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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 22 — Oct. 25, 2010
  • pp: 23133–23146
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Waveguiding mechanism in tube lattice fibers

Luca Vincetti and Valerio Setti  »View Author Affiliations


Optics Express, Vol. 18, Issue 22, pp. 23133-23146 (2010)
http://dx.doi.org/10.1364/OE.18.023133


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Abstract

Waveguiding mechanism and modal characteristics of hollow core fibers consisting of a single or a regular arrangement of dielectric tubes are investigated. These fibers have been recently proposed as low loss, broadband THz waveguides. By starting from a description in terms of coupling between air and dielectric modes in a single tube waveguide, a simple and useful model is proposed and numerically validated. It is able to predict dispersion curves, high and low loss spectral regions, and the conditions to ensure the existence of low loss regions. In addition, it allows a better understanding of the role of the geometrical parameters and of the dielectric refractive index. The model is then applied to improve the tradeoff between low loss and effectively single mode propagation, showing that the best results are obtained with a heptagonal arrangement of the tubes.

© 2010 OSA

1. Introduction

Development and enhancement of low loss waveguide covering the electromagnetic spectrum from 300GHz and 30THz, have been driven by a growing interest in Terahertz (THz) technology [1

1. M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007). [CrossRef]

]. In that spectral region, the development of low loss waveguides with high free space coupling efficiency is a tough feature, due to the high conductivity-losses of metals and the high absorption of dielectrics [2

2. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22(7), 1099–20 (1983). [CrossRef] [PubMed]

7

7. Y. Jin, G. Kim, and S. Jeon, “Terahertz Dielectric Properties of Polymers,” J. Korean Phys. Soc. 49, 513–517 (2006).

]. To overcome that problem, several solutions have been proposed borrowing concepts and techniques from both microwaves and photonics technologies [8

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

16

16. S. Atakaramians, S. Afshar Vahid, H. Ebendorff-Heidepriem, M. Nagel, B. Fischer, D. Abbott, and T. Monro, “THz porous fibers: design, fabrication and experimental characterization,” Opt. Express 17, 14053–14062, http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-16-14053.

]. All of these try to reduce absorption loss by increasing the percentage of electromagnetic power transmitted through the air.

In optical waveguides, this issue can be addressed with fibers having an air cladding and a subwavelength dielectric core [13

13. 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]

] even with one [14

14. C. Zhao, M. Wu, D. Fan, and S. Wen, “Field enhancement and power distribution characteristics of subwavelength-diameter terahertz hollow optical fiber,” Opt. Commun. 281(5), 1129–1133 (2008). [CrossRef]

] or more [15

15. A. Hassani, A. Dupuis, and M. Skorobogatiy, “Porous polymer fibers for low-loss Terahertz guiding” Opt. Express 16, 6340–6351 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-9-6340.

, 16

16. S. Atakaramians, S. Afshar Vahid, H. Ebendorff-Heidepriem, M. Nagel, B. Fischer, D. Abbott, and T. Monro, “THz porous fibers: design, fabrication and experimental characterization,” Opt. Express 17, 14053–14062, http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-16-14053.

] air holes. Alternatively, hollow core (HC) fibers confine electromagnetic field inside an air-core surrounded by a microstructured cladding [17

17. F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, “Light and gas confinement in hollow-core photonic crystal fibre based photonic microcells,” J. Eur. Opt. Soc. 4, 09004 (2009). [CrossRef]

19

19. F. Benabid, “Hollow-core photonic bandgap fibre: new light guidance for new science and technology,” Philos. Trans. R. Soc. London, Ser. A. 364(1849), 3439–3462 (2006). [CrossRef]

]. Two techniques can be applied in order to confine the field in a hollow core. The photonic band gap fibers (PBGFs) operate using a cladding with a periodic refractive index which forbids radial propagation at certain frequencies [20

20. J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998). [CrossRef] [PubMed]

]. Low loss region is spectrally limited in the frequency range where the effective index of the mode lies in the photonic bang gap. Numerical analyses have shown transmission bandwidth in the THz range of a couple of hundreds of GHz [21

21. Y. F. Geng, X. L. Tan, P. Wang, and J. Q. Yao, “Transmission loss and dispersion in plastic terahertz photonic band-gap fibers,” Appl. Phys. B 91(2), 333–336 (2008). [CrossRef]

, 22

22. L. Vincetti, “Hollow core photonic band gap fibre for THz Applications,” Microwave Opt. Technol. Lett. 51(7), 1711–1714 (2009). [CrossRef]

]. In the second technique, fiber cladding does not support photonic band gap [17

17. F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, “Light and gas confinement in hollow-core photonic crystal fibre based photonic microcells,” J. Eur. Opt. Soc. 4, 09004 (2009). [CrossRef]

]. The guided modes confined in the hollow core are prevented from efficiently coupling to cladding due to the weak coupling with cladding modes. Compared to PBGFs, these fibers exhibit much broader low loss regions alternate with high loss regions, and thus they are called broadband HC fibers (BHCFs).

A HC fiber with a cladding formed by a periodic arrangement of Teflon tubes in a triangular lattice (Triangular Tube Lattice - TTL) has been recently demonstrated [23

23. J. Lu, C. Yu, H. Chang, H. Chen, Y. Li, C. Pan, and C. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett. 92(6), 64105 (2008). [CrossRef]

]. Numerical analysis has shown that this fiber falls into the class of the BHCFs [24

24. L. Vincetti, “Numerical analysis of plastic hollow core microstructured fiber for Terahertz applications,” Opt. Fiber Technol. 15(4), 398–401 (2009). [CrossRef]

]. It exhibits very interesting properties such as transmission bandwidth of several hundreds of GHz, low loss, low dispersion and high coupling efficiency with free space propagating beams [24

24. L. Vincetti, “Numerical analysis of plastic hollow core microstructured fiber for Terahertz applications,” Opt. Fiber Technol. 15(4), 398–401 (2009). [CrossRef]

], [25

25. L. Vincetti, “Single-mode propagation in triangular tube lattice hollow-core terahertz fibers,” Opt. Commun. 283(6), 979–984 (2010). [CrossRef]

]. Despite that, further improvements are limited by the lack of a proper understanding of waveguiding mechanism and simple models. Interesting and useful models have been proposed for BHCFs with kagome [26

26. F. Couny and F. Benabid, “P. J. robets, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 118–121 (2007).

28

28. G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. St. Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express 15, 12680–12685 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-15-20-12680.

