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Abstract
Designing a cavity with a high quality factor for omnidirectionally emitting laser (OEL) can extend its potential applications in optical communication and biomedical detection. We demonstrate a method including five steps to design a high-Q cavity for OEL using a one-dimensional topological photonic crystal heterostructure. A Si/SiO2 fiber cavity for OEL with solid gain medium Er-doped SiO2 is designed following our design steps. The designed fiber can axially transmit the pump energy at low confine loss and act as a cavity for the radial emission of the exited beam, simultaneously. The quality factor of this fiber cavity is on the order of magnitude of 108. Moreover, a method of further improving the Q-factor is proposed. The results in this paper are not restricted to the solid gain medium, and they also can be applied to designing a cavity for optofluidic OEL or quantum dot OEL. Our study may provide not only the reference for OEL manufacture, but also a route for improving the performance of OEL.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
On the basis of whispering gallery modes(WGMs), omnidirectional emission was initially achieved mainly by pumping the gain media, which are filled in rotationally symmetric resonator, in the direction perpendicular to the symmetric axis [1]. Fiber cavity for this kind of omnidirectional emission has been applying to fields such as biological detection and optofluidic microring resonator laser for several years [2,3]. Researchers in fields of biological detection and disease diagnostics pay more attentions on designing various microcavities using side-pumping for advantages such as the convenience in changing the fluidic gain medial [4–6]. However, omnidirectional emission realized by the excitation of WGMs is the diffraction loss or the scattering light at the rough surface, which limits the control over light output coupling [7]. In addition, the complexity of the arrangement resulting from side-pumping method becomes another obstacle impeding the application of omnidirectionally emitting laser(OEL). To overcome these drawbacks, part of researchers started to discard WGMs and proposed axial-pumped cylindrical photonic-bandgap fiber cavity, which can support purely radial modes [8]. Then, OEL begins to show the potential in field such as omnidirectional imaging, biomedical detection, and photo- dynamic therapy [9]. The previous studies of OEL without utilizing WGMs concentrated on the gain media [7–9], but few efforts are taken in the design of cavity with higher quality factor(Q-factor) for OEL, which, however, is also of great significance in practical application. Due to kinds of unique properties, topological insulators, especially topological photonic crystal [10,11] and topological phononic crystal [12,13], show outstanding performance in variety of fields, which provides researchers a nice suggestion on the design of high-Q cavity for both traditional laser [14,15] and OEL.
In this paper, we report on not only the method (five steps) to design a high-Q cavity for OEL but also the way to further improve the Q-factor of the cavity taking advantage of one-dimensional topological photonic crystal heterostructure. The design process is guided by the band theory. The performance of the cavity is exhibited by finite element method(FEM), and the results from Bragg reflection theory are also provided as a comparison with those from FEM. The Q-factor of the proposed cavity in this paper is at the order of 108, and it still can be further improved. Although we only show how to use the proposed method to design a cavity for an OEL filled with common solid gain medium (Er-doped SiO2) in this paper, this method also can take effect for designing OEL cavities filled with other gain media.
2. Fiber cavity design
According to Figs. 1(a) and 1(b), it is easy to think of three requirements for the fiber cavity of an OEL: (1) low confinement loss for the pump light at λp, (2) the capability of “leaking” the excited light at λe out though the cladding, and (3) high Q-factor. To simultaneously achieve these, we designed a Bragg fiber with the cladding composed of three types of photonic crystal (PhC) as shown in Fig. 1(c). PhC1 is designed for confining the pump light, and the topological photonic crystal (TPhC) comprising PhC2 and PhC3 is designed to improve the Q-factor. As an example, a high-Q fiber cavity for an OEL with emission at the wavelength of about 1550 nm is designed. Considering that the Er-doped SiO2 is commonly selected as the gain medium for fiber lasers at 1550 nm, the Er-doped SiO2 is still chosen here as the gain medium, thus the wavelength of the pump is determined to be around 980 nm [16–18]. Then, the first two requirements referred become two specific ones: (1) low confinement loss at 980 nm and (2) capability of “leaking” the light around 1550 nm out through the cladding.
Fig. 1 (a) Schematic drawing of the omnidirectionally emitting laser. (b) Schematic drawing of the designed fiber cavity. The cladding of the fiber is composed of three types of PhC. (c) Detail of the layers of the fiber cavity. The function of PhC1 is to confine the pump light. PhC2 and PhC3 form a topological photonic crystal heterostructure to improve the Q-factor of the cavity.
