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High order resonances between fundamental core mode and cladding LP01 supermodes are demonstrated in long period fiber gratings (LPFGs) inscribed in all-solid photonic bandgap fibers for the first time to our knowledge. The resonance wavelengths of the LPFGs calculated by way of photonic bandgap theory agree with the experimental results. The temperature responses of these resonance peaks have been theoretically and experimentally investigated. In addition, the mechanism of LPFG formation has been researched deeply through coupled-mode theory (CMT) and the cutback experiments.
©2010 Optical Society of America
1. Introduction
Long-period fiber gratings (LPFGs), an important kind of optical device, provide an efficient means to resonantly couple light between two copropagating modes in optical fibers, The resonance wavelength of an LPFG can be determined by the phase-matching condition, , where λ res is the resonance wavelength, ΛLPG is the grating period, n eff,1 and n eff,2 are the effective indices of two coupled modes, and the positive integer q represents the order of the resonance. High order resonances (q>1) have been observed in ultra-long period of gratings, which posses potential applications in multi-parameters optical sensors [1,2].
Photonic bandgap fibers (PBGFs), including hollow-core PBGFs, fluid-filled PBGFs, and all-solid PBGFs, confine light in low refraction index core by photonic bandgap effect of the cladding, and exhibit a lot of characteristics different from index-guiding fibers. In recent years, many LPFGs have been fabricated in PBGFs, the inscription methods include Electric-arc discharge [3], acousto-optic interaction [4,5], UV exposure [6], Electrically field [7,8], mechanically stress [7,9,10], femtosecond laser irradiation [11,12], CO2 laser irradiation [13], and so on. These LPFGs can couple light from fundamental core mode to high order core modes or Bloch supermodes in the cladding, and exhibit different properties. The coupling between fundamental core mode and the cladding supermodes can be realized by both fiber Bragg gratings [14] and LPFGs [6] in all-solid PBGFs, which not only extend the applications of fiber gratings in sensor and communication, but also provide a direct method to investigate the structures of supermode bands and bandgaps in PBGFs [15].
In this paper, high order resonances between fundamental core mode and Bloch mode band (LP01 supermode band) in cladding are demonstrated in the LPFGs inscribed in all-solid PBGFs for the first time to our knowledge. The resonance wavelengths of the LPFGs calculated by way of photonic bandgap theory agree with the experimental results well. The temperature responses of these resonance peaks have been theoretically and experimentally investigated. The results show that the temperature sensitivities of the resonance peaks in the same grating are close to each other, while those in different gratings exhibit obviously different temperature sensitivities. What is more, we deeply analyze the mechanism of LPFG formation by way of coupled-mode theory (CMT) and the cutback experiments to observe the resonance and light field propagation.
2. Band structure of the all-solid fiber
The cross section of the all solid fiber used in our experiments, which is fabricated by Yangtze Optical Fiber and Cable Corporation, is shown in the inset of Fig. 1(a) . In the fiber a high index rod surrounded by a low index ring lattice of five layers is embedded in pure silica background, the core is formed by omitting a high index rod and a low index ring. The diameter of the fiber is about 150 μm, and the pitch between adjacent rods Λ is 10.5 μm. The outer radii of a high index rod and a low index ring are 0.181Λ and 0.3786Λ, respectively. Compared with the pure silica background, the average refractive index differences of the high index rods and the low index rings are approximately 0.028 and −0.008. The all-solid PBGF has low transmission loss and less bend sensitivity [16].
Fig. 1 (a) (Color online) The bandgaps, fundamental core mode of the all-solid fiber, spectra before and after inscription of the 240 μm period of LPFG. The inset is the cross section of the all-solid fiber. The red curve represents the dispersion curve of the fundamental core mode. The orange and purple curves represent spectra before and after inscription respectively. (b) The periodic notches on the all-solid fiber after inscription. (c) The cross section of the all-solid fiber at a notched region corresponding to the left end of the fiber shown in Fig. 1(b).
By means of plane wave expansion method [17] and a commercial finite element method, the bandgaps and fundamental core mode are calculated, which are shown in Fig. 1(a), where material dispersion is neglected since it imposes slight impact on the final results. Due to low index rings, above the core line the Bloch modes (supermodes) are better confined in high index rods, so that the supermode bands, such as LP01 and LP02 supermode bands, are very narrow. Whereas below the core line the supermodes, which form the lower edge of the bandgaps, mainly distribute in pure silica area and the supermode bands are wide.
