Abstract

We introduce and present general properties of hybrid terahertz waveguides. Weakly confined Zenneck waves on a metal-dielectric interface at terahertz frequencies can be transformed to a strongly confined yet low-loss subwavelength mode through coupling with a photonic mode of a nearby high-index dielectric strip. We analyze confinement, attenuation, and dispersion properties of this mode. The proposed design is suitable for planar integration and allows easy fabrication on chip scale. The superior waveguiding properties at terahertz frequencies could enable the hybrid terahertz waveguides as building blocks for terahertz integrated circuits.

© 2009 Optical Society of America

1. Introduction

Scaling-down has been the mainstream research thrust in electronics. Likewise, many of the future optical functionality will be realized in a chip format where multiple heterostructures are brought together in a compact manner. Similar arguments can be drawn about the future of terahertz (THz) photonic functionality. Most of current THz photonics are based on bulky free-space optics [1]. Compact integration of THz functionality could benefit a number of applications ranging from THz spectroscopy [2] to THz sensing [3]. To realize highly integrated THz chips, developing subwavelength integrated waveguides is essential. Large length scale in the THz frequency range (λo=300μmat 1 THz) has hindered compact and efficient integrated waveguides. Dielectric-based THz waveguides such as sapphire or plastic fibers [4,5], and plastic ribbon planar waveguides [6] have reasonably low propagation loss for chip scale operation (power absorption coefficient α ~1 cm−1, mainly limited by material absorption). They all, however, are subject to fundamental diffraction limit and the mode area cannot be reduced beyond it. Also, large waveguide dimensions preclude their use in densely integrated devices. More recently, a low-index discontinuity dielectric THz waveguide [7] was proposed, which could have subwavelength mode size but nonetheless requires a large foot print in the lateral dimension and is rather challenging to fabricate. Metal-based THz waveguides have been more successful than their dielectric counterparts. Low-loss waveguiding (α ~0.02 cm−1) at THz frequencies have been demonstrated using cylindrical metal wires with 0.9mm diameter [8]. Such waveguiding is essentially due to propagation of radially polarized surface plasmon polaritons (SPP) along the wire. The field of the SPP mode supported on the wire, however, is highly delocalized in radial direction for the large wire diameter. Although subwavelength mode size can be obtained by substantially reducing the wire diameter, its usefulness for waveguide integration is in question due to low coupling efficiency from polarization mismatch. Parallel-plate waveguides (PPWG) are also widely considered at THz frequencies [9,10]. PPWGs provide low loss (α ~0.2 cm−1) and negligible group velocity dispersion (GVD) but obviously offer no lateral confinement. Lateral confinement can be obtained as done with a metallic slit waveguide [11] and a microstrip waveguide [12] by making the width of both plates or one plate finite, respectively. Such waveguides also suffer from high attenuation if the mode size shrinks down to subwavelength as we will see for the case of the microstrip waveguides.

In this paper, we propose a novel approach for THz waveguiding having subwavelength mode confinement and reasonably low waveguide loss based on a hybrid photonic-plasmonic mode. This concept was originally proposed for optical wavelengths [13]. In the optical wavelength range, the field decay length of a SPP mode into dielectric (defined as 1/Re(κ), where κis the decay constant in the field form ~eκyfor y>0) is already subwavelength or comparable to the wavelength and the mode attenuation is quite high. On the contrary, the decay length of Zenneck waves at THz frequencies is much larger (for instance, ~58λatλ=269μmcompared to ~0.54λat λ=1.55μmat a gold-air interface [14]) with small attenuation of the mode. So, unlike at optical wavelengths, our focus is on obtaining subwavelength mode size by actually reducing the long propagation length of Zenneck waves to a level still suitable for chip-scale waveguiding (~tens of millimeters). Recently, some efforts to achieve tight confinement of Zenneck waves have been made through so called ‘spoof SPPs’ (surface modes on textured metal surfaces) and propagation of Zenneck waves with considerably reduced field decay length on a dielectric coated metal surface was recently demonstrated [1517]. We show that the mode hybridization can enable strong confinement of weakly bound Zenneck waves at THz frequencies without fabricating deep holes in metal. We discuss general properties of hybrid plasmonic waveguides in THz frequency regime in terms of mode confinement, attenuation, and dispersion.

2. One-dimensional properties

A schematic of the hybrid THz waveguide is shown in Fig. 1 . Without the metal substrate, the structure would be identical to a conventional dielectric waveguide consisting of a high permittivity core (εco) and surrounding low permittivity cladding (εd). The core has rectangular shape with thickness t and width w. If the core layer is removed, the SPP mode would be supported at the metal-cladding interface. As was mentioned previously, however, the SPP mode is weakly confined since metals behave more as perfect conductors at THz frequencies. With the dielectric core located at a short distance d above the metal substrate, the two independent photonic and SPP modes couple to form a hybrid mode, which possesses plasmonic and photonic characters simultaneously. While the metal microstrip waveguide, shown in the inset to Fig. 1, can support a highly confined subwavelength mode in the THz range [12], it comes at the cost of a large propagation loss. On the other hand, the hybrid metal-dielectric waveguide displays both low loss and strong mode confinement simultaneously, as will be shown below.

