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Predicting Mode Properties of Porous-Core Honeycomb Bandgap THz Fibers by Semi-Analytical Theory Jintao Fan, Yanfeng Li, Member, OSA, Ximing Zhang, Minglie Hu, Member, OSA, Lu Chai, and Chingyue Wang
Abstract—We describe a semi-analytical approach for modeling fibers which guide THz radiation in a porous core by the photonic bandgap effect due to a honeycomb cladding of air holes. These porous-core honeycomb bandgap THz fibers (PCHBTFs) are modeled as equivalent step-index fibers (SIFs). The effective core index of the SIFs can be solved analytically as that of the fundamental space-filling mode of the triangular lattice comprising the porous core. The cladding index is taken as that of the bandgap edge with the lower index and is computed numerically. The validity of the analogy between PCHBTFs with SIFs is established by comparing the mode profiles, mode number, mode indices, and the fraction of mode power in the core obtained for a 19-cell core multimode THz fiber by the semi-analytical theory with results by finite element method. The subtle difference in mode degeneracy of the PCHBTFs and SIFs is briefly discussed. The semi-analytical theory also predicts that a seven-cell core PCHBTF is effectively single mode with higher-order modes pushed close to the bandgap edge. Index Terms—Finite element method (FEM), honeycomb lattice, photonic bandgap, photonic crystal fiber, porous core, step-index fiber, THz fiber, THz radiation.
I. INTRODUCTION N parallel with the rapid advances in the generation, detection, and application of terahertz (THz) radiation [1]–[3], considerable efforts have been made to develop waveguides for the THz range [4], [5]. The development of low-loss and lowdispersion THz waveguides will lead to more compact THz systems with improved functionalities. A class of THz dielectric waveguides is based on the photonic bandgap (PBG) effect [6] like their optical counterparts [7], [8]. The issue of propagation loss is, nevertheless, more prominent with THz fibers because the material loss is significantly higher in the THz range. Therefore, ideal THz fibers are those that can maximize the mode power inside an air core [9]–[12]. Currently the problem lies in the difficulty to produce good-quality, long-
I
length, and broadband hollow-core PBG THz fibers, which are typically based on polymers. To enable bandgap guidance with low loss while allowing easy fabrication, a porous-core honeycomb bandgap THz fiber (PCHBTF) was proposed in [13] and later demonstrated [5] by the Danish group. Combining a well-studied honeycomb lattice in the optical regime [14]–[17] and a porous core design in the THz field [18]–[21], the PCHBTF has the following advantages [5], [13]: first, the cladding PBG structure is based on a honeycomb lattice, which is known [14]–[17] to allow broad bandgaps for smaller air holes than the triangular lattice; second, although not guiding in air, the porous core containing a significant number of air holes is still able to reduce the loss compared to a solid core; finally, the air holes in the core and cladding have the same size, making fabrication easier. However, the composite porous core of the PCHBTF makes the fiber design complicated and mode classification for this fiber also needs to be investigated. Moreover, Ref. [5], where the design of a PCHBTF and its characterization by THz time-domain spectroscopy is well described, postulated that the existence of higher-order modes played a role in shaping the transmission spectra. For such a heterostructured photonic crystal fiber (PCF) as it is called in [22], neither a comprehensive study of the mode classification nor a simple design strategy has been presented so far to our knowledge. This work presents a semi-analytical analysis of the mode properties of the PCHBTFs to bridge the gap. We demonstrate that a close analogy can be established between the PCHBTF in [5] and an equivalent step-index fiber (SIF) in terms of the mode field profiles, the mode numbers, and even quantitatively the mode indices. Based on the analogy, a seven-cell core design is proposed to reduce the number of higher-order modes.