3. Inscription, spectral characteristics and temperature responses of the LPGs
We inscribe gratings with 40 periods in the PBGFs. The LPFGs are inscribed through a point by point side illumination process using a CO2 inscription platform. About 5 cm of all-solid fibers are spliced to single mode fibers at both ends, in which the accuracy of aligning of cores is achieved as much as possible in order to guarantee the mode in the fiber core is mainly fundamental mode, reduce energy loss and the influence of interference between fundamental core mode and cladding modes or higher order core modes. The fiber is placed as near the focal point of the CO2 laser beam as possible to achieve an effective illumination. The inscription process is similar to that mentioned in [13]. In our experiments, the two most important parameters are the number of hit at a particular point in every scanning cycle and the time of releasing each laser pulse. Generally, we set them 2 and 70 μs respectively, however, for the 80 μm period of grating, the parameters are set 1 and 60 μs in order to reduce the damage to the PBGF. Figure 1(b) shows the periodic notches of a 240 μm period of grating on the fiber surface, in which the left end is at a notched region, whose cross section is shown in Fig. 1(c). During observation the supercontinuum light is launched into the fiber in order to make the image clear. Obviously the CO2 laser induces the ablation of the fiber on the illumination side, and the deformation is very large and near the fiber core.
The measured spectra before and after inscription of the 240 μm period of grating are also given in Fig. 1. The positions of high loss wavelength regions in the transmission spectrum before inscription agree with the breaking points of dispersion curves of fundamental core mode, the slight difference is due to the imperfect structure in the real fiber, inaccurate estimation of parameters in our model and the microbend of the fiber. After inscription two distinct resonant peaks (1313.4nm and 1572.7 nm) can be observed in Bandgap I. Besides, in Bandgap II, there is a very weak peak at 890.2 nm, it may be from coupling between the fundamental core mode and LP11 supermode, and further analysis will be done in the future work.
The normalized spectra of the LPFGs, whose period are 80 μm, 120 μm, 240 μm, 480 μm, are displayed in black solid curves in Fig. 2 . The resonance wavelengths labeled with A-G are given in Table 1 . Using infrared CCD we observe near field images for some of these resonance peaks, which are given as insets in Fig. 2. The PBGFs are cut at a distance of about 2 cm away from the input end of the LPFGs. Evidently, these resonances are mainly between fundamental core mode and guided LP01 supermodes. The judgment is verified through immersing the LPFGs into a high index liquid, the results are shown in red dash curves in Fig. 2. The spectra of the LPFGs after immersing into a high index liquid hardly change, which indicates the light is coupled into guided modes.
Fig. 2 (Color online)The normalized spectra for different periods of LPFGs. The black solid curves and red dash curves represent spectra of the LPFGs before and after being immersed into a high index liquid respectively. The insets are near field images corresponding to these resonance peaks.
Table 1. The resonance order, resonance wavelength and temperature sensitivity corresponding to each resonance peak
Through temperature sensing experiments, the linear temperature responses of these resonance peaks are derived and the specific sensitivities are provided in Table 1. Note that in a grating, the sensitivities of different resonance peaks are very close, while in different gratings, the difference is obvious.
In addition, the smaller peaks near resonance peak E are possibly formed by the resonance between fundamental core mode and the LP11 supermodes, the wide wavelength range should arise from strong index dispersion of LP11 supermode band below the core line. Obviously, compared with other peaks, the peaks F and G are much smaller. It should be attributed to small corresponding Fourier expansion coefficients, which determine the resonance strength [18].
4. Theoretical analysis
We analyze the resonance coupling between the fundamental core mode and LP01 supermode band by the phase-matching condition of the LPFG. The effective index of LP01 supermode band can be represented by the low edge of the band, since the band is very narrow. Figure 3 presents period of a grating against the resonance wavelength, where the resonance wavelengths can be determined from the cross points between the red curves and the black horizontal lines which represent periods. According to the figure, 80 μm, 240 μm and 480 μm period of gratings all have the resonance peak at 1301 nm; 120 μm, 240 μm and 480 μm period of gratings all have the resonance peak at 1576 nm; besides 480 μm period of grating have a resonance peak at 1426 nm. The theoretical results agree with our experimental results well except the slight discrepancy on the resonance wavelengths, which should be due to influences induced by inscription, inaccurate estimation of fiber parameters and imperfect structure in the real fiber. The difference of resonance wavelengths, which are the same in theoretical analysis, results from different extent to which the effective index differences between high-index supermodes and fundamental core mode are reduced. When the period becomes short, the overlap of adjacent inscription points is stronger, leading to increasingly smaller index difference between fundamental core mode and LP01 supermodes and larger blue shift of resonance wavelengths. That the 80 μm period of grating whose resonance wavelength is longer than that of 240 μm period of grating, however, is an exception, this is because the influence caused by CO2 laser is weaker due to less number of hit and low time of releasing each laser pulse mentioned above. Therefore the conclusion can be derived that the resonances given in Fig. 2 are all from the coupling between fundamental core mode and LP01 supermodes.
