IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 24, NO. 11, JUNE 1, 2012
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Understanding All-Solid Honeycomb Photonic Bandgap Fibers Chang Yang, Jinxu Bai, Yanfeng Li, and Aimin Wang
Abstract— Plane-wave expansion calculations show that a broad higher-order bandgap can be formed in all-solid honeycomb photonic bandgap fibers, different from fibers based on the typical triangular lattice. Both density of states plots and Blochmode field distributions reveal that the higher-order bandgap results from a re-ordering of the linearly polarized modes that form the cladding states, that is, those modes with high azimuthal order have more nodal lines than the cladding structure can support and thus are pushed away to high frequencies. Index Terms— Honeycomb lattice, photonic bandgap, photonic bandgap fiber, photonic crystal fiber, plane wave expansion method.
I. I NTRODUCTION
T
HE ability to guide light in a low-index core, air in particular, by the photonic bandgap effect has made photonic bandgap fibers (PBFs) very attractive in a wide range of applications [1, 2]. One form of PBFs is composed of high-index inclusions in a low-index background (most often Ge-doped silica in pure silica), known as all-solid photonic bandgap fibers (ASPBFs) [3]. Having very interesting bend loss property [4, 5] and typically normal-zero-anomalous dispersion profile [3] within the bandgaps, ASPBFs can be used as tunable bandpass filters [6], intracavity wavelengthselective elements [7], dispersion-compensation components [8], and nonlinear optical devices [9]. A broad and robust bandgap is often needed for the above applications. Although ASPBFs are mostly based on the triangular lattice, some new or improved designs such as depressed-index layer [10], ring [11] and broken-ring [12] structures have been proposed to tailor their properties. In this letter, an understanding of the bandgap properties of ASPBFs based on the honeycomb lattice, which has
Manuscript received October 8, 2011; revised January 31, 2012; accepted March 8, 2012. Date of publication March 13, 2012; date of current version April 24, 2012. This work was supported in part by the National Basic Research Program of China under Grant 2010CB327604 and Grant 2011CB808101, in part by the National Natural Science Foundation of China under Grant 61078028, Grant 60838004, Grant 60907040, and Grant 60927010, and in part by the 111 Project under Grant B07014. C. Yang, J. Bai, and Y. Li are with the Ultrafast Laser Laboratory, College of Precision Instruments and Optoelectronics Engineering, Tianjin University, Tianjin 300072, China, and also with the Key Laboratory of Optoelectronics Information Technology (Tianjin University), Ministry of Education, Tianjin 300072, China (e-mail:
[email protected]). A. Wang is with the Institute of Quantum Electronics, State Key Laboratory of Advanced Optical Communication System and Networks, School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China. Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LPT.2012.2190761
been used for hollow-core PBFs [13, 14], referred to as ASHPBFs hereafter, is presented. Numerical results by plane wave expansion methods show that different from triangularlattice ASPBFs, ASHPBFs feature a broad higher-order bandgap, which can be utilized for bandgap guidance within a broad transmission window. It is found that the bandgap is formed from a re-ordering of the linearly polarized cladding modes, namely, those modes with high azimuthal order have more nodal lines than the cladding structure can support and thus are pushed away to high frequencies. This can be understood by considering ASHPBFs as a special case of the broken-ring design [12]. II. R ESULTS AND D ISCUSSION Shown in Fig. 1(a) is the schematic of the ASHPBF. High-index Ge-doped rods with a refractive index