 

Fig. 1 Schematic of a hybrid terahertz waveguide. Inset: metal microstrip waveguide.

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To investigate the general properties of the hybrid THz waveguide, we begin by studying a one-dimensional (1-D) structure taking gap (d) (distance between the metal and the core) and the core thickness (t) as variable parameters. For 1-D structure (core width w is infinite, see the inset to Fig. 2(c) ), we compute the mode area (MA) and the propagation length (PL) as a function of d and t at the frequencyfo=1​    THz. The materials used in the modeling are gold for the metal substrate, GaAs for the core (εco=12.96), and HDPE for the cladding (εcl=2.37). The Drude model is used for the permittivity of gold, ε(ω)=1ωp2/[ω(ω+iωc)], where ωp=1.37×1016Hz and ωc=4.07×1013Hz. The propagation length is defined as the length at which the power drops to 1/e of its input value:

L=12Im{β},
where β is the complex wave vector. The definition of the mode area can be more complicated since the mode profile in plasmonic waveguides often deviates from that of conventional dielectric waveguides due to strong local enhancement of the fields near metallic surfaces. Among the various types of definitions for the mode area [18] we employ the definition given by equation:
A1=[|E(r)|2dA]2|E(r)|4dA,
which is based on the electric field extension and is considered more relevant for the evaluation of the crosstalk between adjacent waveguides [19]. The MA is normalized to the diffraction-limited area in free-space, A0=λ2/4. For 1-D structure, integration in Eq. (2) is performed for the lateral width λ/2for proper normalization. Our modeling was performed using finite element method for the fundamental TM mode. For the PL calculation, we only include ohmic losses in the metal to clarify the effect of confinement. However, in practical applications all material losses should ideally be taken into account [20].

 

Fig. 2 THz Guiding properties for 1-D hybrid structure at 1 THz. (a) Normalized mode area vs. gap, (b) Propagation length vs. gap, (c) Fraction of the mode power in the gap, (d), (e), (f) Normalized power distribution for gap d = 1 μm and core layer thickness t = 20 μm (d), for d = 20 μm and t = 20 μm (e), for d = 10 μm and t = 50 μm (f). The inset in (d) is a schematic of 1-D waveguide. The darker grey and lighter grey areas indicate metal (Au) layer and dielectric core layer, respectively. The black dotted lines are for PPWG.

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Figure 2(a) and 2(b) show the results. The general trend in trade-off between the MA and PL is easily observed. The normalized MA (A1/A0) and the PL tend to decrease with decreasing d and increasing t with exceptions of t = 50 μm. For relatively thin core thicknesses (t = 5 μm and 10 μm), the mode confinement is a weak function of d and the MA is close to A0.

This shows that the character of the hybrid mode with a thin core is more Zenneck-like at the metal-dielectric interface. The thin core layer only weakly perturbs the mode and does not make a significant impact on the mode properties. The character of the hybrid mode changes substantially with increasing the dielectric core layer thickness. The MA shrinks to subwavelength size for the t = 20μm core thickness and the PL also reduces substantially. The power distribution in the mode for the case of t = 20 μm core thickness is shown in Fig. 2(d) for d = 1 μm and 2(e) for d = 20 μm. We observe strongly confined mode power in the low-index layer between the metal and the core layers which clearly differs from the Zenneck waves or dielectric waveguide modes. Also, some portion of energy is localized inside the core and the outer cladding as well. The strong field confinement near the metal is mainly due to the continuity condition of electric displacement’s normal component, εcoEco=εclEcl [21]. The increased electric field at the cladding–core boundary then intensifies weak coupling of Zenneck wave to the electrons in the metal. The fractional mode power in the gap increases with increasing d and t with an exception of t = 50 μm case, as shown in Fig. 2(c). More than 40% of the mode power is confined in the gap for d = 20 μm and t = 20 μm.

For d = 1 μm and t = 20 μm, the MA is about ten times smaller than A0and the propagation length is on the order of several millimeters, which can be further increased to tens of millimeters at the expense of slight increase of the MA with increasing d (see Fig. 2(e) for d = 20 μm). Thus, subwavelength confinement and long PL can be simultaneously realized in a hybrid THz waveguide.

With a thicker core layer (t = 50 μm), the MA and PL exhibit quite different dependence on d: Both are strong functions of d, which suggests that the mode character is transformed to either photonic- or plasmonic-like, depending on the value of d. From Fig. 2(c) and 2(f) for t = 50 μm and d = 10 μm, we notice that a large fraction of the mode power is shifted into the core layer.