] and square lattice [29

29. F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express 16, 20626–20636 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-25-20626.

, 30

30. A. Argyros, S. G. Leon-Saval, J. Pla, and A. Docherty, “Antiresonant reflection and inhibited coupling in hollow-core square lattice optical fibres,” Opt. Express 16, 5642–5648 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-8-5642.

] in visible and near infrared spectral regions. High loss regions correspond to strong resonances between the core modes and particular cladding modes. Since both kagome and square lattice can be seen as an intersection of slab waveguides of infinite width, the resonance frequencies are approximated by the transverse resonance condition [29

29. F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express 16, 20626–20636 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-25-20626.

], corresponding to cut-off frequencies of slab modes. These models are unable to explain some important features of transmission spectrum such as the high loss region at low frequencies and the bandwidth of the high loss regions. These features have been ascribed to the different behavior of TE and TM modes approaching cut-off frequencies [30

30. A. Argyros, S. G. Leon-Saval, J. Pla, and A. Docherty, “Antiresonant reflection and inhibited coupling in hollow-core square lattice optical fibres,” Opt. Express 16, 5642–5648 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-8-5642.

]. Although the model has been successfully applied to predict high loss regions in square lattice [30

30. A. Argyros, S. G. Leon-Saval, J. Pla, and A. Docherty, “Antiresonant reflection and inhibited coupling in hollow-core square lattice optical fibres,” Opt. Express 16, 5642–5648 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-8-5642.

] and TTL fibers [24

24. L. Vincetti, “Numerical analysis of plastic hollow core microstructured fiber for Terahertz applications,” Opt. Fiber Technol. 15(4), 398–401 (2009). [CrossRef]

], it requires to introduce an “effective” cut-off conditions for TM modes whose choice is not rigorously defined.

The transverse resonance condition has been also applied to analyzed tube, also called pipe, waveguides for terahertz waveguiding [31

31. C. Lai, B. You, J. Lu, T. Lu, J. Peng, C. Sun, and H. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18, 309–322 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-1-309.

]. Fibers with a hollow core surrounded by a thin dielectric layer have already analytically investigated in the 60’s, but focusing only on the modes guided by the dielectric layer [32

32. M. Kharadly, and J. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEE 116, 214–224 (1969).

]. Leaky modes propagating in a hollow core have been analytically analyzed by considering a dielectric with infinite extension [33

33. E. Marcatili and R. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 1783–1809 (1964).

].

2. Numerical analysis method: The finite element method

The solver is based on the curl-curl equation:
¯×(p^¯×v¯)k02q^v¯=0,
obtained by decoupling the Maxwell equations. p^and q^ represent ε^r1 and μ^r when v¯ is the magnetic field h¯, and μ^r1 and ε^r when v¯ is the electric field e¯; k0=2πf/c is the wavenumber in the vacuum, being c the light speed in the vacuum and f the frequency.

([A](γk0)2[B]){V}=0.

The eigenvector {V} is the discretized field vector, which provides the mode distribution on the transverse plane. The eigenvalue γ allows to evaluate dispersion and loss properties. In particular loss is calculated starting from α, according to
LOSS=20log10eα8.686α[dB/m]
where α is given in m−1.

In order to enclose the computational domain without affecting the numerical solution, anisotropic perfectly matched layers (PMLs) are placed before the outer boundary [38

38. A. Cucinotta, G. Pelosi, S. Selleri, L. Vincetti, and M. Zoboli, “Perfectly Matched Anisotropic Layers for Optical Waveguides Analysis through the Finite Element Beam Propagation Method,” Microw. Opt. Technol. Lett. 23(2), 67–69 (1999). [CrossRef]

].

3. Tube waveguide

F=2tcn21f.
(1)

Dispersion curves of the dielectric modes have been obtained from their characteristics equations [32

32. M. Kharadly, and J. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEE 116, 214–224 (1969).

]. Leakage loss and dispersion curves of the airy modes have been numerically computed. The discontinuities in the dispersion curves of the airy modes are due to the coupling with dielectric modes which causes anti-crossing and high leakage loss. Similar coupling has been observed in hollow core photonic band gap fibers between core and surface modes [40

40. K. Saitoh, N. A. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express 12, 394–400 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-3-394.

] and in W fibers between core and discrete lossy cladding modes [41

41. P. L. François and C. Vassallo, “Finite cladding effects in W fibers: a new interpretation of leaky modes,” Appl. Opt. 22(19), 3109–3120 (1983). [CrossRef] [PubMed]

]. Since the effective indices of the airy modes are closed to 1, coupling occurs when dielectric modes approach their cut-off frequencies. The normalized cut-off frequencies of the HE1,ν dielectric modes are Fc=ν−1, and they correspond to the cut-off of the TEν−1 modes of a slab waveguide with width t or, equivalently, to transverse resonance condition. They have been used to predict high loss regions in kagome [26

26. F. Couny and F. Benabid, “P. J. robets, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 118–121 (2007).

28

28. G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. St. Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express 15, 12680–12685 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-15-20-12680.