2.1. Basic design
According to previous reports on Bragg fiber, a Bragg fiber with the layers that satisfy quarter-wave stack condition is easier to confine the light at λ (in vacuum) [19,20]:
where n1(2), l1(2) and λ1(2) are the refractive index and thickness of dielectric 1(2), and the wavelength of the light in dielectric 1(2). In this case, Si and SiO2 are chosen to compose the layers only because they are commonly used. Then, the thickness of Si (71.7 nm) and SiO2 (166.9 nm) in PhC1 are determined. The number of period can be estimated according to the practical requirements for the confinement loss through the equations as follow [21,22]: where NTE and NTM are the numbers of the layers for TE mode and TM mode, respectively. In Eqs. (2) and (3), n1, n2 are the refractive index of the two layers in a period, and.2.2. Design for high-Q
As is reported in previous researches, if two PhCs have different topological properties in their overlapping gaps, there will be topological edge modes in the 1D PhC heterostructure interface (Fig. 1(b)) [15]. The topological properties can be determined by [23]
where the integer l is the number of crossing points under the nth gap, the Zak phase [23,24] of the m-th isolated(no crossing) band is given by Eq. (5), andis given by.where ε(z) gives the dielectric distribution, un,q(z) is the Bloch electric field eigen-function of a state on the nth band with wave vector q and the integrand is the Berry connection [25]. To assure topological edge state exist around 1550 nm, PhC2 and PhC3 adopt the same thickness of Si (111.4 nm) and SiO2 (263.6 nm), which guarantees PhC2 and PhC3 have overlap gap around 1550 nm. Herein, the thickness of Si and SiO2 are also determined by Eq. (1). Besides, the inversed arrangement of Si and SiO2 in PhC2 (Si-SiO2) and PhC3 (SiO2-Si) (Fig. 1(c)) can bring different topological properties to PhC2 and PhC3 (Figs. 2(a) and (b)) according to Eq. (4). On the basis of Bragg reflection theory [26], transmission spectra of the multilayer composed of five periods of PhC2 and five periods of PhC3 is obtained to illustrate the existence of the topological edge state. As shown in Fig. 2(c), the sharp peak appearing in the spectra demonstrates the topological edge state. The zoom-in of the sharp peak shown in Fig. 2(d) illustrate the center wavelength and the full width at half maxima (FWHM) are about 1546.3 nm and 0.15 nm, respectively, which indicates that this multilayer structure has the potential of composing a high-Q cavity at the wavelength around 1546.3 nm.Fig. 2 (a,b) Band structures of PhC2 and PhC3. The magenta strip indicates the gap with positive topological property, and the cyan trip indicates gap with negative topological property. Here, λ = 1550 nm and Λ = 375 nm. (c) Transmission spectra of the whole multilayer structure (PhC2 + PhC3). The numbers of period of PhC2 and PhC3 are both 5. (d) Zoom-in of the transmission peak of the photonic topological edge state in (c).
3. Simulation results
With the help of COMSOL Multiphysics finite-element based software, the fiber cavity is studied in two steps: analyzing the confinement loss of the pump light with wavelength of 980 nm and analyzing the Q-factor of the cavity. When analyzing the Q-factor, we set dense point sources in the defect region of the crystal to represent for the excited gain medium. The details of the simulation models are introduced in the Appendix 1.
As seen in Fig. 3(a), the designed fiber nicely confines the light with wavelength of 980 nm. The curves in Fig. 3(b) illustrate that the confinement loss for different numbers of period of PhC2 and PhC3 keeps at a low level when the wavelength ranges from 970 nm to 990 nm. The confinement loss can be further decreased by increasing the number of period of PhC1 if necessary [27].
Fig. 3 (a) Field distribution of the fundamental mode. The cladding comprises PhC1(10 periods), PhC2(5 periods) and PhC2(5 periods). The radius is 1 μm. (b) Confinement loss as a function of wavelength in vacuum for different numbers of period of PhC2 and PhC3. N2 and N3 are the numbers of period of PhC2 and PhC3, respectively.
As seen in Figs. 4(a) and 4(d), the majority of excited light at the resonant wavelength enters into the defect region, which acts as a cavity. The sharp peaks at about 1546.7 nm in the spectrum of the output beam shown in Figs. 4(b) and 4(e) agree well with the result shown in Fig. 2(d) indicating the high Q-factor of this designed cavity. Moreover, the angular emission pattern demonstrates that the output beam from this cavity is omnidirectional. The curves of Fig. 4(f) illustrate that the output beam is concentrated. The concentration of the output beam shown in Fig. 4(f) is similar with the far-field phenomenon exhibited in the previous study [6].