Fig. 3 (Color online) The period of a grating and dispersion factor against the resonance wavelength. The red curves represent the phase-matching condition of an LPFG, the black horizontal lines represent the periods of LPFGs, the blue curve represents the relation between dispersion factor and the resonance wavelength.
The temperature responses for our gratings are different from those in fluid-filled PBGFs whose refractive index difference between high-index rods and background strongly depends on temperature, the LPFG in the all-solid PBGF have the temperature responses more similar to those in conventional fibers. Therefore, the temperature sensitivities of the gratings can be expressed by [19]:
where the effective index difference Δne = n eff, s-n eff, c, n eff, s and n eff, c, are the effective indices of low edge of LP01 supermode band and fundamental core mode, the dispersion factor , and the group index difference, α is thermal expansion coefficient, and represent thermo-optic coefficients of LP01 supermode band and fundamental core mode. The dependence of the temperature sensitivity on the resonance order q is implicitly in λ res by the phase-matching condition. According to the dispersion curves shown in Fig. 1(a), γ monotonically decreases when the wavelength increases, as shown in the blue curve in Fig. 3. Consequently, the variation of λ res⋅γ is slight at different resonance wavelengths, which induces that the resonance peaks in the same grating have close temperature sensitivities. Whereas, the values of thermo-optic coefficients are not only dependent on the material component of rods but also closely relevant to inscription condition, which contributes that the resonance peaks in different periods of gratings have different temperature sensitivities.5. Deep discussion of mechanism of LPFG formation
In order to make clear the mechanism of LPFG formation, we investigate the evolution of light field in the grating and the PBGF after grating by way of a cutback experiment with another 240 μm period of grating, whose resonance wavelengths are 1317.2 nm and 1575.8 nm respectively. The schematic diagram in the experiment is shown in Fig. 4 . Figure 5(a) is the light intensity distribution at 1575.8 nm before the light enters the grating, it is well confined in the core, the slight leak into the cladding high-index rods is due to inaccuracy of splicing between single mode fiber (SMF) and PBFG and mode field mismatch between the fundamental core modes in the SMF and in the PBGF. Note that all the coordinates given in the Fig. 5 indicate the distances between the observation points and the input end of the LPFG. Figure 5(b) shows the intensity distribution at 1600 nm at z = 8 mm, which is far away from resonance wavelengths, the background modes in silica area can be observed. Figure 5(c), (d) are the intensity distributions at resonance wavelength 1317.2 nm at z = 30 mm and z = 60 mm. Obviously, there is little change except that in Fig. 5(c) there is weak light coupled into the LP11 supermode, whereas in Fig. 5(d) the supermode disappears, which should be due to leak from fundamental core mode to the highly lossy cladding modes below the core line because 1317.2 nm is near the short wavelength edge of Bandgap I. At 1317.2 nm the light is nearly confined in one high-index rod. Using finite element analysis, the coupling length between the LP01 modes in adjacent high-index rods at 1317.2 nm is calculated, that is about 33 cm, so the coupling between two adjacent rods is weak. Thus the evolution of light field at 1317.2 nm is not obvious and is not investigated further. For another resonance peak (1575.8 nm) the representative light field distributions at different locations are given in Fig. 5(e)-(j). The illumination side is in the top-left. With propagating in the grating, the light field is coupled into the first layer of high-index rods gradually and begins to spread to outer high-index rods in the non-illumination side [Fig. 5(e)-(i)]. The LP11 mode in the fiber core should result from the inaccuracy of splicing between single mode fiber (SMF) and PBFG. It can be observed when most of the light energy in LP01 core mode is coupled into cladding supermodes. Note that the light energy coupled into the rods in the illumination side is stronger than that in the opposite side, which should be ascribed to the stronger resonance in the illumination side. After the grating, the coupling among the rods continues and the light spreads gradually, at z = 70 mm the light field distribution is recorded [Fig. 5(j)]. When the observation points are in and near the grating, background modes can also be seen [Fig. 5(e)-(i)], which is the same with that observed in Fig. 5(b) for the light intensity distribution at 1600 nm, then the modes disappears due to high loss when the observation point is far away from grating [Fig. 5(j)]. Thus the occurrence of the background modes is not from resonance discussed in the paper, maybe these modes result from other resonances introduced by the grating, the resonance wavelength range is wide and the resonance peaks are not evident because of low density of state (DOS) of these background modes.