], and square fibers [29

29. F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express 16, 20626–20636 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-25-20626.

,30

30. A. Argyros, S. G. Leon-Saval, J. Pla, and A. Docherty, “Antiresonant reflection and inhibited coupling in hollow-core square lattice optical fibres,” Opt. Express 16, 5642–5648 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-8-5642.

], and in pipe waveguides [31

31. C. Lai, B. You, J. Lu, T. Lu, J. Peng, C. Sun, and H. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18, 309–322 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-1-309.

]. However they cannot explain neither the spectral width of the high loss regions nor the high loss at low frequencies. To overcome this lack, an “effective” cut-off condition for TM slab modes has been introduced to analyze square lattice BHCF [30

30. A. Argyros, S. G. Leon-Saval, J. Pla, and A. Docherty, “Antiresonant reflection and inhibited coupling in hollow-core square lattice optical fibres,” Opt. Express 16, 5642–5648 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-8-5642.

].

Figure 2 highlights that, in the tube waveguide, in addition to HE1,ν, there are other dielectric modes with the same radial dependence but higher azimuthal dependence which interact with airy modes. Since their cut-off frequencies are higher, they extend high loss regions. Moreover, cut-off of modes with ν=1 are spread from F=0 to F=0.5 explaining high loss region at low frequency. Actually, for a fixed value of ν, there exist much more dielectric modes with higher azimuthal index μ than those reported in Fig. 2. They have been omitted, because their interaction with airy modes is very weak. In fact, the higher is the azimuthal dependence compared to that of airy modes, the lower is the field overlap and thus the coupling strength [17

17. F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, “Light and gas confinement in hollow-core photonic crystal fibre based photonic microcells,” J. Eur. Opt. Soc. 4, 09004 (2009). [CrossRef]

]. Dielectric mode cut-off can thus be used to estimate high loss spectral width. Cut-off frequencies mainly depend on the ratio ρ between inner and outer tube diameter: ρ=D/d=12t/d [32

32. M. Kharadly, and J. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEE 116, 214–224 (1969).

]. Figure 3
Fig. 3 Normalized cut-off frequencies versus the inner and outer diameter ratio ρ, for two different dielectric refractive indices: n=1.44, 2.5.
shows the normalized cut-off frequencies of the dielectric modes with low azimuthal dependence versus ρ, for two different values of the dielectric refractive index n. As ρ reduces, the cut-off frequencies spread out over a wider range.

Below a critical value ρc, some curves cross each other and there are no more free cut-off spectral regions. Since spreading depends on radial index ν, ρc reduces as ν increases. However this value does not significantly depend on the refractive index. On the contrary, for high ρ values, the spreading increases as n increases. This means that fibers made of dielectrics with higher refractive index exhibit wider high loss regions. Finally, notice that, the higher the radial index ν and the ratio ρ, the narrower is the high loss region and the better is the approximation of the transverse resonant condition. In [31

31. C. Lai, B. You, J. Lu, T. Lu, J. Peng, C. Sun, and H. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18, 309–322 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-1-309.

] ρ varies from 0.875 to 0.947 and n between 1.4 and 1.6. Far from resonances, airy and dielectric modes are decoupled. Airy modes are strongly confined inside the HC and their intensities inside the dielectric negligible. This suggests that their characteristics do not significantly depend on cladding parameters. In Fig. 2 dashed lines represent the dispersion curves of the modes of a fiber with an circular air core surrounded by a dielectric with infinite extension obtained from the approximated expression [33

33. E. Marcatili and R. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 1783–1809 (1964).

]:
neffμν(f)=112(uμνcπ2Rf)2,
(2)
being R the air core radius, and uμνis the ν−th root of the equation Jμ1(uμν)=0. It has been obtained under the assumption that π2Rf>>uμνc. Despite that, the approximation of the airy mode dispersion curves is very good, except at the anti-crossing points.

4. Tube lattice waveguide

Despite the confinement mechanism does not depend on the particular arrangement of the tube lattice, a fiber with triangular tube lattice (TTL) will be initially considered [23

23. J. Lu, C. Yu, H. Chang, H. Chen, Y. Li, C. Pan, and C. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett. 92(6), 64105 (2008). [CrossRef]

, 24

24. L. Vincetti, “Numerical analysis of plastic hollow core microstructured fiber for Terahertz applications,” Opt. Fiber Technol. 15(4), 398–401 (2009). [CrossRef]

]. The cross section is reported in Fig. 4(a)
Fig. 4 (a) Transverse section of a TTL fiber. (b) Intensity distribution of the first four core modes. (c) Leakage loss (top) and dispersion curves (bottom) of the first core modes of a TTL fiber with ρ=0.9. Solid lines show curves given by Eq. (2) with R=(R1+R2)/2 being R1 and R2 computed through Eq. (3)
. The hollow core is obtained by removing the seven innermost tubes. Since the tubes surrounding the core are centered on the vertices and on the middle points of a hexagon with side length 2l, the core shape is approximately a hexagon whose apothem R1 and circumradius R2 are:

R1=d(3ld12),andR2=d(2ld12).
(3)

The analysis does not depend by the number of tube rings surrounding the core, thus, for sake of simplicity, here a fiber with two tube rings is considered. The guiding mechanism is the same of kagome and square lattice fibers recently developed for visible and near infrared applications [24

24. L. Vincetti, “Numerical analysis of plastic hollow core microstructured fiber for Terahertz applications,” Opt. Fiber Technol. 15(4), 398–401 (2009). [CrossRef]