Fig. 4 (a) Normalized field distribution at 1546.7 nm in x-y plane. (b) Summation of the power outflow at the boundary in (a) as a function of wavelength. (c) Normalized angular emission pattern measured around the fiber in x-y cross section. (d) Normalized field distribution at 1546.7 nm in x-z plane. (e) Summation of the power outflow at the upmost boundary in (d) as a function of wavelength (f) Far field distribution of the light with wavelength of 1546.6 nm, 1546.7 nm and 1546.8 nm. The far field distribution is measured at the upmost boundary in (d).
4. Further improve Q-factor
As shown in the previous section, the topological edge state is like a “wound” in the gap though which light in a small size of wavelength range can escape out. Because this “wound,” in fact, results from the defect in the TPhC, it is reasonable to speculate that if the size of the TPhC increases with the defect keeping the same size, the impact bringing from the defect to the whole multilayer structure may be reduced, which means part of the “wound” may “heal” resulting in decreasing FWHM, thus the Q-factor can further increase. To verify this speculation, by regarding the whole TPhC as a supercell, the band structures of TPhC with different numbers of period of PhC2 and PhC3 are approximately calculated on the basis of supercell method [28].
Figures 5(a)-5(h) illustrates how the “wound” changes with the number of period of PhC2 and PhC3 increasing from 4 to 9. Due to the defect, the bands appear around 1550 nm in the gap giving the reason for the phenomena shown in Fig. 4. The details of the variation of the bands are shown in Figs. 5(g) and 5(h). It is clear that the band becomes flat with the number of period of PhC2 and PhC3 increasing, which verifies the speculation and suggests a straight way to further improve the Q-factor of the fiber cavity. By the way, the supercell method as an approximate method is enough to provide a hint but not a quite precise result. The details about this statement are illustrated in the Appendix 2. Therefore, more accurate results for Q-factor are obtained also by FEM and Bragg reflection theory.
Fig. 5 (a-f) Band structures of PhC2 + PhC3 calculated on the basis of supercell method. From (a) to (f), the numbers of period of PhC2 and PhC3 changes from 4 to 9. (g) Zoom-in of the crossed bands in (a,b). (h) Zoom-in of the lower band of the two appeared in the gap in (c-f). Λi is the period of the supercell comprising i periods of PhC2 and PhC3.
As seen in Figs. 6(a) and 6(b), with the number of period of PhC2 and PhC3 increasing, the peak narrows down, which is consistent with the results illustrated in Fig. 5. Besides, the variation trend of the curves in Figs. 6(a) and 6(b) agree with each other demonstrating the reliability of the results. According to the definition of Q-factor Q = λ/Δλ, where λ is central wavelength and Δλ is FWHM, the Q-factor is calculated and shown in Fig. 5(c). Curves of Fig. 6(c) illustrate sustaining increase of the Q-factor with the number of period of PhC2 and PhC3 increasing. According to Fig. 6(c), the Q-factor calculated on the basis of Bragg reflection theory and FEM for N2 = N3 = 9 is at the order of magnitude of 107 and 108, respectively. A possible reason for the difference between the two Q-factors obtained from the two methods is that the transmission spectra are the results for only the incident light perpendicular to the layers, but, in fact, the light in the fiber arrives at the layers with arbitrary incident angle. Therefore, the Q-factor obtained from FEM is more reliable. However, the two curves in Fig. 6(c) are consistent on the trend demonstrating that the Q-factor can be further improved by simply increasing N2 and N3. The Q-factor of the cavity proposed here far exceeds the theoretical Q-factor (~15000) announced in [6], Xu et al, indicating the superiority of our method for designing high-Q cavity for OEL.
Fig. 6 (a) Transmission spectrum for different numbers of period of PhC2 and PhC3 based on Bragg reflection theory. (b) Sum of the power outflow for different numbers of period of PhC2 and PhC3. The power out flow is measured at the outmost boundary of the models in x-y cross section (e.g. Fig. 4(b)). Power outflows are normalized by the maximum power outflows measured in each case with different N2 and N3. N2 and N3 are the numbers of period of PhC2 and PhC3, respectively. (c) Quality factor calculated with the help of FEM and Bragg reflection theory.