II. SEMI-ANALYTICAL THEORY Manuscript received September 22, 2014; revised January 25, 2015; accepted February 11, 2015. Date of publication February 15, 2015; date of current version March 16, 2015. This work was supported in part by the National Basic Research Program of China (Grants 2014CB339800, 2010CB327604, and 2011CB808101), in part by the National Natural Science Foundation of China (Grants 61377047, 61377041, 61322502, 61077083, and 61027013), and in part by the Program for Changjiang Scholars and Innovative Research Team in University (Grant IRT13033). (Corresponding author: Yanfeng Li.). The authors are with Ultrafast Laser Laboratory, College of Precision Instrument and Optoelectronics Engineering, Key Laboratory of Optoelectronic Information Technology (Ministry of Education), Tianjin University, Tianjin 300072, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2015.2404354
Fig. 1 shows schematically the idealized PCHBTF structure under study and its working principle. The PCHBTF is composed of a honeycomb lattice, which does not form bandgaps crossing the air line unless for very large values of the air hole size [22]. Therefore, on the index scale shown in Fig. 1(b) for a single frequency, the effective indices of the bandgaps (two are illustrated) fall between those of the material (nTOPAS ) and air (nair ). To achieve waveguiding exclusively based on the PBG effect [6]–[8], the core index (ncore ) must fall below the effective index of the cladding (nclad ) and at the same time between those of a bandgap, defined by nhigh and nlow . It should be noted
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Fig. 2. In analytical solution of n c o re , the hexagonal unit cell in (a) is approximated as a circular one with the same area in (b).
Fig. 1. (a) Idealized structure considered in this work. The circles represent air holes and the shaded region (19 air holes here) is the porous core. The parameters used in this work are the same as in [5]: d = 165 μm, Λ = 360 μm, Dc = 0.8 mm, and D = 3.65 mm. (b) Illustration of the principle of the PCHBTF as shown on the left. Two bandgaps are schematically show. n T O PA S is the index of the fiber material (TOPAS in this work) and n a ir = 1 is that of air. The honeycomb cladding has an effective index n c la d and the porous core n c o re . n h ig h and n low define, respectively, the edges of a bandgap. Within a bandgap, there could be a number of modes (shown as yellow lines) whose effective index falls between n h ig h and n low . See text for further detail.
that in what follows we use nhigh and nlow to refer to the bottom and top edges of a bandgap (in terms of frequency), respectively. The porous core (the central shaded area in Fig. 1(a) including the air holes) fulfils this condition: for the same air hole size, the triangular lattice of air holes has a higher porosity than the honeycomb lattice and hence ncore < nclad . A confined mode is defined [23] as one whose effective index nm o de lies within the bandgap and meanwhile below ncore , that is, nlow < nm o de < nhigh , nm o de < ncore .
(1a) (1b)
Precise modeling and predicting the mode properties of the complex PCHBTF structure as shown in Fig. 1(a) requires numerical methods. However, ever since the first PCF was reported, there have been continuous efforts to establish an analogy between PCFs and conventional fibers, e.g., for solid-core PCFs [24]–[26], hollow-core PCFs [23], [27], and all-solid PBG fibers [28], [29]. In all these cases, the core region is homogeneous [22] and it has been shown [27], [29], [30] that the core modes in these PCFs share similar mode profiles with their conventional counterparts and therefore the modes can be classified and designated accordingly. The analogy can even go so far [23], [29] as to quite accurately predict the number of modes, the mode indices and cutoffs. In this work, we extend this analogy further to the study of PCHBTFs or heterostructured PCFs as they are referred to as in [22]. The semi-analytical theory is to model the complicated PCHBTF as an SIF, whose cladding index is determined by the top edge of the bandgap, nlow . Since no analytical solution exists to determine the bandgap edges of the honeycomb lattice, the PBGs are computed numerically using the MIT
Photonic- Bands package (MPB) [31]. For the porous core of the PCHBTF, the core index can be defined as the effective index of the lowest mode that could propagate in an infinite triangular lattice of air holes, which happens to be the fundamental space-filling mode (FSM) commonly defined for a PCF cladding [6]–[8], [24]. ncore can be solved analytically as outlined below [26], [32]. For a hexagonal unit cell shown in Fig. 2(a), symmetry constrains [32], [33] require that the longitudinal components of the electromagnetic fields Ez and Hz of the FSM both be zero at P, a point at the intersection of two symmetry axes. The hexagonal unit cell is most accurately approximate by a circular unit √ cell with the same area, i.e., R = Λ[ 3/(2π)]1/2 [26], [33], as shown in Fig. 2(b). In the circular unit cell, the zero condition for the longitudinal components is extended to all points on the outer circumference. It follows that in the cylindrical coordinate system, the solutions for both Ez and Hz outside the air hole have the following form: Pl (ur) = Jl (ur)Yl (uR) − Yl (ur)Jl (uR) .