]. In this kind of fibers the cladding does not exhibit photonic band gap. Cladding modes can be still classified in airy modes and dielectric modes. An example of the two kind of modes is reported in Fig. 5(a)
Fig. 5 (a) Intensity distribution of two cladding modes: a dielectric mode (top) and a HE11-like airy mode (bottom). (b) Effective indices of cladding modes versus the normalized frequency F of a TTL fiber with ρ=0.9. Top: dielectric modes around F=1; with high (green crosses) and low (red points) azimuthal dependence; squares show effective index of the dielectric mode of the single tube waveguide analytically computed. Bottom: the airy mode HE11-like (red points); dotted black line shows dispersion curves computed through Eq. (2) with R=D/2.
. Intensity and electric field distribution of the core modes are reported in Fig. 4(b). The field confinement inside the hollow core is due to the weak coupling between the core modes and the cladding modes [26

26. F. Couny and F. Benabid, “P. J. robets, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 118–121 (2007).

30

30. A. Argyros, S. G. Leon-Saval, J. Pla, and A. Docherty, “Antiresonant reflection and inhibited coupling in hollow-core square lattice optical fibres,” Opt. Express 16, 5642–5648 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-8-5642.

]. Core modes can propagate with low loss if the difference between their effective indices with those of cladding modes is enough high and the field overlap is low. Figure 4(c) shows the propagation characteristics of the first 14 core modes (considering that some modes have a doublet polarization). The leakage loss quickly increases at the resonances with dielectric modes and the dispersion curves are perturbed by the anti-crossing phenomenon. As in the single tube waveguide, this occurs when dielectric modes are closed to the their cut-off frequencies. Far from them, the differences between effective indices of core and dielectric modes are high enough to make low the leakage loss. The irregular spectral behavior of the leakage loss in the low loss regions is due to the weak coupling of the core modes with dielectric modes having high azimuthal dependence whose cut-off fall in this range of frequencies. The High Order Mode (HOM) with the lowest leakage loss is the TE01 mode.

Since the core modes are very confined within the core, it is reasonable to think that their dispersion characteristics are similar to those of the airy modes of a tube waveguide. In Fig. 4, the curves obtained through Eq. (2) with core radius R=(R1+R2)/2 are reported with solid lines, and they agree very well with the core mode dispersion curves, except around resonances due to anti-crossing.

By observing the cladding modes reported in Fig. 5(a), it clearly appears that the field distributions can be seen as composed by multiple replicas of those of a single tube waveguide. This simply observation suggests that the properties of the cladding modes can be obtained from those of a single tube waveguide. The model here proposed follows along the lines of [30

30. A. Argyros, S. G. Leon-Saval, J. Pla, and A. Docherty, “Antiresonant reflection and inhibited coupling in hollow-core square lattice optical fibres,” Opt. Express 16, 5642–5648 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-8-5642.

]. The cladding of the fiber can be considered as composed by several tube waveguides. By considering them as isolated waveguides and by neglecting any coupling between them, the dispersion curves of the cladding modes can be approximated by those of the modes of the single tube waveguide. This assumption is confirmed by the results reported in Fig. 5(b). On the top, the dispersion curves of the dielectric cladding modes are compared with those of the single tube waveguide around the normalized frequency F=1. Red crosses show dielectric cladding modes with ν=2 and low μ, whereas the green ones show modes with ν=1, 2 and high μ. Squares refer to single tube dielectric modes. Despite the dispersion curve splitting due to the weak coupling between dielectric cladding modes, the agreement is good. The effective indices of the airy cladding modes are reported on the bottom of Fig. 5(b). Red points show effective indices of the HE11-like airy modes. Dashed black line shows Eq. (2) by using the inner radius of the tubes of the cladding R=D/2 and u11. In further confirmation of the assumption, in Fig. 4(c), black solid vertical lines show the cut-off frequencies of the first dielectric modes with ν=1, 2, 3 of the single tube waveguide. They accurately predict high loss regions.

This value is about ρc=0.65 and ρc=0.45 respectively for the first two low loss regions located between the resonances with modes with ν=1, 2 and with ν=2, 3. In Fig. 6
Fig. 6 Leakage loss (top) and dispersion curves (bottom) for n=1.44 and three different values of the ratio ρ: 0.4 (left), 0.65 (middle), and 0.75 (right). Dashed black lines show dispersion curves given by Eq. (2) with u11 and R=(R1+R2)/2 being R1 and R2 computed through Eq. (3).
the leakage loss of the FM is reported for three different values of ρ and n=1.44. According with Fig. 3, with ρ=0.4 there are not low loss regions in the considered normalized frequency range and the dispersion curves exhibit several anti-crossing perturbations due to coupling with dielectric cladding modes. By increasing the ratio to ρ=0.65, a low loss region clearly appears between F=1 and F=2, whereas between F=0 and F=1 the low loss region is only slightly sketched. Finally, with ρ=0.75 the two low loss regions are clearly depicted and the anti-crossing perturbations concentrated around F=1, and 2. Dispersion curves computed through Eq. (2) and (3) with R=(R1+R2)/2 are also shown in Fig. 6 with dashed black lines. They are always in good agreement with FM dispersion curves.

Leakage loss and dispersion curves have been also analyzed by varying the refractive index of the cladding tubes. Three different values have been considered: n=1.44, 2.0, and 2.5. Tube thickness and diameters are constant with ρ=0.9. Even in this case, the high loss regions change in agreement with that predicted by Fig. 3. The higher is the refractive index, the wider are the high loss regions and their enlargement is towards the high frequencies. By observing Eq. (1) and (2), normalized frequency depends on refractive index whereas the effective index does not. This is the reason why in Fig. 7
Fig. 7 Leakage loss (top) and dispersion curves (bottom) for ρ=0.9 and three different values of the refractive index n: 1.44 (red), 2.0 (green), and 2.5 (blue). Solid black lines shows dispersion curves computed through Eq. (2) with R=(R1+R2)/2 being R1 and R2 computed through Eq. (3).
, by increasing the refractive index the dispersion curves reduce and the leakage loss increases.