5. Conclusions
In summary, we propose a method to design a high-Q fiber cavity of an OEL only if the gain medium and the material of the layers are determined in advance according to practical demand. Then, the way to design an appropriate Bragg fiber acting as a high-Q cavity of an OEL can be summarized in five steps: (a) figure out the thicknesses of the layers in PhC1 by substituting the wavelength of the pump and the two refractive indexes of the materials to Eq. (1); (b) determine the number of period of PhC1 by Eq. (2) or Eq. (3); (c) figure out the thicknesses of the layers in PhC2 and PhC3 by substituting the wavelength of the excited light and the two refractive indexes of the selected materials to Eq. (1); (d) arrange the layers of PhC2 (ABAB) and PhC3 (BABA) in inversed way to generate topological edge state; (e) increase the numbers of period of PhC2 and PhC3 to improve Q-factor. Thus, the Bragg fiber composed of PhC1 + PhC2 + PhC3 can serve as a high-Q fiber cavity of the required OEL. By taking Er-doped SiO2 as gain medium and taking Si and SiO2 as the material of the layers, we demonstrate the five steps and design a Bragg fiber cavity as an example. This fiber can transmit the pump under low confinement loss and act as a cavity for the excited light. The Q-factor of this cavity is at the order of magnitude of 108, and it can be further improved by simply increasing the layers of PhC2 and PhC3. These steps may significantly shorten the time for designing a high-Q fiber cavity of a required OEL. Because these steps are not limited by the gain medium and the materials of the layers, they also can be used to design cavity for advanced OEL such as optofluidic OEL or quantum dot OEL. Besides, OEL with higher Q-factor may show more new potential in various research fields.
Appendix 1 Methods
Appendix 1.1 Band structure
The band structures shown in this paper are carried out with the help of RSoft on the basis of plane wave expansion method and supercell method.
Appendix 1.2 Transmission spectra
The transmission spectra shown in this paper are carried out with the help of Matlab on the basis of Bragg reflection theory.
Appendix 1.3 Simulation model set-up
The simulations of the power outflow and the electromagnetic field distributions are carried out with the help of COMSOL Multiphysics finite-element based software. Only the “Wave optics” module is utilized. The optical data for Si and SiO2 are taken from COMSOL’s Optical Materials database where they are named as Si (Salzberg) and SiO2 (Lemarchand), and the region of air is defined by setting the real part of the refractive index equal to 1. Perfectly matched layer (PML) setting at outmost region of each model was constructed to act as absorber of the light escaping out the concerned region.
Appendix 1.4 Confinement loss calculation
In this case, the core region is filled with air, and there is no region between the outmost layers of the fiber and the PML. The “Mode analysis” function in the “Wave optics” module is utilized to carry out the mode field distribution and the effective refractive index of the fiber. With the help of Matlab, the confinement loss is calculated by [29]
where Im(β) is the imaginary part of the complex propagation constant.Appendix 1.5 Power outflow, far-field distribution and Q-factor calculation
In this case, dense point sources representing for the excited Er-doped SiO2 are set at the region topological edge state. Two types of point sources including line current and magnetic current are used to approximately simulate excited light with Hz=0 or Ez=0. Here Hz means the z component of magnetic field and Ez means the z component of electric field. The results under these two conditions are almost the same, thus we only put results for Hz=0 in the main body, and those for Ez=0 are exhibited in the Appendix 3. Far-field domain filled with air and located between the outmost layer of the fiber and PML is applied. The summation of the power outflow is calculated by integrating power outflow at the boundary between Far-field region and PML with the wavelength in vacuum varying. Figures 4(e) and 6(b) are calculated from model in x-y plane and in y-z plane, respectively. The far-field distribution is obtained at the boundary between Far-field region and PML. The Q-factor is calculated by Q=λ/Δλ.
Appendix 2 Explanations about supercell method
It may be confusing that there are two bands appearing in the gap shown in Fig. 5, however there is only one peak in the transmission spectrum Fig. 2(c) with the wavelength ranging from 1400 nm to 1600 nm. Actually, this results from the supercell method itself. Here, explanations are given for this.
