光子晶体光纤（带隙型）

Porous fibers: a novel approach to low loss THz waveguides Shaghik Atakaramians1,2, Shahraam Afshar V.1, Bernd M. Fischer2, Derek Abbott2 and Tanya M. Monro1 1Centre of Expertise in Photonics, School of Chemistry & Physics, 2Centre for Biomedical Engineering, School of Electrical & Electronic Engineering, The University of Adelaide, SA 5005 Australia shaghik@eleceng.adelaide.edu.au Abstract: We propose a novel class of optical fiber with a porous trans- verse cross-section that is created by arranging sub-wavelength air-holes within the core of the fiber. These fibers can offer a combination of low transmission loss and high mode confinement in the THz regime by ex- ploiting the enhancement of the guided mode field that occurs within these sub-wavelength holes. We evaluate the properties of these porous fibers and quantitatively compare their performance relative to that of a solid core air cladded fiber (microwire). For similar loss values, porous fibers enable improved light confinement and reduced distortion of a broadband pulse compared to microwires. © 2008 Optical Society of America OCIS codes: (230.7370) Waveguides; (130.2790) Guided waves; (260.3090) Infrared, far. References and links 1. D. Abbott and X.-C. Zhang, “Scanning the issue: T-ray imaging, sensing, and retection,” Proceedings of the IEEE 95, 1509–1513 (2007). 2. K. Sakai, Terahertz Optoelectronics (Springer, Berlin, and Heidlberg, 2005). 3. W. Withayachumnankul, G. M. Png, X. Yin, S. Atakaramians, I. Jones, H. Lin, B. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, and D. Abbott, “T-ray sensing and imaging,” Proceedings of the IEEE 95, 1528–1558 (2007). 4. B. M. Fischer, M. Hoffmann, H. Helm, R. Wilk, F. Rutz, T. Kleine-Ostmann, M. Koch, and P. U. Jepsen, “Tera- hertz time-domain spectroscopy and imaging of artificial RNA,” Opt. Express 13, 5205–5215 (2005). 5. R. W. McGowan, G. Gallot, and D. 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Kuo, J.-Y. Lu, L.-J. Chen, and C.-K. Sun, “Investigation on spectral loss characteristics of subwavelength terahertz fibers,” Opt. Lett. 32, 1017–1019 (2007). 1. Introduction The terahertz (THz) or T-ray region of electromagnetic spectrum, located between millimeter wave and infrared frequencies, has attracted much interest over the last decade. The frequency range is loosely defined as 0.1-10 THz [1]. Terahertz spectroscopic techniques have many ap- plications, as for example in detection of biological and chemical materials [2, 3]. Many of these preliminary proof-of-principle studies have been carried out in spectroscopy systems, where terahertz radiation propagates in free space. For biomedical applications, however, these systems have limitations such as large diffraction limited focal spot size leading to the need for inconveniently large biosamples [1], and the systems are quite large thus not easy to inte- grate with optical and infrared techniques. Terahertz waveguides provide a promising approach for overcoming these hurdles and potentially provide low-cost and robust THz integrated sys- tems. However, efficient low-loss transmission of THz radiation within waveguides is still a challenge. Waveguide solutions based on technologies from both electronics and photonics have been studied such as the hollow metallic circular waveguide [5], hollow metallic rectangu- #94241 - $15.00 USD Received 25 Mar 2008; revised 30 May 2008; accepted 30 May 2008; published 2 Jun 2008 (C) 2008 OSA 9 June 2008 / Vol. 16, No. 12 / OPTICS EXPRESS 8846 lar waveguide [6], sapphire fiber [7], plastic ribbon waveguide [8], air-filled parallel-plate waveguide [9, 10], plastic photonic crystal fiber [11], coaxial waveguide [12], metal wire waveguide [13, 14], parallel-plate photonic waveguide [15], metal sheet waveguide [16], sub- wavelength plastic fibers [17], the dielectric-filled parallel-plate waveguide [18], low-index dis- continuity terahertz waveguides [19], and the metallic slit waveguide [20]. Metallic waveguide solutions based on bare metal wires, and dielectric fiber solutions based on sub-wavelength plastic fibers (described as THz microwires [21] in analogy with optical nanowires) and air-core microstructure fibers, with attenuation constants less than 0.03 cm−1 [13], 0.01 cm−1 [17], and 0.01 cm−1 [22] respectively, have the lowest loss reported in the literature. The guiding mechanism in sub-wavelength plastic fibers is based on the total internal refection, while in the air-core microstructure fiber is based on band gap effect. Low losses are achieved in bare metal wires and microwires, because almost all the field propagates in the medium surrounding the structure. This medium is usually air, which is transparent to THz. In other words, these structures act as rails for guiding THz radiation rather than as pipes that provide confinement. This, however, results in weak confinement of the guided field within the structure that makes the guided field susceptible to any small perturbation on the