(2)
In the above equation, J and Y are Bessel functions of order l, r 1/2 is cylindrical coordinate,u = k n2TOPAS − n2core , and k = 2π/λ is wave number in vacuum with λ being the wavelength. Inside the air hole, the longitudinal fields are expressed by modified Bessel function I of order l. The next steps are, as is for SIFs [34], to express the transverse electromagnetic components as a combination of Ez and Hz and then impose the continuity conditions of the tangential field components at the air hole boundary r = a = d/2. The characteristic equation for the FSM (with l = 1) is finally obtained as [26], [32]: 2 P1 (ua) I1 (wa) nTOPAS P1 (ua) n2air I1 (wa) + + uaP1 (ua) waI1 (wa) uaP1 (ua) waI1 (wa) 2 2 2 1 1 = + n2core , (3) ua wa where the primes denote differentiation with respect to the ar 1/2 gument and w = k n2core − n2air . Once the effective core index ncore is obtained from (3), the well-known results [34] of mode equations (HE, EH, TE and
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TM) and mode power distributions for SIFs can be used to calculate the quantities of interest. III. RESULTS AND ANALYSIS To establish the validity of the semi-analytical approach proposed for the analysis of PCHBTFs, we first compare the analytical results with those obtained using commercial finite element method (FEM)-based software COMSOL multi- physics with a perfectly matched layer outer boundary. The PCHBTF parameters are the same as in [5]: d = 165 μm, Λ = 360 μm, and D = 3.65 mm. Because there is no clear core-cladding boundary for the equivalent SIF, the SIF core size Dc is taken to be 0.8 mm as in [5]. In addition, nTOPAS is assumed to be 1.5235 for all frequencies. A. V Parameter and Number of Modes The normalized frequency V can well predict the number of modes of an SIF [34]. For a bandgap fiber, however, the prediction of the number of modes based on the V parameter needs special consideration, which is described in detail in [23]. For the PCHBTF with the parameters given above, there exist two wide bandgaps between 0.5 and 2 THz that can be utilized for waveguiding. Even for a constant value of nTOPAS , ncore , nhigh , and nlow are a strong function of frequency, as illustrated in Fig. 3(a) for the second bandgap. Because the mode indices have to satisfy both conditions in (1), each bandgap can be divided into two regions. In the region above f0 , at which the core line (ncore line) crosses the bottom edge (nhigh line) of the bandgap, (1) can be combined into a single equation since ncore < nhigh : nlow < nm o de < ncore
(f ≥ f0 ).
(4)
A single V parameter can, therefore, be defined for the PCHBTF in the same way as for an SIF in this region, namely, Dc n2core − n2low . (5) V =k 2 In the region below f0 , (1) still applies. However, since nhigh < ncore , modes whose effective index falls above nhigh and below ncore are not guided by the PBG effect. Therefore, in this region the acceptance angle of the propagating modes is a hollow cone formed between the V defined in (5) and another V [23] defined as: Dc 2 ncore − n2high . (6) V =k 2 The dependence of the two V parameters on the frequency for the second bandgap is shown in Fig. 3(b), and the number of modes reaches maximum near f0 [23]. V is about 4.8 near f0 = 1.4 THz, implying that, according to SIF theory [34], in addition to the fundamental HE11 modes, the group of HE21 , TE01 and TM01 modes and the next group of higher-order HE12 , HE31 and EH11 modes could also propagate within the second bandgap. For the first bandgap, V is about 2.9 near f0 = 0.7 THz, implying that both the fundamental HE11 modes and the group of HE21 , TE01 and TM01 modes could propagate within
Fig. 3. (a) Second bandgap of the PCHBTF under study calculated by MPB and the effective core index calculated analytically based on (3). (b) Frequencydependent V parameters as defined in (5) and (6) for the second bandgap using index values from (a).