Fig. 5 The evolution of light field for a 240 μm period of grating. (a)-(j) present the experimental results, (k)-(p) present the simulation results. The observation wavelength and the distance between the observation point and the input end of the LPFG are given in each figure.
As shown in Fig. 1(b), after inscription there is no bend observed with the microscope, so the mechanism should not be ascribed to microbend. As shown in Fig. 1(c), part of the fiber cross section is ablated during inscription, the deformation is very large and near the fiber core, but the influence on mode distribution located in high-index rods and fiber core can be neglected, which is verified through the analysis by way of the finite element method. The deduction can also be derived by the experiment where the spectra of the gratings hardly change after the gratings are immersed into the high-index liquid. It is believed that this is partly due to the existence of low-index rings, which improve the confinement of the rod modes and fundamental core mode greatly. In addition, through calculation the coupling length between LP01 modes in adjacent high-index rods is 6.56 cm, whereas between the modes in other rods the coupling is very weak, thus it can be assumed that the coupling for LP01 modes in the high-index rods only occurs between adjacent rods. What is more, we believe that the resonance only occur between fundamental core mode and LP01 modes in the first layer of high-index rods, on the one hand, this can be observed in the light field resonance process shown in Fig. 5(e)-(i), on the other hand, the resonances in our gratings are not influenced by external high-index liquid. Therefore, we believe the mechanism of LPFG formation in our experiments mainly originate from refractive index modulation in the first layer of high-index rods, which is introduced by CO2 laser illumination. According to the theory in [20], we speculate the indices in the high-index rods decrease, which produce the blue shift of the bandgaps, although the same result can be induced by increasing background refractive index or shrinking the rods. The ablation of the fiber mainly causes transmission loss. It is assumed that the index modulation in the background can be neglected. In our experiments the shift of the bandgap is about 30 nm, by way of theory in [21] index decrease in high-index rods is , which is very large and should be related to strong inscription as can be seen from the large deformation of fiber cross section in Fig. 1(c).
By means of CMT we research the resonance and light propagation in the grating and PBGF after grating. The coupled equation applied here can be expressed by:
where,u is a vector representing the complex amplitudes of unperturbed mode fields in the fiber core and the high-index rods, in the right the first term only represents interaction between high-index rods in the cladding, the second term only represents the resonance between the fiber core and the first layer of high-index rods. In the first term is the coupling matrix for the PBGF, determining whether there is an interaction between LP01 modes in two rods, the elements in it are equal to 1 if their indices refer to adjacent rods or the same rod, otherwise they are zero. Note that in the grating, the ablation of the fiber cross section is considered in through setting corresponding elements zero. For the PBGF the coupling coefficient matrix κ is calculated through the other two matrices and c, and can be expressed by:The element represents the interaction of two LP01 modes in adjacent rods in the index perturbation region, which can be obtained by the following expression:where A represents the entire fiber cross section, represents the refractive index variation. The element represents overlap coefficient of two LP01 modes in adjacent rods and can be given by:Generally, the element is not equal to zero if m is not equal to n, because the modes of adjacent rods are not necessarily orthogonal to each other. In the second term of the right side in Eq. (2), , have the similar meaning with , κ, they describe the light field resonance in the grating and only consider the interaction between the LP01 modes in the fiber core and each rod in the first layer. The positive integer q represents the resonance order. In only the elements representing the interaction between the fiber core and the rods in the first layer are 1, the others are zero. The is expressed as follow: and have the similar meaning with and c. The element in can be expressed by:where represents the refractive index variation in the grating, for the interaction between the fiber core and one high-index rod in the first layer, the variation region