(1-V/V _{0})π/2* with the

*x*-axis. The electric field can be derived from the gradient of the potential:

*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.

Figure 3(a) shows the equipotential curves and the electric field for a slot-line with 10 μm separation. Around the gap between the plates, the potential function varies significantly with a small spatial change, resulting in a high field amplitude, specially at the edges. Far from the gap, on the other hand, the potential function has a slow spatial change, leading to a small field amplitude as shown in Fig. 3(b). These results are consistent with the ones from numerical simulations using the finite-difference frequency domain (FDFD) method reported in [14].

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

Electromagnetic fields inside a good, but not perfect conductor are attenuated exponentially in a characteristic length *δ*, the skin depth, and relate to the tangential magnetic field just outside the conductor surface by [15]

*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

Equations (5),6) give the tangential magnetic field at the surface of the conductors with infinitesimal thicknesses. For small, but not zero thicknesses of the conductors, a good estimation can be obtained by approximating the planar metal plates with two branches of a hyperbola whose asymptotes make very small angles with the *x*-axis, like the ones shown in Fig. 4
. This hyperbola corresponds to the lines *u = ± u _{0}* in the

*uv*-plane when

*u*is very close to

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

The power loss per unit length of the transmission line is then

*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:

The plate width is assumed to be much larger than the separation (*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, is

*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

The attenuation constant is then

where*P*is the power loss per unit length and

_{l}*L*is the length of the transmission line. Substituting Eq. (14) and (18) into Eq. (19) gives:

*v*depends on

_{0}*s*and

*w*by Eq. (15), and

*u*depends on the thickness of the metal plates.

_{0}## 3. Slot-lines on a layered substrate

A slot-line on a layered substrate, shown in Fig. 1(c), can also solve the problem of shock-wave radiation. The slot-line on a layered substrate has a continuous translational symmetry in the z-direction leading to the modes with the function form *e ^{ikz}* as an eigenfunction [16]. Therefore, the modes must have the form

*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]. The frequency at the middle of the gap ω

_{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}The air-substrate interface breaks the translational symmetry, making the structure sustain localized modes within the photonic band gap. These are evanescent modes on both sides of the surface with their field spatially bounded in the direction perpendicular to the substrate as shown in Fig. 5(a)
. The metal plates of the slot-line also confine the waves in the *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

Figure 6
shows simulation results for slot-lines obtained from full-wave 3D FEM simulations using the Ansoft HFSS frequency-domain solver. Three cases are considered: a conventional slot-line on a GaAs substrate, a slot-line in GaAs, and a slot-line on a periodic Si/SiO_{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.

For the slot-line on the half-space GaAs, the substrate/superstrate mismatch causes large radiation loss into the substrate as clearly shown in Fig. 6(a). This radiation loss changes in proportion to the third power of frequency. Simulation results show 23.9 cm^{−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).

Figure 7(a)
shows the conductor loss versus *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]. In that paper, the relationship *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}.

Figure 7(b) shows the conductor loss versus *u _{0}* for a slot-line with

*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.

For the waveguide that is completely filled with a homogenous dielectric, the dielectric loss can be calculated from the propagation constant regardless of the type of guide [20]. For instance, the dielectric loss for crystalline high-resistivity GaAs is measured to be 0.5 cm^{−1} at 1THz [21]. The total loss is the sum of dielectric and conductor loss. For a slot-line with more than 10 μm separation and more than 100 nm thickness of the plates, the total loss would be about 2.5 cm^{−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 [9,10,22].

For the slot-line on multilayer substrate, shown in Fig. 6(c), a quarter-wave stack of Si/SiO_{2} (4d_{1} = λ / n_{1}, 4d_{2} = λ / n_{2}) is chosen to maximize the photonic band gap size [16,17]. Figure 6(c) shows that the multilayer film can effectively prevent wave radiation into the substrate, making this slot-line efficient for guiding terahertz waves. Figure 7(a) (purple circles) shows the conductor loss versus *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}.

The focus of this paper is to present the theoretical and simulation results to evaluate the performance of slot-lines for terahertz waves. However, the fabrication of a slot-line in a homogeneous medium could be done through the following steps: 1) growing 1 μm-thick Low-Temperature-Grown GaAs (LTG-GaAs) layer on Semi-Insulating GaAs (SI-GaAs) substrate by Molecular Beam Epitaxy (MBE), 2) depositing 100 nm-thick slot-line metal plates on LTG-GaAs, 3) depositing a 200 nm-thick layer of photoresist on the LTG-GaAs as an adhesion layer by a spin coater, and 3) putting another SI-GaAs or Float-Zone (FZ) silicon wafer on top of the photoresist as a superstrate. Since the photoresist layer thickness is much smaller than the wavelength, it would not affect the performance of the transmission line.

## 5. Summary

The structures of planar transmission lines are strongly compatible with terahertz photoconductive sources, but significant radiation loss into the substrate limits the applicability of conventional planar lines for terahertz waves. However, the slot-line in a homogeneous medium offers unique advantages like low loss, supporting a TEM mode, and excellent coupling of electromagnetic energy into the transmission line. We present an analytical method for determining the field distribution and estimating the absorption coefficient in terms of the dimensions. The results agree with both numerical simulation data and previously reported experimental data. We also propose using slot-lines on a layered substrate to avoid radiation loss for narrowband applications. Numerical simulations were presented to validate the performance of slot-lines on a layered substrate. Works continue on fabrication and testing of the devices.

## Acknowledgement

This work was supported by funding from the Natural Science and Engineering Research Council (NSERC) Canada.

## References and links

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

**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 **28**(3), 558–566 (2011). [CrossRef]

**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]

**11. **J. V. Jelley, *Cherenlov radiation and its applications* (Pergamon, New York, 1958).

**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]

**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]

**15. **J. D. Jackson, *Classical electrodynamics* (John Wiley & Sons, 1999), pp. 352–356.

**16. **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.

**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]

**18. **P. Yeh, *Optical waves in layered media* (Wiley,1988), Chap.6.

**19. **A. Yariv and P. Yeh, *Optical Waves in Crystals: Propagation and Control of Laser Radiation* (Wiley,1984), Chap. 6.

**20. **D. M. Pozar, “*Microwave engineering* (John Wiley & Sons, 2005), pp. 97-98.

**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]

**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]