the first bandgap. These predications agree well with simulated results, as will be shown below. B. Mode Classification As described above, the V parameters properly defined for the PCHBTF can be used to predict the number of core modes and it turns out that the PCHBTF reported in [5] is multi-mode in both bandgaps. Here, the guided modes are classified by examining the transverse electric field patterns in comparison with those of their SIM counterparts [27], [29], [30], as shown in Fig. 4 for the core modes at 1.6 THz within the second bandgap. As can be observed, striking similarities are found between the electric field profiles of the core modes of the PCHBTF and those of the corresponding SIFs, the subtle difference being that the electric field will try to avoid the air holes and be concentrated in the material regions [6] in the case of the porous core fiber. This observation is not surprising in view of the fact that similar observations have already been made in other PCFs [27], [29], [30]. As a result, the PCHBTF modes are designated accordingly. As has already been established by symmetry analysis [27]–[30], the PCHBTF shown in Fig. 1 have C6v symmetry, whose modes are either non-degenerate or occur in degenerate pairs. The modes of the PCHBTF that can
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be found within the second bandgap have the same degeneracy with their SIF counterparts except the HE31 modes. These two modes are non-degenerate because each of the field can exhibit the full fiber symmetry. The non-degeneracy is also borne out by numerical results, where the effective index difference between the two HE31 modes is several orders of magnitude larger than that of a degenerate pair. Consequently, in Fig. 4, these two near-degenerate modes are both shown and named with a superscript while only one of the degenerate pairs of the other modes is shown. C. Mode Indices and Fraction of Mode Power in the Core
Fig. 4. Calculated transverse electric fields of core modes of the PCHBTF of Fig. 1 (left column) and of the equivalent SIF (right column) at 1.6 THz within the second bandgap. Each PCHBTF mode is labeled based on the field similarity with the corresponding SIF mode profile. See text for more detail.
Having verified the number of modes that can propagate in the PCHBTF and established their analogy with the corresponding SIF modes, we now examine the accuracy of the semi-analytical method in terms of mode indices and mode confinement in comparison with numerical results. Fig. 5(a) and (b) show, respectively, the effective indices of the HE11 mode and the first higher-order group of TM01 , HE21 , and TE01 modes obtained by semi-analytical and numerical means. These are all the modes that can propagate within the first bandgap, as discussed previously. The next higher-order group of HE31 , EH11 , and HE12 modes are shown in Fig. 5(c) for the second bandgap. The two near-degenerate HE31 modes are hardly distinguishable in Fig. 5(c) because of the proximity of their indices. The PCHBTF modes group as in an SIF. The most noticeable feature is that the cut-off property of the modes is different at the two ends of a bandgap. Due to the nature of bandgap guidance and the PBG shapes, the higher-order modes are guided while the fundamental modes cut off at the low frequency end, in accordance with the guidance cone [23] described earlier and shown in Fig. 3(b). At the high frequency end within each bandgap, the higher-order modes cut off first because this region is described by a single V as in an SIF. Secondly, the semi-analytical theory yields results in good agreement with the numerical simulations. The agreement is especially good for the fundamental modes and the maximum index difference (at bandgap edges) obtained by the two methods is below 2% for all groups of higher-order modes. The fraction of the mode power localized in the core (Dc ) for the fundamental HE11 mode is a quantity measuring the degree of mode confinement, and for the porous-core fiber here, is also closely tied with the fiber loss. A comparison of this quantity obtained by the two approaches is made in Fig. 6, where the trend of the mode power in the core is quantitatively well predicted. The discrepancy is larger at the gap edges, as is observed when the SIF analogy is used to estimate the mode radius of a hollowcore PBG fiber [23]. The reason, we believe, is the same and is due to the fundamentally different guiding mechanism of the PCHBTF from an SIF [23]: PBG guiding is based upon multiple reflections in the photonic crystal cladding where mode properties depend sensitively on the field distribution within the composite materials, whereas in an SIF the mode field expands or shrinks more quickly because of index guiding. A monotonic increase of the mode power in the core for the SIF at the low frequency ends is a direct consequence of the increase of the V parameter as shown in Fig. 3(b). In contrast, below f0 the
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Fig. 6. Comparison of fraction of mode power in the core obtained by semianalytical theory (solid lines) and by FEM (dashed lines) for the two bandgaps of the 19-cell core PCHBTF and the second bandgap of a seven-cell core PCHBTF.
Fig. 7. Comparison of effective indices of all modes within the second bandgap of the seven-cell core PCHBTF obtained by semi-analytical theory (solid lines) and by FEM (dashed lines).
Fig. 5. Comparison of effective mode indices for the 19-cell core PCHBTF as shown in Fig. 1 (a) obtained by semi-analytical theory (solid lines) and by FEM (dashed lines). (a) All modes within the first bandgap, and modes within the second bandgap: (b) HE1 1 modes and the group of TM0 1 , HE2 1 , and TE0 1 modes and (c) group of HE3 1 , EH1 1 , and HE1 2 modes. Note that two HE3 1 modes are shown but their differences are indistinguishable on the scale shown here.