is only limited in that rod. is a wavenumber induced by the periodic structure. The expression form of element is the same with , so it is not given. Because the variation of refractive index is a periodic function of length, the element of is calculated by Fourier series expansion and it is q order expansion coefficient. δ is phase mismatch matrix, is the element of δ, represents phase mismatch and can be expressed aswhen is larger than , otherwise . In our experiment, the resonance at 1575.8 nm is second order resonance. The index variation is seen as a periodic rectangular function. The average width of a notch, e.g. the index variation region, is about 41 μm. The simulation results are shown in Fig. 5(k)-(p). In these figures, the left side is the illumination side. Figure 5(k)-(o) present the evolution of intensity distribution in the grating, Fig. 5(p) presents the distribution at z = 70 mm. The results agree with the experimental observation, but there are two main different points between them: (1) in the simulation the coupling from inner rods to outer rods is weaker than that in experiments, (2) compared with the experimental observation, in the simulation the spread of light field over the cross section at z = 70 mm is stronger, as shown in Fig. 5(j) and Fig. 5(p). We believe the differences may be resulted from some influences induced during inscription, imperfect fiber structure in the real fiber, the difference between our simulation parameters and the real fiber parameters. Besides, in our model we assume that the coupling between the fiber core and each of the six rods in the first layer is the same, which should be different from the real case. Further reasons will be analyzed in the future work.In order to make a comparison, we do the same experiment with anther grating inscribed in a different all-solid PBGF, whose parameters are similar to the fiber used previously, only the pitch is smaller, that is about 9.5 μm, which can cause stronger coupling. The period of the grating is 260 μm, and the longer resonance wavelength is 1563 nm. The experimental and simulation results at the wavelength are shown in Fig. 6 , where the illumination sides are left sides. The coordinate given in each figure indicates the distance between the observation point and the input end of the grating. Obviously, the agreement between the two groups of results is well. Compared with the gratings discussed above, in this grating, the resonance between fiber core and the rods in the first layer and coupling between adjacent rods are stronger as our expectation, so the asymmetric light intensity distribution is more obvious, which shows the influence of pitch on the light field distribution.
Fig. 6 The evolution of light field for a 260 μm period of grating inscribed in another PBGF whose pitch is about 9,5 μm. (a)-(e) present the experimental results, (f)-(j) present the simulation results. The observation wavelength and the distance between the observation point and the input end of the LPFG are given in each figure.
Therefore, it is reasonable to believe that the LPFG formation is mainly due to the index modulation in the first layer of high-index rods. From the calculation results and the large deformation of the PBGF cross section, we know that the index variation is large, inducing large coupling coefficients and the occurrence of so many strong high order resonances which is rarely observed in others’ works on LPFGs inscribed in PBGFs.
6. Conclusion
In summary, LPFGs are inscribed in all-solid PBGFs, in which every resonance peak arises from coupling between fundamental core mode and guided LP01 supermode band and corresponds to different coupling order. The temperature sensitivities of the peaks in the same grating are close to each other, while the resonance peaks in different periods of gratings exhibit different temperature responses. The mechanism of LPFG formation is investigated deeply to explain the occurrence of these high order resonances. The work provide theory base for generating high order resonances, the application of high order resonance LPFGs in the area of multi-parameter sensing through a pair of LPFGs or other methods and the potential to realize transferring light energy between signal core waveguide and discrete waveguide array.
Acknowledgments
This work was supported by the National Key Basic Research and Development Program of China under Grant No.2010CB327605, the National Natural Science Foundation of China under Grant Nos. 50802044, 10774077, and 60736039, and Doctoral Fund of Ministry of Education of China under Grant No. 200800551025. Prof. Xiaoying Li and Mr. Liang Cui are acknowledged for some experimental measurements.