With increasing d, the mode character rapidly transitions to a photonic-like and the mode power in the gap decreases. The sharp increase in propagation length with increasing d indicates that the mode is mostly index-guided with significant power propagating inside the core. On the other hand, with decreasing d, the local field strength in the gap greatly enhances [9]. The localized mode power in the gap also increases up to ~5μm with decreasing d and slowly decreases since the absolute area of the gap decreases with decreasing d. For the hybrid plasmonic waveguide, interplay between plasmonic and photonic mode is crucial to obtain tailored properties. Our results show that this can be achieved by controlling the gap and the core thicknesses.

It is instructive to compare the results with the metal PPWG structure, in which case the core layer is replaced with 1 μm thick gold layer. Since PPWG supports fully plasmonic mode (fundamental even TM mode with no cutoff) with almost all the mode power confined between two metal layers, the MA is the smallest. As a result of strong mode confinement, the PL is significantly reduced to only ~500 μm for d = 1 μm, which could considerably limit the practical use of the PPWGs. Note that the same level of strong mode confinement as in PPWG with d = 10 μm can be achieved using the hybrid waveguide with d = 1 μm and t = 50 μm while the PL (~15 mm) is much longer than that of the PPWG (~5 mm).

Dispersion is another important characteristic of THz waveguides. For broadband THz spectroscopy applications, it’s desirable to transmit THz pulses without significant pulse broadening. GVD of dielectric-based waveguides is caused by waveguide dispersion as well as material dispersion. For metal-based waveguides, GVD becomes significant near the cutoff frequency [10]. The PPWGs have negligible GVD since there is no cut-off for the fundamental mode and the MA is determined by the distance between the metal plates. For the hybrid THz waveguides, the MA is a function of the frequency, so we expect a certain degree of waveguide dispersion.

The MA and PL as a function of frequency in the range 0.5–5 THz are plotted in Fig. 3(a) and 3(b) for the fixed core thickness of t = 20 μm and gap d = 0.5, 2, and 5 μm. In the whole frequency range considered the mode confinement is stronger and the propagation length is shorter for smaller gaps. Both properties do not monotonically change with frequency and the minima always exist at specific frequencies. This is consistent with the results shown in Fig. 2. When the frequency approaches either lower or higher limit, the mode confinement becomes weaker due to different physical origins. At lower frequencies, the mode becomes more Zenneck-like at a single interface. At higher frequencies, the power tends to be confined inside the dielectric core as an index-guided mode, leading to larger MA.

 

Fig. 3 Frequency-dependent characteristics of the 1-D hybrid structure with t = 20 μm, d = 0.5, 2, and 5 μm. (a) normalized mode area, (b) propagation length, (c) effective index, (d) group and phase velocities in the hybrid waveguide of t = 20 μm (e) group and phase velocities in the hybrid waveguide of t = 50 μm.

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Frequency-dependent effective index curves shown in Fig. 3(c) also confirm such mode transitions. The effective index of the mode approaches the refractive index of the cladding (n = 1.54) at lower frequencies and that of the core (n = 3.6) at higher frequencies. Figure 3(d) shows group velocity (vg) and phase velocity (vϕ) as a function of frequency for t = 20 μm and three values of d. From the group velocity curves, we observe relatively small dispersion at higher frequencies while strong dispersion is present at lower frequencies. The dispersion can be suppressed by decreasing the gap. The strong low frequency dispersion is inevitable due to the transition of the mode character. However, we can minimize the dispersion by increasing the core thickness as shown in Fig. 3(e). In the figure, t is increased to 50 μm. We observe the significantly suppressed dispersion, especially for d = 2 μm, even at lower frequencies.

3. Two-dimensional properties

Although 1-D analysis presented above captures most of the important physics, 2-D waveguides with lateral confinement need to be investigated for practical applications in planar integration. The effects of the waveguide width w on the MA and PL is shown in Fig. 4(a) and 4(b) for fixed t = 20 μm (solid lines). For w = 60 μm, the MA and PL are similar to those of the infinite width waveguide (compare with the red lines in Fig. 2(a) and (b)).

 

Fig. 4 THz Guiding properties for 2-D hybrid structure at 1 THz. (a) Normalized mode area vs. gap for w = 20, 40, and 60 μm. Inset: normalized mode area (A2/A0) defined by Eq. (2) vs. gap, (b) Propagation length vs. gap for w = 20, 40, and 60 μm. Inset: magnified view for dotted lines. Solid lines: hybrid waveguides, dotted lines: all-metal microstrip waveguides, (c),(d),(e) mode power distributions for w = 60, 40, and 20 μm, respectively with fixed t = 20 μm and d = 10 μm.

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As seen in Fig. 4(c), the mode power is mainly located under the core layer with good lateral confinement providing subwavelength MA and long PL (13-43 mm, depending on the gap). With decreasing width, the MA and PL increase rapidly indicating poor mode confinement as seen in Fig. 4(d) and 4(e) for w = 40 and 20 μm, respectively. The dotted lines in Fig. 4(a) and 4(b) are the MA and PL for the metal microstrip waveguide with the same d and w (see the inset in Fig. 1). Similar to the 1-D case, the metal microstrip waveguides show deep subwavelength mode areas, which results in much shorter PL (0.5–5 mm).