As is known to all, supercell method is an approximate method to carry out the band structure of the periodic dielectric with defect by regarding some whole as a supercell. As an example, Fig. 7 illustrates the supercell selected to solve the band structure for the case N2=N3=5. Because the supercell is regarded as a basic periodic unit, actually, the band structure obtained (e.g. Fig. 5) is the result of the multilayer dielectric composed of PhC2+PhC3+PhC2+……PhC3+ PhC2+PhC3. However, the transmission spectrum is the results of the multilayer dielectric composed of only PhC2+PhC3. That is why the different results appear. To show this more clearly, the transmission spectra of multilayer dielectric composed of 50×(PhC2+PhC3) for different N2 and N3 are shown in Fig. 8. Now, the transmission bands are almost consistent with those in Fig. 5. Moreover, the downtrend of width of the bands in Fig. 8 is corresponding to the planarization of band in Fig. 5(h). In a word, supercell method is enough good to exhibit the property of the structure with defect though it provides results that may not quite accurate.
Fig. 7 Schematic drawing of the supercell chosen for calculating the band structure with N2 = N3 = 5.
Fig. 8 Transmission spectra of multilayer structure composed of 50 periods of supercells with different N2 and N3: (a) N2 = N3 = 5; (b) N2 = N3 = 6; (c) N2 = N3 = 7; (d) N2 = N3 = 8.
Appendix 3 Simulation results for the light source: line current
The results for Ez=0 are shown in Fig. 9. Both Fig. 4 and Fig. 9 exhibit the performance of the designed cavity. Furthermore, the Q-factors in this case are also calculated with the period of PhC2 and PhC3 increasing. As seen in Fig. 10, the results from two sources are consistent with each other.
Fig. 9 (a) Normalized field distribution at 1544.9 nm in x-y plane. (b) Summation of the power outflow at the boundary in (a) as a function of wavelength. (c) Normalized angular emission pattern measured around the fiber in x-y cross section. (d) Normalized field distribution at 1544.4 nm in x-z plane. (e) Summation of the power outflow at the upmost boundary in (d) as a function of wavelength (f) Far field distribution of the light with wavelength of 1544.3 nm, 1544.4 nm and 1544.5 nm. The far field distribution is measured at the upmost boundary in (d).
Funding
National Natural Science Foundation of China (11574042, 61271167) and the Young Scientists Fund of the National Natural Science Foundation of China (61307093).
References
1. H. J. Moon, Y. T. Chough, and K. An, “Cylindrical microcavity laser based on the evanescent-wave-coupled gain,” Phys. Rev. Lett. 85(15), 3161–3164 (2000). [CrossRef] [PubMed]
2. Z.-L. Li, W.-Y. Zhou, Y.-G. Liu, M. Yan, and J.-G. Tian, “Simplified Hollow-core microstructural optical fiber laser with intense output and polarized radial emission,” SPIE Proc. 8960, 89600K (2014). [CrossRef]
3. Z.-L. Li, Y.-G. Liu, M. Yan, W.-Y. Zhou, C.-F. Ying, Q. Ye, and J.-G. Tian, “A simplified hollow-core microstructured optical fibre laser with microring resonators and strong radial emission,” Appl. Phys. Lett. 105(7), 71902 (2014). [CrossRef]
4. D. Yan, H. Zhang, B. Liu, B. Huang, J. Sun, and D. Wang, “Optofluidic Microring Resonator Laser Based on Cavity-Assisted Energy Transfer in Dye-Infiltrated Side-Hole Microstructured Optical Fibers,” J. Lightwave Technol. 35(19), 4153–4158 (2017). [CrossRef]
5. C. Y. Gong, Y. Gong, W. L. Zhang, Y. Wu, Y. J. Rao, G. D. Peng, and X. Fan, “Fiber Optofluidic Microlaser With Lateral Single Mode Emission,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–6 (2018). [CrossRef]
6. Y. Xu, C. Gong, Q. Chen, Y. Luo, Y. Wu, Y. Wang, G. D. Peng, Y. J. Rao, X. Fan, and Y. Gong, “Highly Reproducible, Isotropic Optofluidic Laser Based on Hollow Optical Fiber,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–6 (2019). [CrossRef]
7. A. M. Stolyarov, L. Wei, O. Shapira, F. Sorin, S. L. Chua, J. D. Joannopoulos, and Y. Fink, “Microfluidic directional emission control of anazimuthally polarized radial fibre laser,” Nat. Photonics 6(4), 229–233 (2012). [CrossRef]