5. Effectively single mode fiber design

In hollow core fibers, the overlap between mode field and dielectric material plays a key role in determining absorbing loss. This is true especially in THz region due to high dielectric absorption. In order to reduce it, the hollow core size must be significantly larger than the wavelength. Unfortunately this makes the fibers multimode [25

25. L. Vincetti, “Single-mode propagation in triangular tube lattice hollow-core terahertz fibers,” Opt. Commun. 283(6), 979–984 (2010). [CrossRef]

]. The multimode propagation can affect the output beam quality, and the signal quality, especially in time domain applications. A way to reduce the detrimental effect of the high order modes is to impair them by increasing the differential loss:
Δα=αFMαHOM,
where αFM is the propagation loss coefficient of the fundamental mode, and αHOM is that of the HOM with the lowest loss. To address this issue, it is possible to reduce the core size at the expense of higher αFM [25

25. L. Vincetti, “Single-mode propagation in triangular tube lattice hollow-core terahertz fibers,” Opt. Commun. 283(6), 979–984 (2010). [CrossRef]

]. The purpose of this section is to show that, by using the proposed model, it is possible to improve the tradeoff between Δα and αFM, simply by changing the arrangement of the tubes around the core.

A technique to increases the HOM loss without affecting the FM loss is coupling the HOMs with the cladding modes. This technique has been already proposed to obtain effectively single mode operation in solid and hole assisted fibers [42

42. J. Fini, “Design of solid and microstructure fibers for suppression of higher-order modes,” Opt. Express 13, 3477–3490 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-9-3477.

] and in HC-PBG fibers [43

43. K. Saitoh, N. Florous, T. Murao, and M. Koshiba, “Design of photonic band gap fibers with suppressed higher-order modes: Towards the development of effectively single mode large hollow-core fiber platforms,” Opt. Express 14, 7342–7352 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-16-7342.

]. The coupling was obtained, respectively, by changing the hole spacing between the first and second hole ring or by introducing smaller hole cores around the central one. In the present case, it is not necessary to change cladding structure, because it is possible to directly exploit the airy cladding modes [35

35. L. Vincetti, V. Setti, and M. Zoboli, “Terahertz Tube Lattice Fibers With Octagonal Symmetry,” IEEE Photon. Technol. Lett. 22(13), 972–974 (2010). [CrossRef]

]. Furthermore, since their dispersion characteristics are quite similar to those of the core modes (both are well represented by Eq. (2)), phase matching condition is verified over a wide frequency range. The interaction between core and airy cladding modes has been already experimentally observed in [26

26. F. Couny and F. Benabid, “P. J. robets, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 118–121 (2007).

], and [29

29. F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express 16, 20626–20636 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-25-20626.

], but their role in determining fiber properties has not been thoroughly investigated. Since in THz spectral region the material absorption loss is extremely high, the leakage loss is negligible compared to absorption loss even with just one ring of tubes around the hollow core [25

25. L. Vincetti, “Single-mode propagation in triangular tube lattice hollow-core terahertz fibers,” Opt. Commun. 283(6), 979–984 (2010). [CrossRef]

]. This opens up the use of new geometries to arrange the tubes. In particular, as shown in Fig. 8
Fig. 8 Left: Fibers geometries obtained by arranging tube on the vertices on a polygon with N sides. Right: on the top detail on the core geometry; on the bottom number N of tubes necessary to guarantee the resonance between core mode TE01 and the airy mode HE11 versus the ratio diameters ρ, for three different l/d.
, it is possible to arrange the tubes on the vertices of a polygon with N sides having length l. The relationship between tube diameter d and fiber core radius R depends on N. With some simply geometrical considerations, it is possible to show that:

R=12d(ld1sin(πN)1).
(4)

Similarly, the effective index of the first airy cladding mode, namely the HE11-like, can be estimated as:

nAiryHE11(f)=112(u11cπDf)2.
(6)

By equating Eq. (5) and (6), replacing Eq. (4), and recalling that D=d2t, the number N of tubes necessary to guarantee the index matched coupling between core mode TE01 and the airy mode HE11–like is:

N=πarcsin[ldu11u11+u01ρ].

N versus ρ for three different values of l/d is reported in the right panel of Fig. 8. By excluding values too high and too low of ρ, the best value of N to maximize the coupling is N=7.

To bear out this prediction, three different kind of fibers have been considered: a TTL fiber, a Heptagonal fiber with N=7, and an Octagonal one with N=8. The dielectric material has been assumed to be Teflon, with a refractive index n=nr-jni. In the THz spectral region Teflon dispersion is negligible [3

3. M. Naftaly and R. E. Miles, “Terahertz Time-Domain spectroscopy for Material Characterization,” Proc. IEEE 95(8), 1658–1665 (2007). [CrossRef]

7

7. Y. Jin, G. Kim, and S. Jeon, “Terahertz Dielectric Properties of Polymers,” J. Korean Phys. Soc. 49, 513–517 (2006).

], so that nr=1.44 for all the considered frequencies. About imaginary part, several values have been reported in the literature. At f=1THz, ni varies from 0.69E-3, corresponding to about 120dB/m [4

4. M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon photonic crystal fiber as terahertz waveguide,” Jpn. J. Appl. Phys. 43(2B), 317–319 (2004). [CrossRef]

], to 4.5E-3 corresponding to 870dB/m [7

7. Y. Jin, G. Kim, and S. Jeon, “Terahertz Dielectric Properties of Polymers,” J. Korean Phys. Soc. 49, 513–517 (2006).

]. In the present analysis it has been assumed ni=1.2E-3 corresponding to 220dB/m [5