of 1.48614 in a silica background (refractive index 1.457) are used for bandgap calculations. This amounts to an index contrast of 2%. The density of states (DOS) map for this fiber structure is calculated by a fixed-frequency plane wave expansion method [15] and is given in Fig. 2(a), where the bandgaps are shown in red. The results are normalized to , the distance between two-nearest rods [13, 14]. As is well understood from the antiresonant reflecting optical waveguide (ARROW) model [16, 17], the photonic bands in ASPBFs can be considered as arising from the coupled modes of the individual high-index rods, linearly polarized LPlm modes in the weak-guidance approximation, where l counts the number of angular nodes and m-1 counts the number of radial nodes. Light is guided in the low-index core when the high-index rods are anti-resonant, corresponding to the bandgaps. In the DOS map for a triangular-lattice ASPBF, these LPlm modes would appear in order of frequency because the higherorder modes cut off at high frequencies. For D = 0.45, there are only the first few LP modes (LP01 , LP11 , LP21 and LP02 ) within the normalized frequency range of k0 up to 50. However, in the DOS map for the ASHPBF shown in Fig. 2(a), there are only two broad photonic bands, both of which seem to be a merge of several narrower ones (note the variation of the black and white density scales). Therefore, the ASHPBFs behave differently from triangular-lattice ASPBFs. To understand the bandgap formation in the ASHPBF, we notice the similarity between the ASHPBF and the broken-ring ASPBF in [12]: if the parameters of the six rods in a ring in the broken-ring ASPBF shown in Fig. 1(b) are
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Fig. 1. Schematics of two fiber structures. (a) ASHPBF, where D is the rod diameter and is the distance between two-nearest rods. (b) Brokenring ASPBF as in [12], where D is the rod diameter, is the pitch in this triangular lattice of rings of rods, and d is the ring diameter. The dotted circle in (a) can be considered a corresponding six ring unit in (b) and can also be the core region when the six rods are removed.
Fig. 2. DOS maps calculated for (a) ASHPBF with D = 0.45 , (b) broken-ring ASPBF with D = 0.45 and d = 0.5 , and (c) a brokenring ASPBF with D = 0.45 and d = 0.6 . Grey scale represents DOS of propagating states (white = high DOS) and the bandgaps are shown in red. The rod modes from which the bands arise are labeled along the top in (b) and (c). All the results are expressed in effective index β/k0 as a function of normalized frequency k0 . The horizontal line is the silica line 1.457.
chosen such that d = 2 and = 3 (D remains the same), the broken-ring design becomes the honeycomb lattice in Fig. 1(a) with the dotted circle showing a corresponding six rod unit. Hence, the ASHPBF can be treated as a special case of the broken-ring ASPBF, as will be demonstrated by a study of the change of the DOS map of the broken-ring ASPBF. Figures 2 (b) and (c) show how the variation of the distance parameter d affects the DOS map of the broken-ring ASPBF. Because and are different, the results are normalized to for easy comparison. In Fig. 2(b), we can clearly
IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 24, NO. 11, JUNE 1, 2012
Fig. 3. Real parts of the E x field at a normalized propagation constant β = 113.10, illustrating LP-like Bloch modes at various Brillouin zone points and normalized frequencies k0 calculated using a broken-ring ASPBF in Fig. 1(b) with d = 2/3 . (a) LP01 mode at point and k0 = 77.10. (b) LP11 mode at M point and k0 = 77.14. (c) LP21 mode at point and k0 = 77.16. (d) LP31 mode at M point and k0 = 77.19. Blue and red represent positive and negative field maxima, respectively, with white being zero. The circled region can be regarded as a six-rod unit in the corresponding ASHPBF.