surface or vicinity of the structure, since a large portion of the guided power can be readily coupled into radiation modes. Furthermore, as a result of weak confinement, the guided modes within these structures suffer strong bend loss [19, 23]. To produce practical waveguide structures for THz, it is critical to find a means of improving the mode confinement while retaining the reduction in material loss that is associated with locating most of the guided field in air. An approach to improve field confinement in the structure has been demonstrated by Nagel et al. [19]. They have shown that a low index discontinuity in dielectric waveguides (split rec- tangular and tube waveguides) has increased field in the low-index central region and reduced field in the air-clad surrounding region, resulting in increased confinement. Nagel et al. [19] demonstrated that a sub-wavelength air gap in a dielectric slab waveguide can trap 55% of the mode power in the vicinity of the gap between the two slabs, and a sub-wavelength hole in a fiber can trap 26% of the power in the sub-wavelength discontinuity. The former is difficult to handle because it needs another structure to keep the slabs together and the latter is not a promising low-loss waveguide structure for terahertz since a large portion of the power of the guiding mode is still propagating inside the material [24]. In this paper, we propose a novel class of fiber for THz, based on introducing sub-wavelength low index discontinuities within air-clad fibers. Instead of having one sub-wavelength air-hole in the core of the fiber [25, 26], we consider a pattern of sub-wavelength air-holes in the core, and show that this leads to a better confinement of the field to the structure while still allowing THz propagation in the sub-wavelength air-holes. In the optical regime, elongated void regions have previously been used to improve the transmission efficiency simply by reducing the ma- terial used within the core [27]. This paper considers a porous fiber core with sub-wavelength holes to enable localization and enhancement of the guided mode within the holes, and the use of such structures for improving the effective material loss and confinement. Here, we theo- retically investigate the physics of THz propagation in porous fibers and examine the effective material loss and the confinement of THz propagation as a function of fiber porosity and di- ameter, in comparison with those of THz microwires (straight dielectric rods). We demonstrate that for similar loss values, our porous core fiber leads to better confinement than microwires. 2. Porous fiber Porous fibers in this study are created by including a distribution of sub-wavelength air holes within the core of an air-clad fiber. A typical example is shown in Fig. 1(a). The distribution, shape, and size of the holes determine the porosity of the structure, which is defined as the #94241 - $15.00 USD Received 25 Mar 2008; revised 30 May 2008; accepted 30 May 2008; published 2 Jun 2008 (C) 2008 OSA 9 June 2008 / Vol. 16, No. 12 / OPTICS EXPRESS 8847 fraction of the air holes to core area. The structure shown in Fig. 1(a) has a triangular lattice of circular holes with radii of 20 μm, core radius of 200 μm, which results in a porosity of 37%. This sparse porosity value is only chosen for purposes of clarity for displaying the concept in Fig. 1. The arrangement of the holes within the core of porous fiber can be based on different varieties of lattice structures. We chose a triangular lattice because it leads to higher porosity when compared to a rectangular lattice of circular holes. Polymethyl methacrylate (PMMA) is considered as the host material for all of simulations here where we use the THz properties of PMMA (refractive index and attenuation constant as a function of frequency) [28] measured by THz time-domain spectroscopy (THz-TDS). To find the propagation constant and field distributions for the porous fibers, we solve the full vectorial form of Maxwell’s equations since, for the sub-wavelength scales considered here, a scalar approximation gives inaccurate results [29]. To solve Maxwell’s equations for this geometry we use the Finite Element Modeling (FEM) technique instantiated in the com- mercial FEM package COMSOL 3.2. Considering the symmetry of the structure, we employ a quarter-plane of the fiber’s cross-section and appropriate combination of perfect electric and magnetic conductor boundary conditions in order to determine the propagation constant and mode distribution. Different mesh densities are employed in different regions within the cross section in order to achieve convergence for the calculated parameters. The calculated propaga- tion constant of the fiber is accurate to five significant figures and all other parameters presented in this paper are accurate to three significant figures. For calculating the THz microwire para- meters, we use the solution of the vectorial Maxwell’s equation in cylindrical