5. J. R. Birch, J. D. Dromey, and J. Lesurf, “The optical constants of some common low-loss polymers between 4 and 40 cm−1,” Infrared Phys. 21(4), 225–228 (1981). [CrossRef]

, 6

6. C. Winnewisser, F. Lewen, and H. Helm, “Transmission characteristics of dichroic filters measured by THz time-domain spectroscopy,” Appl. Phys., A Mater. Sci. Process. 66(6), 593–598 (1998). [CrossRef]

]. By fixing t=0.1mm, there is a low loss region centered around 1THz and the resonance for ν=2 is at f=1.45THz. In the three fibers, the tube diameters have been chosen to have the same core radius R. In order to evaluate only the leakage loss, firstly a lossless material has been considered by assuming ni=0. In Fig. 9
Fig. 9 Leakege loss and dispersion curves for TTL (left), Heptagonal (middle), and Octagonal (right) fibers with a core radius R=1.2mm and tube thickness t=0.1mm. Solid red lines show dispersion curves of the HE11 airy cladding modes.
leakage loss and the dispersion curves of the FM and the first four HOMs for the three kind of fibers are reported in case of R=1.2mm. The dispersion curves are approximately the same for the three fibers, according to the theory. Also the minimum of the leakage loss of the FM does not significantly change by changing the fiber. On the contrary, the leakage loss of HOMs significantly increases passing from TTL to Octagonal and, finally, to Heptagonal fiber, according to the theory. In the Heptagonal fiber, the improvement is about a order of magnitude if compared to TTL one. In fact, as shown in Fig. 9, in the Heptagonal fiber, the dispersion curve of the cladding airy mode HE11–like (solid red line) is much closer to those of HOMs than in the other fibers. Furthermore this phase matching condition is maintained over a broad range. In order to show more clearly the improvement, the differential lossΔαversus the minimum of the FM loss αFM for different core radius is shown in Fig. 10
Fig. 10 Differential loss versus the minimum of the FM loss for different core radius in case of lossless dielectric (left) and lossy dielectric (right) with ni=1.2E-3. By starting form bottom-left side the core radii are: R=1.6mm, 1.2mm, 0.97mm, 0.73mm.
by considering ni=0 and ni=1.2E-3. All fibers exhibit a quasi-linear relationship. The Heptagonal fiber exhibits the sharpest curves in case of both lossless and lossy medium. As the core size increases, ρ tends to 1, because d increases and t is constant to 0.1 mm. By observing Fig. 8, this means that the optimum value of N changes from 7 to 8. This is why, with R=1.6 mm, the values of the Octagonal and the Heptagonal fibers are the same.

The material absorption shifts the curves toward higher FM loss. In this case, loss is composed by leakage and absorption loss. The effect of material absorption on the total loss is different for the different fibers: the lower is the leakage loss, the higher is the sensitivity towards material absorption, whereas the larger is the core, the lower is the sensitivity. In the Heptagonal fiber, absorption increases the FM loss, whereas the differential loss is substantially unchanged. In the Octagonal fiber, the differential loss is unchanged only for large core. Finally, in TTL fibers the absorption effects on the FM loss and on the differential loss are comparable. Although the threshold over which the fiber can be considered effectively single mode depends on the particular application, to fix the ideas, let assuming a differential loss of 20 dB/m. Figure 10 shows that with TTL fiber a core of 0.73 mm is required, and the FM loss is about 8 dB/m. The same condition is obtained with a core around of 1 mm and a FM loss of 1 dB/m in case of Heptagonal fiber, that is almost a decade lower than TTL one.

Finally, in order to show that effectively single mode operation can be obtained over a broadband, Fig. 11
Fig. 11 Leakege loss and dispersion curves of a heptagonal fiber with a core radius R=0.73mm and tube thickness t=0.1mm. Solid red lines show dispersion curves of the HE11-like airy cladding modes.
shows loss and dispersion characteristics versus the frequency of a Heptagonal fiber with R=0.73 mm, cladding tubes with diameter d=1.18 mm and thickness t=0.1 mm. The differential loss is always higher than 70 dB/m, whereas the FM loss is lower 10 dB/m over a band 400 GHz with a minimum of 2.8 dB/m at 1.07 THz.

6. Conclusion

In this paper the waveguiding mechanism in hollow core fibers composed by a single dielectric tube or a regular arrangement of tubes has been thoroughly analyzed. The high loss regions are due to the resonant coupling between core modes and dielectric modes like in the kagome and square BHCFs.

In the single tube waveguides, the high loss spectral regions and the dispersion curves of the core modes can be accurately predicted by starting from analytical equations. In the tube lattice fibers, a simple and useful model has been proposed. In the model, both the core and the cladding modes are described in terms of the modes of a single tube waveguide. Numerical results show that the model is able to predict loss and dispersion properties with the change of geometrical and physical parameters, allowing a better understanding of the role of tube diameter, thickness, and refractive index into design process. High loss regions mainly depend on ratio between inner and outer tube diameters, whereas the dielectric refractive index dependence is weaker. The model has been then applied to improve effectively single mode operation of the fiber by enhancing the coupling between high order core modes and airy cladding modes. It has been analytically shown, and then numerically verified, that fibers made of tubes arranged in a heptagonal symmetry exhibit the best tradeoff between high HOMs loss and low FM loss.

Finally, since there is a similarity between a circle and a hexagon, it is possible that the proposed model could be also applied to the analysis of BHCFs with a kagome lattice. These aspects will be subject of future investigations.

References and links

1.

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007). [CrossRef]

2.

M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22(7), 1099–20 (1983). [CrossRef] [PubMed]

3.

M. Naftaly and R. E. Miles, “Terahertz Time-Domain spectroscopy for Material Characterization,” Proc. IEEE 95(8), 1658–1665 (2007). [CrossRef]

4.