We also calculate the MA using a different definition, which is the ratio of the total mode energy to the peak energy density:

A2=1max{W(r)}W(r)dA
W(r)=12Re{d[ωε(r)]dω}|E(r)|2+12μ0|H(r)|2.

In our simulation the corners of the core are rounded with the radius of 100 nm to avoid extremely high peak energy density at the sharp corners. The inset in Fig. 4(a) is the normalized MA (A2/A0) based on Eq. (3). Compared to (A1/A0), (A2/A0) is almost one order of magnitude smaller. Both measures confirm that subwavelength confinement of the mode at THz frequencies can be achieved with finite width of the core while having long enough PL for chip scale integration. Broadband properties of the 2-D waveguides for t = 20 μm, w = 40 μm are shown in Fig. 5 . We observe behavior similar to the 1-D case. At lower and higher frequencies, the mode confinement becomes poorer for the same reasons. Figure 5(d) shows group velocity (vg) and phase velocity (vϕ) as a function of frequency for t = 20 μm, w = 40 μm and three d values. The dispersion properties also follow the same trends: Smaller dispersion at higher frequencies and stronger dispersion at lower frequencies. The dispersion can be suppressed by decreasing the gap distance and reducing the MA. As for the 1-D structure, while the strong low frequency dispersion is inevitable, we can minimize the effects by allowing larger dimensions for the width and thickness as shown in Fig. 5(e) for the waveguide of t = 40 μm and w = 60 μm.

 

Fig. 5 Frequency-dependent characteristics of the 2D hybrid waveguide with t = 20μm, w = 40μm, d = 0.5, 2, and 5μm. (a) Normalized mode area, (b) propagation length, (c) effective index, blue: d = 0.5 μm, green: d = 2 μm, red: d = 5 μm, (d) group and phase velocities, (e) group and phase velocities in the waveguide of t = 40 μm and w = 60 μm.

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4. Conclusion

We have investigated general properties of hybrid metal-dielectric terahertz waveguides in terms of mode area, propagation length, and dispersion. The resulting hybrid mode overcomes both weak confinement of a plasmonic mode at THz frequencies and the diffraction limit of a photonic mode. With the proper choice of waveguide dimensions, the mode area can be made of subwavelength size at the expense of reduced propagation length, but still exhibiting much lower attenuation than that possible with PPWGs or microstrip waveguides. Even for broadband applications reasonably good performances with tight mode confinement can be achieved. The fabrication of the hybrid THz waveguide is fully compatible with current micro-fabrication technologies owing to much lager dimensions than those required for optical wavelengths. The results presented prove the viability of hybrid waveguides as building blocks for highly integrated THz circuits through right combination of optics and plasmonics, which could enable compact THz integrated devices and allow more efficient use of THz energy due to significantly enhanced energy density.

References and links

1. B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 ( 2002). [CrossRef]   [PubMed]  

2. M. B. Byrne, J. Cunningham, K. Tych, A. D. Burnett, M. R. Stringer, C. D. Wood, L. Dazhang, M. Lachab, E. H. Linfield, and A. G. Davies, “Terahertz vibrational absorption spectroscopy using microstrip-line waveguides,” Appl. Phys. Lett. 93(18), 182904 ( 2008). [CrossRef]  

3. M. Brucherseifer, M. Nagel, P. H. Bolivar, and H. Kurz, “Label-free probing of the binding state of DNA by time-domain terahertz sensing,” Appl. Phys. Lett. 77(24), 4049–4051 ( 2000). [CrossRef]  

4. 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(15), 1987–1989 ( 2000). [CrossRef]  

5. L.-J. Chen, H.-W. Chen, T.-F. Kao, J.-Y. Lu, and C.-K. Sun, “Low-loss subwavelength plastic fiber for terahertz waveguiding,” Opt. Lett. 31(3), 308–310 ( 2006). [CrossRef]   [PubMed]  

6. R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88(7), 4449–4451 ( 2000). [CrossRef]  

7. M. Nagel, A. Marchewka, and H. Kurz, “Low-index discontinuity terahertz waveguides,” Opt. Express 14(21), 9944–9954 ( 2006). [CrossRef]   [PubMed]  

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

9. R. Mendis, “Nature of subpicosecond terahertz pulse propagation in practical dielectric-filled parallel-plate waveguides,” Opt. Lett. 31(17), 2643–2645 ( 2006). [CrossRef]   [PubMed]  

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

11. M. Wächter, M. Nagel, and H. Kurz, “Metallic slit waveguide for dispersion-free low-loss terahertz signal transmission,” Appl. Phys. Lett. 90(6), 061111 ( 2007). [CrossRef]  

12. X. Shou, A. Agrawal, and A. Nahata, “Efficient Broadband Terahertz Microstrip Waveguide,” OSA CLEO/QELS CMM5 (2008).

13. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 ( 2008). [CrossRef]  

14. B. G. Ghamsari and A. H. Majedi, “Terahertz transmission lines based on surface waves in plasmonic waveguides,” J. Appl. Phys. 104(8), 083108 ( 2008). [CrossRef]  

15. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 ( 2004). [CrossRef]   [PubMed]  

16. C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 ( 2008). [CrossRef]  

17. M. Gong, T.-I. Jeon, and D. Grischkowsky, “THz surface wave collapse on coated metal surfaces,” Opt. Express 17(19), 17088–17101 ( 2009). [CrossRef]   [PubMed]  

18. R. F. Oulton, G. Bartal, D. F. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” N. J. Phys. 10(10), 105018 ( 2008). [CrossRef]  

19. J. A. Conway, S. Sahni, and T. Szkopek, “Plasmonic interconnects versus conventional interconnects: a comparison of latency, crosstalk and energy costs,” Opt. Express 15(8), 4474–4484 ( 2007). [CrossRef]   [PubMed]  

20. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17(5), 851–863 ( 2000). [CrossRef]  

21. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 ( 2004). [CrossRef]   [PubMed]  

References

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  1. B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 ( 2002).
    [Crossref] [PubMed]
  2. M. B. Byrne, J. Cunningham, K. Tych, A. D. Burnett, M. R. Stringer, C. D. Wood, L. Dazhang, M. Lachab, E. H. Linfield, and A. G. Davies, “Terahertz vibrational absorption spectroscopy using microstrip-line waveguides,” Appl. Phys. Lett. 93(18), 182904 ( 2008).
    [Crossref]
  3. M. Brucherseifer, M. Nagel, P. H. Bolivar, and H. Kurz, “Label-free probing of the binding state of DNA by time-domain terahertz sensing,” Appl. Phys. Lett. 77(24), 4049–4051 ( 2000).
    [Crossref]
  4. 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(15), 1987–1989 ( 2000).
    [Crossref]
  5. L.-J. Chen, H.-W. Chen, T.-F. Kao, J.-Y. Lu, and C.-K. Sun, “Low-loss subwavelength plastic fiber for terahertz waveguiding,” Opt. Lett. 31(3), 308–310 ( 2006).
    [Crossref] [PubMed]
  6. R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88(7), 4449–4451 ( 2000).
    [Crossref]
  7. M. Nagel, A. Marchewka, and H. Kurz, “Low-index discontinuity terahertz waveguides,” Opt. Express 14(21), 9944–9954 ( 2006).
    [Crossref] [PubMed]
  8. K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 ( 2004).
    [Crossref] [PubMed]
  9. R. Mendis, “Nature of subpicosecond terahertz pulse propagation in practical dielectric-filled parallel-plate waveguides,” Opt. Lett. 31(17), 2643–2645 ( 2006).
    [Crossref] [PubMed]
  10. R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26(11), 846–848 ( 2001).
    [Crossref] [PubMed]
  11. M. Wächter, M. Nagel, and H. Kurz, “Metallic slit waveguide for dispersion-free low-loss terahertz signal transmission,” Appl. Phys. Lett. 90(6), 061111 ( 2007).
    [Crossref]
  12. X. Shou, A. Agrawal, and A. Nahata, “Efficient Broadband Terahertz Microstrip Waveguide,” OSA CLEO/QELS CMM5 (2008).
  13. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 ( 2008).
    [Crossref]
  14. B. G. Ghamsari and A. H. Majedi, “Terahertz transmission lines based on surface waves in plasmonic waveguides,” J. Appl. Phys. 104(8), 083108 ( 2008).
    [Crossref]
  15. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 ( 2004).
    [Crossref] [PubMed]
  16. C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 ( 2008).
    [Crossref]
  17. M. Gong, T.-I. Jeon, and D. Grischkowsky, “THz surface wave collapse on coated metal surfaces,” Opt. Express 17(19), 17088–17101 ( 2009).
    [Crossref] [PubMed]
  18. R. F. Oulton, G. Bartal, D. F. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” N. J. Phys. 10(10), 105018 ( 2008).
    [Crossref]
  19. J. A. Conway, S. Sahni, and T. Szkopek, “Plasmonic interconnects versus conventional interconnects: a comparison of latency, crosstalk and energy costs,” Opt. Express 15(8), 4474–4484 ( 2007).
    [Crossref] [PubMed]
  20. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17(5), 851–863 ( 2000).
    [Crossref]
  21. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 ( 2004).
    [Crossref] [PubMed]

2009 (1)

2008 (5)

R. F. Oulton, G. Bartal, D. F. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” N. J. Phys. 10(10), 105018 ( 2008).
[Crossref]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 ( 2008).
[Crossref]

B. G. Ghamsari and A. H. Majedi, “Terahertz transmission lines based on surface waves in plasmonic waveguides,” J. Appl. Phys. 104(8), 083108 ( 2008).
[Crossref]

M. B. Byrne, J. Cunningham, K. Tych, A. D. Burnett, M. R. Stringer, C. D. Wood, L. Dazhang, M. Lachab, E. H. Linfield, and A. G. Davies, “Terahertz vibrational absorption spectroscopy using microstrip-line waveguides,” Appl. Phys. Lett. 93(18), 182904 ( 2008).
[Crossref]