modes in the PCHBTF can only be guided in a cone of the V parameter defined by (5) and (6) and therefore the power of the HE11 mode in the fiber core will decline as a result of its being close to cut-off. D. A Seven-Cell Core Design The results above show that the 19-cell core PCHBTF as reported in [5] is indeed multimode and excitation of higherorder modes would degrade the signal quality. For the PCHBTF,
the core and cladding air holes have the same size, an advantage for fabrication. Therefore, the strategy to reduce the number of higher-order modes is to use a smaller-core size, the natural choice in this case being a seven-cell porous core. The semianalytical approach can be pushed further to predict and design a seven-cell core PCHBTF for transmission. For a seven-cell porous core, all the relevant quantities used in the semi-analytical theory are available: the cladding index nlow , the core index ncore , and the new core size (chosen as half of the 19-cell core, i.e., Dc = 0.4 mm). All other parameters are the same as in Fig. 1. The results for the mode indices are given in Fig. 7. If the core size is reduced, all the mode indices shown in Fig. 5 for the 19-cell core fiber will be pushed down toward the upper bandgap edges with the highest-order modes being cut off first. For the smaller seven-cell core fiber considered here, within the first bandgap the HE11 mode indices turn out to be very close to nlow and hence even these fundamental modes will not be considered as well-guided modes. The SIF analogy, however, always predicts the existence of a fundamental mode pair for V < 2.405 [34]. These results is a reminder that the SIF analogy should be used with caution when the modes are not well-guided or very close to the bandgap edge. Within the second bandgap, in addition to the fundamental modes the
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second higher-order group of modes remain but they are also very close to the bandgap edge, as shown in Fig. 7. As a result, these higher-order modes are also poorly guided and suffer from great loss during transmission. The seven-core PCHBTF is in a sense single mode. It should be noted that more than 80% of the mode power is still in the seven-cell core (Fig. 6) at the center of the bandgap even though the core size is half the original size, indicating that there would be no significant rise of the fiber loss. The whole working frequencies are red-shifted within the bandgap, as would be expected from a reduction of the mode indices. A related issue is the fraction of mode power inside all the air holes in the core region, which is very important for the design of THz dielectric fibers to minimize material absorption [18]–[21]. It is possible to compute the fraction of mode power inside a single hole in the process of computing the effective core index and use it as an average to compute the overall mode power inside the air holes. The equations, however, are lengthy and we plan to present the procedure elsewhere in an effort to study porous THz fibers. Modeling the heterostructured waveguide geometries as described here is nontrivial and the semi-analytical approach requires numerical computation of the bandgap edges only once for a particular lattice structure and all the remaining calculations are based on conventional fiber theory [34], thus providing a fast and simple design approach. IV. CONCLUSION We have established the validity of using conventional fiber optics to model complicated THz fibers, where both the core and cladding are non-homogeneous. The honeycomb cladding of the fibers considered enables bandgap guiding and the composite core increases the porosity of air to reduce absorption loss. The semi-analytical theory described in this work provides quantitatively accurate results in terms of mode number, mode indices and fraction of mode power in the core. We believe that this SIF-based semi-analytical method can facilitate the design of similar complex waveguides as well. REFERENCES [1] P. H. Siegel, “Terahertz technology,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 910–928, Mar. 2002. [2] B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nature Mater., vol. 1, no. 1, pp. 26–33, Sep. 2002. [3] M. Tonouchi, “Cutting-edge terahertz technology,” Nature Photon., vol. 1, no. 2, pp. 97–105, Feb. 2007. [4] S. Atakaramians, S. Afshar V., T. M. Monro, and D. Abbott, “Terahertz dielectric waveguides,” Adv. Opt. Photon., vol. 5, no. 2, pp. 169–215, Jun. 2013. [5] H. Bao, K. Nielsen, H. K. Rasmussen, P. U. Jepsen, and O. Bang, “Fabrication and characterization of porous-core honeycomb bandgap THz fibers,” Opt. Exp., vol. 20, no. 28, pp. 29507–29517, Dec. 2012. [6] J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. Princeton, NJ, USA: Princeton Univ. Press, 2008. [7] J. C. Knight, “Photonic crystal fibres,” Nature, vol. 424, pp. 847–851, Aug. 2003. [8] P. St. Russell, “Photonic-crystal fibers,” J. Lightw. Technol., vol. 24, no. 12, pp. 4729–4749, Dec. 2006. [9] J.-Y. Lu, C.-P. Yu, H.-C. Chang, H.-W. Chen, Y.-T. Li, C.-L. Pan, and C.-K. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett., vol. 92, no. 6, p. 064105, Feb. 2008.
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Authors’ biographies not available at the time of publication.