M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon photonic crystal fiber as terahertz waveguide,” Jpn. J. Appl. Phys. 43(2B), 317–319 (2004). [CrossRef]

5.

J. R. Birch, J. D. Dromey, and J. Lesurf, “The optical constants of some common low-loss polymers between 4 and 40 cm−1,” Infrared Phys. 21(4), 225–228 (1981). [CrossRef]

6.

C. Winnewisser, F. Lewen, and H. Helm, “Transmission characteristics of dichroic filters measured by THz time-domain spectroscopy,” Appl. Phys., A Mater. Sci. Process. 66(6), 593–598 (1998). [CrossRef]

7.

Y. Jin, G. Kim, and S. Jeon, “Terahertz Dielectric Properties of Polymers,” J. Korean Phys. Soc. 49, 513–517 (2006).

8.

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

9.

T. Ito, Y. Matsuura, M. Miyagi, H. Minamide, and H. Ito, “Flexible terahertz fiber optics with low bend-induced losses,” J. Opt. Soc. Am. B 24(5), 1230–1235 (2007). [CrossRef]

10.

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

11.

K. Wang and D. M. Mittleman, “Guided propagation of terahertz pulses on metal wires,” J. Opt. Soc. Am. B 22, 2001–2008 (2005). [CrossRef]

12.

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

13.

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]

14.

C. Zhao, M. Wu, D. Fan, and S. Wen, “Field enhancement and power distribution characteristics of subwavelength-diameter terahertz hollow optical fiber,” Opt. Commun. 281(5), 1129–1133 (2008). [CrossRef]

15.

A. Hassani, A. Dupuis, and M. Skorobogatiy, “Porous polymer fibers for low-loss Terahertz guiding” Opt. Express 16, 6340–6351 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-9-6340.

16.

S. Atakaramians, S. Afshar Vahid, H. Ebendorff-Heidepriem, M. Nagel, B. Fischer, D. Abbott, and T. Monro, “THz porous fibers: design, fabrication and experimental characterization,” Opt. Express 17, 14053–14062, http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-16-14053.

17.

F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, “Light and gas confinement in hollow-core photonic crystal fibre based photonic microcells,” J. Eur. Opt. Soc. 4, 09004 (2009). [CrossRef]

18.

P. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006). [CrossRef]

19.

F. Benabid, “Hollow-core photonic bandgap fibre: new light guidance for new science and technology,” Philos. Trans. R. Soc. London, Ser. A. 364(1849), 3439–3462 (2006). [CrossRef]

20.

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998). [CrossRef] [PubMed]

21.

Y. F. Geng, X. L. Tan, P. Wang, and J. Q. Yao, “Transmission loss and dispersion in plastic terahertz photonic band-gap fibers,” Appl. Phys. B 91(2), 333–336 (2008). [CrossRef]

22.

L. Vincetti, “Hollow core photonic band gap fibre for THz Applications,” Microwave Opt. Technol. Lett. 51(7), 1711–1714 (2009). [CrossRef]

23.

J. Lu, C. Yu, H. Chang, H. Chen, Y. Li, C. Pan, and C. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett. 92(6), 64105 (2008). [CrossRef]

24.

L. Vincetti, “Numerical analysis of plastic hollow core microstructured fiber for Terahertz applications,” Opt. Fiber Technol. 15(4), 398–401 (2009). [CrossRef]

25.

L. Vincetti, “Single-mode propagation in triangular tube lattice hollow-core terahertz fibers,” Opt. Commun. 283(6), 979–984 (2010). [CrossRef]

26.

F. Couny and F. Benabid, “P. J. robets, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 118–121 (2007).

27.

A. Argyros, and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-15-12-7713.

28.

G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. St. Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express 15, 12680–12685 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-15-20-12680.

29.

F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express 16, 20626–20636 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-25-20626.

30.

A. Argyros, S. G. Leon-Saval, J. Pla, and A. Docherty, “Antiresonant reflection and inhibited coupling in hollow-core square lattice optical fibres,” Opt. Express 16, 5642–5648 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-8-5642.

31.

C. Lai, B. You, J. Lu, T. Lu, J. Peng, C. Sun, and H. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18, 309–322 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-1-309.

32.

M. Kharadly, and J. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEE 116, 214–224 (1969).

33.

E. Marcatili and R. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 1783–1809 (1964).

34.

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM Modal Solver of Optical Waveguides with PML Boundary Conditions,” Opt. Quantum Electron. 33(4/5), 359–371 (2001). [CrossRef]

35.

L. Vincetti, V. Setti, and M. Zoboli, “Terahertz Tube Lattice Fibers With Octagonal Symmetry,” IEEE Photon. Technol. Lett. 22(13), 972–974 (2010). [CrossRef]

36.

L. Vincetti, “Confinement losses in honeycomb fibers,” IEEE Photon. Technol. Lett. 16(9), 2048–2050 (2004). [CrossRef]

37.

L. Vincetti, “Hollow core photonic band gap fiber for THz applications,” Microw. Opt. Technol. Lett. 51(7), 1711–1714 (2009). [CrossRef]

38.

A. Cucinotta, G. Pelosi, S. Selleri, L. Vincetti, and M. Zoboli, “Perfectly Matched Anisotropic Layers for Optical Waveguides Analysis through the Finite Element Beam Propagation Method,” Microw. Opt. Technol. Lett. 23(2), 67–69 (1999). [CrossRef]

39.

D. Chen, and H. Chen, “A novel low-loss Terahertz waveguide: Polymer tube,” Opt. Express 18, 3762–3767 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-18-4-3762.

40.

K. Saitoh, N. A. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express 12, 394–400 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-3-394.

41.