8. O. Shapira, K. Kuriki, N. D. Orf, A. F. Abouraddy, G. Benoit, J. F. Viens, A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, Y. Fink, and M. M. Brewster, “Surface-emitting fiber lasers,” Opt. Express 14(9), 3929–3935 (2006). [CrossRef] [PubMed]
9. N. Zhang, H. Liu, A. M. Stolyarov, T. Zhang, K. Li, P. P. Shum, Y. Fink, X. W. Sun, and L. Wei, “Azimuthally Polarized Radial Emission from a Quantum Dot Fiber Laser,” ACS Photonics 3(12), 2275–2279 (2016). [CrossRef]
10. S. Stützer, Y. Plotnik, Y. Lumer, P. Titum, N. H. Lindner, M. Segev, M. C. Rechtsman, and A. Szameit, “Photonic topological Anderson insulators,” Nature 560(7719), 461–465 (2018). [CrossRef] [PubMed]
11. Y. Kang, X. Ni, X. Cheng, A. B. Khanikaev, and A. Z. Genack, “Pseudo-spin-valley coupled edge states in a photonic topological insulator,” Nat. Commun. 9(1), 3029 (2018). [CrossRef] [PubMed]
12. H. He, C. Qiu, L. Ye, X. Cai, X. Fan, M. Ke, F. Zhang, and Z. Liu, “Topological negative refraction of surface acoustic waves in a Weyl phononic crystal,” Nature 560(7716), 61–64 (2018). [CrossRef] [PubMed]
13. Y. Meng, X. Wu, R.-Y. Zhang, X. Li, P. Hu, L. Ge, Y. Huang, H. Xiang, D. Han, S. Wang, and W. Wen, “Designing topological interface states in phononic crystals based on the full phase diagrams,” New J. Phys. 20(7), 073032 (2018). [CrossRef]
14. Y. Yang and Z. H. Hang, “Topological whispering gallery modes in two-dimensional photonic crystal cavities,” Opt. Express 26(16), 21235–21241 (2018). [CrossRef] [PubMed]
15. W. Gao, X. Hu, C. Li, J. Yang, Z. Chai, J. Xie, and Q. Gong, “Fano-resonance in one-dimensional topological photonic crystal heterostructure,” Opt. Express 26(7), 8634–8644 (2018). [CrossRef] [PubMed]
16. C. R. Giles and E. Desurvire, “Modeling Erbium-Doped Fiber Amplifiers,” J. Lightwave Technol. 9(2), 271–283 (1991). [CrossRef]
17. W. J. Miniscalco, “Erbium-Doped Glasses for Fiber Amplifiers at 1500 nm,” J. Lightwave Technol. 9(2), 234–250 (1991). [CrossRef]
18. L. H. Slooff, A. V. Blaaderen, G. A. Hebbink, S. I. Klink, F. C. J. M. V. Veggel, D. N. Reinhoudt, and J. W. Hofstraat, “Rare-earth doped polymers for planar optical amplifiers,” Appl. Phys. Rev. 91(7), 3954–3980 (2002).
19. J. Sakai, “Hybrid modes in a Bragg fiber: general properties and formulas under the quarter-wave stack condition,” J. Opt. Soc. Am. B 22(11), 2319–2330 (2005). [CrossRef]
20. A. Kitagawa and J. Sakai, “Bloch theorem in cylindrical coordinates and its application to a Bragg fiber,” Phys. Rev. A 80(3), 033802 (2009). [CrossRef]
21. H. Yang, C. Liu, Y. Wang, S. Su, and J. Li, “Simulation design for the structures of Bragg fibers,” Optik (Stuttg.) 123(1), 63–66 (2012). [CrossRef]
22. Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, “Asymptotic Matrix Theory of Bragg Fibers,” J. Lightwave Technol. 20(3), 428–440 (2002). [CrossRef]
23. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems,” Phys. Rev. X 4(2), 21017 (2014). [CrossRef]
24. J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62(23), 2747–2750 (1989). [CrossRef] [PubMed]
25. M. V. Berry and M. Wilkinson, “Diabolical points in the spectra of triangles,” Proc. R. Soc. Lond. A Math. Phys. Sci. 392(1802), 15–43 (1984). [CrossRef]
26. S. K. Awasthi, U. Malaviya, and S. P. Ojha, “Enhancement of omnidirectional total-reflection wavelength range by using one-dimensional ternary photonic bandgap material,” J. Opt. Soc. Am. B 23(12), 2566–2571 (2006). [CrossRef]
27. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68(9), 1196–1201 (1978). [CrossRef]
28. W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and S. Guo, “Compact supercell method based on opposite parity for Bragg fibers,” Opt. Express 11(26), 3542–3549 (2003). [CrossRef] [PubMed]
29. J. Olszewski, M. Szpulak, and W. Urbańczyk, “Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers,” Opt. Express 13(16), 6015–6022 (2005). [CrossRef] [PubMed]