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 ( 2008).
[Crossref]

2007 (2)

M. Wächter, M. Nagel, and H. Kurz, “Metallic slit waveguide for dispersion-free low-loss terahertz signal transmission,” Appl. Phys. Lett. 90(6), 061111 ( 2007).
[Crossref]

J. A. Conway, S. Sahni, and T. Szkopek, “Plasmonic interconnects versus conventional interconnects: a comparison of latency, crosstalk and energy costs,” Opt. Express 15(8), 4474–4484 ( 2007).
[Crossref] [PubMed]

2006 (3)

2004 (3)

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

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 ( 2004).
[Crossref] [PubMed]

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 ( 2004).
[Crossref] [PubMed]

2002 (1)

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 ( 2002).
[Crossref] [PubMed]

2001 (1)

2000 (4)

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88(7), 4449–4451 ( 2000).
[Crossref]

M. Brucherseifer, M. Nagel, P. H. Bolivar, and H. Kurz, “Label-free probing of the binding state of DNA by time-domain terahertz sensing,” Appl. Phys. Lett. 77(24), 4049–4051 ( 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(15), 1987–1989 ( 2000).
[Crossref]

G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17(5), 851–863 ( 2000).
[Crossref]

Almeida, V. R.

Andrews, S. R.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 ( 2008).
[Crossref]

Barrios, C. A.

Bartal, G.

R. F. Oulton, G. Bartal, D. F. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” N. J. Phys. 10(10), 105018 ( 2008).
[Crossref]

Bolivar, P. H.

M. Brucherseifer, M. Nagel, P. H. Bolivar, and H. Kurz, “Label-free probing of the binding state of DNA by time-domain terahertz sensing,” Appl. Phys. Lett. 77(24), 4049–4051 ( 2000).
[Crossref]

Brucherseifer, M.

M. Brucherseifer, M. Nagel, P. H. Bolivar, and H. Kurz, “Label-free probing of the binding state of DNA by time-domain terahertz sensing,” Appl. Phys. Lett. 77(24), 4049–4051 ( 2000).
[Crossref]

Burnett, A. D.

M. B. Byrne, J. Cunningham, K. Tych, A. D. Burnett, M. R. Stringer, C. D. Wood, L. Dazhang, M. Lachab, E. H. Linfield, and A. G. Davies, “Terahertz vibrational absorption spectroscopy using microstrip-line waveguides,” Appl. Phys. Lett. 93(18), 182904 ( 2008).
[Crossref]

Byrne, M. B.

M. B. Byrne, J. Cunningham, K. Tych, A. D. Burnett, M. R. Stringer, C. D. Wood, L. Dazhang, M. Lachab, E. H. Linfield, and A. G. Davies, “Terahertz vibrational absorption spectroscopy using microstrip-line waveguides,” Appl. Phys. Lett. 93(18), 182904 ( 2008).
[Crossref]

Chen, H.-W.

Chen, L.-J.

Conway, J. A.

Cunningham, J.

M. B. Byrne, J. Cunningham, K. Tych, A. D. Burnett, M. R. Stringer, C. D. Wood, L. Dazhang, M. Lachab, E. H. Linfield, and A. G. Davies, “Terahertz vibrational absorption spectroscopy using microstrip-line waveguides,” Appl. Phys. Lett. 93(18), 182904 ( 2008).
[Crossref]

Davies, A. G.

M. B. Byrne, J. Cunningham, K. Tych, A. D. Burnett, M. R. Stringer, C. D. Wood, L. Dazhang, M. Lachab, E. H. Linfield, and A. G. Davies, “Terahertz vibrational absorption spectroscopy using microstrip-line waveguides,” Appl. Phys. Lett. 93(18), 182904 ( 2008).
[Crossref]

Dazhang, L.

M. B. Byrne, J. Cunningham, K. Tych, A. D. Burnett, M. R. Stringer, C. D. Wood, L. Dazhang, M. Lachab, E. H. Linfield, and A. G. Davies, “Terahertz vibrational absorption spectroscopy using microstrip-line waveguides,” Appl. Phys. Lett. 93(18), 182904 ( 2008).
[Crossref]

Ferguson, B.

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 ( 2002).
[Crossref] [PubMed]

Fernández-Domínguez, A. I.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 ( 2008).
[Crossref]

Gallot, G.

Garcia-Vidal, F. J.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 ( 2004).
[Crossref] [PubMed]

García-Vidal, F. J.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 ( 2008).
[Crossref]

Genov, D. A.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 ( 2008).
[Crossref]

Ghamsari, B. G.

B. G. Ghamsari and A. H. Majedi, “Terahertz transmission lines based on surface waves in plasmonic waveguides,” J. Appl. Phys. 104(8), 083108 ( 2008).
[Crossref]

Gong, M.

Grischkowsky, D.

Jamison, S. P.