distinguish between four narrow bands below k0 = 30 and another group of bands above k0 = 30 when d = 0.5 . When d is further increased to 0.6 , which is close to the case of the honeycomb lattice (d = 2/3 ), the first four modes come closer due to a stronger coupling between rings of rods, and so do the next group of modes. When d is set for the honeycomb lattice, we recover Fig. 2(a), where the two groups of modes have merged into their respective broad band because of a further enhanced coupling. The re-grouping of the cladding states can be understood by studying the distributions of the four modes in the first group of Fig. 1(b) or (c), as shown in Fig. 3, where the real parts of the fields are used to reveal the relative phase information of the modes. These Bloch mode field distributions are calculated using the block-iterative plane wave expansion method [18] for a broken-ring ASPBF with d = 2/3 , which means that a supercell of the ASHPBF is used. This is done because the high-symmetry Brillouin zone points for the supercell structure used in Fig. 3 will no longer be such points in the corresponding ASHPBF. According to the ARROW model [16, 17], the photonic bands arise from the coupling of LP modes. Above the silica line, they still retain their identities as LPlm modes. In a ring structure, the LPlm modes with m >1 are pushed to high frequencies because mode fields tend to be concentrated in a high-index region whereas the central low index part in the ring structure means that there is less dielectric space for those modes with more radial variation [11]. Similarly, the distribution of the six rods in a ring imposes a constraint on the azimuthal order l of the LPlm modes: the spaces between two adjacent rods can be viewed as possible nodal lines for those modes with l ≤ 3 (see Fig. 3), and the higher-l order modes with l > 3 have to be shifted to higher frequencies by the ring structure because there is not much dielectric space for their azimuthal variation [12]. As a result, the various LPlm modes are re-grouped: the swept-away family of high-m order LPlm modes, the frequency-shifted family of LPl1 modes with l > 3, and the less-affected family of LPl1 modes with l ≤ 3. The mode designations are then
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that the broad higher-order bandgap in ASHPBFs result from a re-grouping of the LP modes that form the cladding states. The validity of the treatment of the ASHPBF as a special case of the broken-ring ASPBF is established. The ASHPBF may find applications in nonlinear optics, dispersion compensation, and other situations, where a broad transmission window is desirable. ACKNOWLEDGMENT
Fig. 4. Variation of the bandgap of the ASHPBF as a function of normalized wavelength λ/ for three relative sizes of high-index rods D/ (left and bottom axes) and the GVD of a particular ASHPBF with D/ = 0.45 and = 3.0 μm where a ring of six rods is removed to form the fiber core (right and top axes). The two wavelengths λ L and λU , where the bandgap boundaries intersect the horizontal silica line, define roughly the lower and upper wavelengths of the range of transmission, respectively.
labeled as in Fig. 2. The LP31 mode defines the upper limit of the family of modes that are less affected, namely, LPl1 modes with l ≤ 3. Above it, a broad higher-order bandgap is formed because LPl1 modes with l > 3 are grouped and pushed to higher frequencies. The variation of the bandgap of the ASHPBF for different rod diameters D is shown in Fig. 4 (left and bottom axes). It is observed that the gaps become deeper when D is increased because of stronger mode coupling and that the gaps are moved to longer wavelengths due to the increased fraction of high-index materials. As with triangular-lattice ASPBFs, medium-sized high-index rods lead to relatively broad bandgaps. The relative size of the bandgap, defined in terms of the ratio of the gap width to the mid-gap wavelength, i.e., 2(λU − λ L )/( λU + λ L ), is 56% for D/ = 0.45 and the midgap wavelength is around λ/ = 0.25. Using these values, we can get the approximate scale for an ASHPBF that can transmit femtosecond pulses centered at 800 nm for possible nonlinear effects [9]: ≈3 μm. The bandgap data would roughly imply a transmission range of 540 nm to 960 nm. The group velocity dispersion (GVD) with material dispersion taken into account for the ASHPBF with = 3 μm, where six high-index rods as indicated by the dotted circle in Fig. 1(a) are removed to form the core, is also shown in Fig. 4 (right and top axes). Due to the large material dispersion at short wavelengths and also the large core of the fiber and hence weak waveguide dispersion, the GVD is mostly negative. Besides, the large core makes the mode line shallow in the gap and interest the long-wavelength gap edge at a shorter wavelength than expected. A better core design (with size variation or even through doping change) [13, 14] can be envisaged but will not be pursued here. III. C ONCLUSION The bandgap properties of ASHPBFs are studied by numerical simulations. DOS maps and mode field distributions reveal
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