P. L. François and C. Vassallo, “Finite cladding effects in W fibers: a new interpretation of leaky modes,” Appl. Opt. 22(19), 3109–3120 (1983). [CrossRef] [PubMed]

42.

J. Fini, “Design of solid and microstructure fibers for suppression of higher-order modes,” Opt. Express 13, 3477–3490 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-9-3477.

43.

K. Saitoh, N. Florous, T. Murao, and M. Koshiba, “Design of photonic band gap fibers with suppressed higher-order modes: Towards the development of effectively single mode large hollow-core fiber platforms,” Opt. Express 14, 7342–7352 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-16-7342.

OCIS Codes
(060.2400) Fiber optics and optical communications : Fiber properties
(060.4005) Fiber optics and optical communications : Microstructured fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: August 10, 2010
Revised Manuscript: October 6, 2010
Manuscript Accepted: October 11, 2010
Published: October 19, 2010

Citation
Luca Vincetti and Valerio Setti, "Waveguiding mechanism in tube lattice fibers," Opt. Express 18, 23133-23146 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-23133


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References

  1. M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007). [CrossRef]
  2. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22(7), 1099–20 (1983). [CrossRef] [PubMed]
  3. M. Naftaly and R. E. Miles, “Terahertz Time-Domain spectroscopy for Material Characterization,” Proc. IEEE 95(8), 1658–1665 (2007). [CrossRef]
  4. M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon photonic crystal fiber as terahertz waveguide,” Jpn. J. Appl. Phys. 43(2B), 317–319 (2004). [CrossRef]
  5. J. R. Birch, J. D. Dromey, and J. Lesurf, “The optical constants of some common low-loss polymers between 4 and 40 cm−1,” Infrared Phys. 21(4), 225–228 (1981). [CrossRef]
  6. C. Winnewisser, F. Lewen, and H. Helm, “Transmission characteristics of dichroic filters measured by THz time-domain spectroscopy,” Appl. Phys., A Mater. Sci. Process. 66(6), 593–598 (1998). [CrossRef]
  7. Y. Jin, G. Kim, and S. Jeon, “Terahertz Dielectric Properties of Polymers,” J. Korean Phys. Soc. 49, 513–517 (2006).
  8. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17(5), 851–863 (2000). [CrossRef]
  9. T. Ito, Y. Matsuura, M. Miyagi, H. Minamide, and H. Ito, “Flexible terahertz fiber optics with low bend-induced losses,” J. Opt. Soc. Am. B 24(5), 1230–1235 (2007). [CrossRef]
  10. R. Mendis, “THz transmission characteristics of dielectric-filled parallel-plate waveguides,” J. Appl. Phys. 101(8), 083115 (2007). [CrossRef]
  11. K. Wang and D. M. Mittleman, “Guided propagation of terahertz pulses on metal wires,” J. Opt. Soc. Am. B 22, 2001–2008 (2005). [CrossRef]
  12. T. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfield wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 (2005). [CrossRef]
  13. 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]
  14. C. Zhao, M. Wu, D. Fan, and S. Wen, “Field enhancement and power distribution characteristics of subwavelength-diameter terahertz hollow optical fiber,” Opt. Commun. 281(5), 1129–1133 (2008). [CrossRef]
  15. A. Hassani, A. Dupuis, and M. Skorobogatiy, “Porous polymer fibers for low-loss Terahertz guiding” Opt. Express 16, 6340–6351 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-9-6340 .
  16. S. Atakaramians, S. Afshar Vahid, H. Ebendorff-Heidepriem, M. Nagel, B. Fischer, D. Abbott, and T. Monro, “THz porous fibers: design, fabrication and experimental characterization,” Opt. Express 17, 14053–14062, http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-16-14053 .
  17. F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, “Light and gas confinement in hollow-core photonic crystal fibre based photonic microcells,” J. Eur. Opt. Soc. 4, 09004 (2009). [CrossRef]
  18. P. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006). [CrossRef]
  19. F. Benabid, “Hollow-core photonic bandgap fibre: new light guidance for new science and technology,” Philos. Trans. R. Soc. London, Ser. A. 364(1849), 3439–3462 (2006). [CrossRef]
  20. J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998). [CrossRef] [PubMed]
  21. Y. F. Geng, X. L. Tan, P. Wang, and J. Q. Yao, “Transmission loss and dispersion in plastic terahertz photonic band-gap fibers,” Appl. Phys. B 91(2), 333–336 (2008). [CrossRef]
  22. L. Vincetti, “Hollow core photonic band gap fibre for THz Applications,” Microwave Opt. Technol. Lett. 51(7), 1711–1714 (2009). [CrossRef]
  23. J. Lu, C. Yu, H. Chang, H. Chen, Y. Li, C. Pan, and C. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett. 92(6), 64105 (2008). [CrossRef]
  24. L. Vincetti, “Numerical analysis of plastic hollow core microstructured fiber for Terahertz applications,” Opt. Fiber Technol. 15(4), 398–401 (2009). [CrossRef]
  25. L. Vincetti, “Single-mode propagation in triangular tube lattice hollow-core terahertz fibers,” Opt. Commun. 283(6), 979–984 (2010). [CrossRef]
  26. F. Couny and F. Benabid, “P. J. robets, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 118–121 (2007).
  27. A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-15-12-7713 .
  28. G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. St. Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express 15, 12680–12685 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-15-20-12680 .
  29. F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express 16, 20626–20636 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-25-20626 .
  30. A. Argyros, S. G. Leon-Saval, J. Pla, and A. Docherty, “Antiresonant reflection and inhibited coupling in hollow-core square lattice optical fibres,” Opt. Express 16, 5642–5648 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-8-5642 .
  31. C. Lai, B. You, J. Lu, T. Lu, J. Peng, C. Sun, and H. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18, 309–322 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-1-309 .
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