G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17(5), 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(15), 1987–1989 ( 2000).
[Crossref]

Jeon, T.-I.

Kao, T.-F.

Kurz, H.

M. Wächter, M. Nagel, and H. Kurz, “Metallic slit waveguide for dispersion-free low-loss terahertz signal transmission,” Appl. Phys. Lett. 90(6), 061111 ( 2007).
[Crossref]

M. Nagel, A. Marchewka, and H. Kurz, “Low-index discontinuity terahertz waveguides,” Opt. Express 14(21), 9944–9954 ( 2006).
[Crossref] [PubMed]

M. Brucherseifer, M. Nagel, P. H. Bolivar, and H. Kurz, “Label-free probing of the binding state of DNA by time-domain terahertz sensing,” Appl. Phys. Lett. 77(24), 4049–4051 ( 2000).
[Crossref]

Lachab, M.

M. B. Byrne, J. Cunningham, K. Tych, A. D. Burnett, M. R. Stringer, C. D. Wood, L. Dazhang, M. Lachab, E. H. Linfield, and A. G. Davies, “Terahertz vibrational absorption spectroscopy using microstrip-line waveguides,” Appl. Phys. Lett. 93(18), 182904 ( 2008).
[Crossref]

Linfield, E. H.

M. B. Byrne, J. Cunningham, K. Tych, A. D. Burnett, M. R. Stringer, C. D. Wood, L. Dazhang, M. Lachab, E. H. Linfield, and A. G. Davies, “Terahertz vibrational absorption spectroscopy using microstrip-line waveguides,” Appl. Phys. Lett. 93(18), 182904 ( 2008).
[Crossref]

Lipson, M.

Lu, J.-Y.

Maier, S. A.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 ( 2008).
[Crossref]

Majedi, A. H.

B. G. Ghamsari and A. H. Majedi, “Terahertz transmission lines based on surface waves in plasmonic waveguides,” J. Appl. Phys. 104(8), 083108 ( 2008).
[Crossref]

Marchewka, A.

Martín-Moreno, L.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 ( 2008).
[Crossref]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 ( 2004).
[Crossref] [PubMed]

McGowan, R. W.

G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17(5), 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(15), 1987–1989 ( 2000).
[Crossref]

Mendis, R.

Mittleman, D. M.

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

Nagel, M.

M. Wächter, M. Nagel, and H. Kurz, “Metallic slit waveguide for dispersion-free low-loss terahertz signal transmission,” Appl. Phys. Lett. 90(6), 061111 ( 2007).
[Crossref]

M. Nagel, A. Marchewka, and H. Kurz, “Low-index discontinuity terahertz waveguides,” Opt. Express 14(21), 9944–9954 ( 2006).
[Crossref] [PubMed]

M. Brucherseifer, M. Nagel, P. H. Bolivar, and H. Kurz, “Label-free probing of the binding state of DNA by time-domain terahertz sensing,” Appl. Phys. Lett. 77(24), 4049–4051 ( 2000).
[Crossref]

Oulton, R. F.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 ( 2008).
[Crossref]

R. F. Oulton, G. Bartal, D. F. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” N. J. Phys. 10(10), 105018 ( 2008).
[Crossref]

Pendry, J. B.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 ( 2004).
[Crossref] [PubMed]

Pile, D. F.

R. F. Oulton, G. Bartal, D. F. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” N. J. Phys. 10(10), 105018 ( 2008).
[Crossref]

Pile, D. F. P.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 ( 2008).
[Crossref]

Sahni, S.

Sorger, V. J.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 ( 2008).
[Crossref]

Stringer, M. R.

M. B. Byrne, J. Cunningham, K. Tych, A. D. Burnett, M. R. Stringer, C. D. Wood, L. Dazhang, M. Lachab, E. H. Linfield, and A. G. Davies, “Terahertz vibrational absorption spectroscopy using microstrip-line waveguides,” Appl. Phys. Lett. 93(18), 182904 ( 2008).
[Crossref]

Sun, C.-K.

Szkopek, T.

Tych, K.

M. B. Byrne, J. Cunningham, K. Tych, A. D. Burnett, M. R. Stringer, C. D. Wood, L. Dazhang, M. Lachab, E. H. Linfield, and A. G. Davies, “Terahertz vibrational absorption spectroscopy using microstrip-line waveguides,” Appl. Phys. Lett. 93(18), 182904 ( 2008).
[Crossref]

Wächter, M.

M. Wächter, M. Nagel, and H. Kurz, “Metallic slit waveguide for dispersion-free low-loss terahertz signal transmission,” Appl. Phys. Lett. 90(6), 061111 ( 2007).
[Crossref]

Wang, K.

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

Williams, C. R.

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 ( 2008).
[Crossref]

Wood, C. D.

M. B. Byrne, J. Cunningham, K. Tych, A. D. Burnett, M. R. Stringer, C. D. Wood, L. Dazhang, M. Lachab, E. H. Linfield, and A. G. Davies, “Terahertz vibrational absorption spectroscopy using microstrip-line waveguides,” Appl. Phys. Lett. 93(18), 182904 ( 2008).
[Crossref]

Xu, Q.