coordinates for a circular cross-section step-index fiber with an infinite air cladding [28]. At any interface between two materials with no surface charge, since the normal components of the electric displacement field are continuous, there is a discontinuity in the electric-field strength; i.e., the electric-field enhances on the low refractive index side. The strength of the enhanced electric field depends on the square of the ratio of refractive indices and the electric field strength at the high index side. As a result, a higher enhanced electric field is formed at a low-index discontinuity interface, if it is introduced within a region of a waveguide structure where the electric field is originally stronger than the other regions. The enhanced electric field at the low index side of any discontinuity rapidly decays away from the interface discontinuity. However, if the dimension of the low index discontinuity is at the sub-wavelength scale, the decay of the evanescent field within the discontinuity is minimal. Thus a localized intensity enhancement can be achieved throughout the discontinuity region [25, 26]. The concept of field enhancement within low-index discontinuity (as explained above) is demonstrated in Fig. 1(b) which shows the normalized z-component of the Poynting vector (Sz = 12�E× �H · zˆ) profile of the fundamental mode of the porous fiber along the arrow shown in Fig. 1(a). The refractive index profile along the same axis is also shown, demonstrating where the sub-wavelength air-holes are located. The refractive indices n 1 = 1.6 and n0 = 1 refer to material (PMMA) and air refractive indices, respectively. The field is enhanced at the each air-material interface and stays localized in the sub-wavelength air-holes, where the refractive index is n0. This phenomenon occurs for all the sub-wavelength air-holes in the structure as can be seen in Fig. 1(c). Since the enhancement coefficient at each interface is constant, n 21, the intensity of the localized field depends on the location of the sub-wavelength air-hole posi- tion. The closer the sub-wavelength air-hole to the center of the fiber, where the electric field intensity is stronger, the stronger the localized electric field intensity. Therefore the envelope of the intensified profile of the field has the same profile pattern of the air-clad fiber without sub-wavelength holes. For the chosen porous fiber dimensions, the power profile envelope has a Gaussian shape, Fig. 1(d). To find out the effect of these sub-wavelength air-holes on the fiber parameters, we evaluate the loss (effective material loss) and the confinement of THz propaga- #94241 - $15.00 USD Received 25 Mar 2008; revised 30 May 2008; accepted 30 May 2008; published 2 Jun 2008 (C) 2008 OSA 9 June 2008 / Vol. 16, No. 12 / OPTICS EXPRESS 8848 air cladding dhole dcore (a) 0 100 200 300 400 0 0.2 0.4 0.6 0.8 1 Radial distance (μm) N or m al iz ed S z n1 n0 (b) (c) (d) Fig. 1. (a)- Cross section and geometrical definitions of the triangular lattice porous fiber. (b) The normalized z-component of the Poynting vector, Sz, profile along the cross-section shown in (a), of the fundamental mode of a polymer porous fiber with core radius of dcore/2 = 200 μm, hole radii of dcore/2 = 20 μm and 37% porosity at f = 0.5 THz (λ = 600 μm). The dashed line represents the core to cladding interface and the lower solid line represents the refractive index profile. (c) 2D and (d) 3D view of the normalised Sz of the porous fiber. tion in the following section. 3. Loss and confinement In this section we calculate the optical parameters—i.e. power fraction, effective material loss and fraction of power lost as radiation in bend described below—of the porous fiber and com- pare them with those of a microwire. For convenience we consider three different regions within a porous fiber; sub-wavelength air-holes (region I), solid core material (region II) and the air cladding (region III). The fraction of the guided mode power (η) that is located within each #94241 - $15.00 USD Received 25 Mar 2008; revised 30 May 2008; accepted 30 May 2008; published 2 Jun 2008 (C) 2008 OSA 9 June 2008 / Vol. 16, No. 12 / OPTICS EXPRESS 8849 region can be defined as: ηx = Px Ptotal = ∫ Ax SzdA∫ A∞ SzdA , (1) where, Sz is the z-component of the Poynting vector, the subscript x = {I, II, III} represents each region, dA = rdrdφ , and A∞ and Ax are the infinite and region x cross-sections, respec- tively. Figure 2 shows the power fractions versus fiber diameter of three porous fibers with fixed porosities—61% (black lines), 70% (green lines) and 74% (blue lines)—at f = 0.5 THz (λ = 600 μm). At this frequency the optical parameters of the PMMA are n = 1.6 and αm = 4.2 cm−1. The solid, dashed and dotted lines demonstrate the power fraction in the re- gions I, II, and III, respectively. To maintain a fixed porosity, while the diameter changes for each structure, we keep the ratio of the air-holes to core diameter constant (0.1, 0.8 and