Zhang, X.

R. F. Oulton, G. Bartal, D. F. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” N. J. Phys. 10(10), 105018 ( 2008).
[Crossref]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 ( 2008).
[Crossref]

Zhang, X.-C.

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 ( 2002).
[Crossref] [PubMed]

Appl. Phys. Lett. (4)

M. B. Byrne, J. Cunningham, K. Tych, A. D. Burnett, M. R. Stringer, C. D. Wood, L. Dazhang, M. Lachab, E. H. Linfield, and A. G. Davies, “Terahertz vibrational absorption spectroscopy using microstrip-line waveguides,” Appl. Phys. Lett. 93(18), 182904 ( 2008).
[Crossref]

M. Brucherseifer, M. Nagel, P. H. Bolivar, and H. Kurz, “Label-free probing of the binding state of DNA by time-domain terahertz sensing,” Appl. Phys. Lett. 77(24), 4049–4051 ( 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(15), 1987–1989 ( 2000).
[Crossref]

M. Wächter, M. Nagel, and H. Kurz, “Metallic slit waveguide for dispersion-free low-loss terahertz signal transmission,” Appl. Phys. Lett. 90(6), 061111 ( 2007).
[Crossref]

J. Appl. Phys. (2)

B. G. Ghamsari and A. H. Majedi, “Terahertz transmission lines based on surface waves in plasmonic waveguides,” J. Appl. Phys. 104(8), 083108 ( 2008).
[Crossref]

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88(7), 4449–4451 ( 2000).
[Crossref]

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

N. J. Phys. (1)

R. F. Oulton, G. Bartal, D. F. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” N. J. Phys. 10(10), 105018 ( 2008).
[Crossref]

Nat. Mater. (1)

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 ( 2002).
[Crossref] [PubMed]

Nat. Photonics (2)

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 ( 2008).
[Crossref]

C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernández-Domínguez, L. Martín-Moreno, and F. J. García-Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics 2(3), 175–179 ( 2008).
[Crossref]

Nature (1)

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

Opt. Express (3)

Opt. Lett. (4)

Science (1)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 ( 2004).
[Crossref] [PubMed]

Other (1)

X. Shou, A. Agrawal, and A. Nahata, “Efficient Broadband Terahertz Microstrip Waveguide,” OSA CLEO/QELS CMM5 (2008).

Cited By

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Figures (5)

Fig. 1
Fig. 1

Schematic of a hybrid terahertz waveguide. Inset: metal microstrip waveguide.

Fig. 2
Fig. 2

THz Guiding properties for 1-D hybrid structure at 1 THz. (a) Normalized mode area vs. gap, (b) Propagation length vs. gap, (c) Fraction of the mode power in the gap, (d), (e), (f) Normalized power distribution for gap d = 1 μm and core layer thickness t = 20 μm (d), for d = 20 μm and t = 20 μm (e), for d = 10 μm and t = 50 μm (f). The inset in (d) is a schematic of 1-D waveguide. The darker grey and lighter grey areas indicate metal (Au) layer and dielectric core layer, respectively. The black dotted lines are for PPWG.

Fig. 3
Fig. 3

Frequency-dependent characteristics of the 1-D hybrid structure with t = 20 μm, d = 0.5, 2, and 5 μm. (a) normalized mode area, (b) propagation length, (c) effective index, (d) group and phase velocities in the hybrid waveguide of t = 20 μm (e) group and phase velocities in the hybrid waveguide of t = 50 μm.

Fig. 4
Fig. 4

THz Guiding properties for 2-D hybrid structure at 1 THz. (a) Normalized mode area vs. gap for w = 20, 40, and 60 μm. Inset: normalized mode area (A2/A0 ) defined by Eq. (2) vs. gap, (b) Propagation length vs. gap for w = 20, 40, and 60 μm. Inset: magnified view for dotted lines. Solid lines: hybrid waveguides, dotted lines: all-metal microstrip waveguides, (c),(d),(e) mode power distributions for w = 60, 40, and 20 μm, respectively with fixed t = 20 μm and d = 10 μm.

Fig. 5
Fig. 5

Frequency-dependent characteristics of the 2D hybrid waveguide with t = 20μm, w = 40μm, d = 0.5, 2, and 5μm. (a) Normalized mode area, (b) propagation length, (c) effective index, blue: d = 0.5 μm, green: d = 2 μm, red: d = 5 μm, (d) group and phase velocities, (e) group and phase velocities in the waveguide of t = 40 μm and w = 60 μm.

Equations (4)

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

L = 1 2 Im { β } ,
A 1 = [ | E ( r ) | 2 d A ] 2 | E ( r ) | 4 d A ,
A 2 = 1 max { W ( r ) } W ( r ) d A
W ( r ) = 1 2 Re { d [ ω ε ( r ) ] d ω } | E ( r ) | 2 + 1 2 μ 0 | H ( r ) | 2 .

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