References and links
1. X. W. Shu, L. Zhang, and I. Bennion, “Fabrication and characterisation of ultra-long-period fibre gratings,” Opt. Commun. 203(3-6), 277–281 (2002). [CrossRef]
2. T. Zhu, Y. J. Rao, and J. L. Wang, “Characteristics of novel ultra-long-period fiber gratings fabricated by high-frequency CO2 laser pulses,” Opt. Commun. 277(1), 84–88 (2007). [CrossRef]
3. T. B. Iredale, P. Steinvurzel, and B. J. Eggleton, “Electric-arc-induced long-period gratings in fluid-filled photonic bandgap fibre,” Electron. Lett. 42(13), 739–740 (2006). [CrossRef]
4. D. I. Yeom, P. Steinvurzel, B. J. Eggleton, S. D. Lim, and B. Y. Kim, “Tunable acoustic gratings in solid-core photonic bandgap fiber,” Opt. Express 15(6), 3513–3518 (2007). [CrossRef] [PubMed]
5. Q. Shi and B. T. Kuhlmey, “Optimization of photonic bandgap fiber long period grating refractive-index sensors,” Opt. Commun. 282(24), 4723–4728 (2009). [CrossRef]
6. L. Jin, Z. Wang, Y. G. Liu, G. Y. Kai, and X. Y. Dong, “Ultraviolet-inscribed long period gratings in all-solid photonic bandgap fibers,” Opt. Express 16(25), 21119–21131 (2008). [CrossRef] [PubMed]
7. D. Noordegraaf, L. Scolari, J. Lægsgaard, L. Rindorf, and T. T. Alkeskjold, “Electrically and mechanically induced long period gratings in liquid crystal photonic bandgap fibers,” Opt. Express 15(13), 7901–7912 (2007). [CrossRef] [PubMed]
8. L. Wei, J. Weirich, T. T. Alkeskjold, and A. Bjarklev, “On-chip tunable long-period grating devices based on liquid crystal photonic bandgap fibers,” Opt. Lett. 34(24), 3818–3820 (2009). [CrossRef] [PubMed]
9. P. Steinvurzel, E. D. Moore, E. C. Mägi, and B. J. Eggleton, “Tuning properties of long period gratings in photonic bandgap fibers,” Opt. Lett. 31(14), 2103–2105 (2006). [CrossRef] [PubMed]
10. P. Steinvurzel, E. D. Moore, E. C. Mägi, B. T. Kuhlmey, and B. J. Eggleton, “Long period grating resonances in photonic bandgap fiber,” Opt. Express 14(7), 3007–3014 (2006). [CrossRef] [PubMed]
11. M. W. Yang, D. N. Wang, Y. Wang, and C. R. Liao, “Long period fiber grating formed by periodically structured microholes in all-solid photonic bandgap fiber,” Opt. Express 18(3), 2183–2189 (2010). [CrossRef] [PubMed]
12. C. R. Liao, Y. Wang, D. N. Wang, and L. Jin, “Femtosecond Laser Inscribed Long-Period Gratings in All-Solid Photonic Bandgap Fibers,” IEEE Photon. Technol. Lett. 22(6), 425–427 (2010). [CrossRef]
13. Y. P. Wang, W. Jin, J. Ju, H. F. Xuan, H. L. Ho, L. M. Xiao, and D. N. Wang, “Long period gratings in air-core photonic bandgap fibers,” Opt. Express 16(4), 2784–2790 (2008). [CrossRef] [PubMed]
14. L. Jin, Z. Wang, Q. Fang, Y. Liu, B. Liu, G. Y. Kai, and X. Y. Dong, “Spectral characteristics and bend response of Bragg gratings inscribed in all-solid bandgap fibers,” Opt. Express 15(23), 15555–15565 (2007). [CrossRef] [PubMed]
15. B. T. Kuhlmey, F. Luan, L. B. Fu, D. I. Yeom, B. J. Eggleton, A. M. Wang, and J. C. Knight, “Experimental reconstruction of bands in solid core photonic bandgap fibres using acoustic gratings,” Opt. Express 16(18), 13845–13856 (2008). [CrossRef] [PubMed]
16. G. B. Ren, P. Shum, L. R. Zhang, X. Yu, W. J. Tong, and J. Luo, “Low-loss all-solid photonic bandgap fiber,” Opt. Lett. 32(9), 1023–1025 (2007). [CrossRef] [PubMed]
17. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef] [PubMed]
18. V. Grubsky, A. Skorucak, D. S. Starodubov, and J. Feinberg, “Fabrication of long-period fiber gratings with no harmonics,” IEEE Photon. Technol. Lett. 11(1), 87–89 (1999). [CrossRef]
19. X. W. Shu, L. Zhang, and I. Bennion, “Sensitivity characteristics of long-period fiber gratings,” J. Lightwave Technol. 20(2), 255–266 (2002). [CrossRef]
20. B. H. Kim, Y. Park, T. J. Ahn, D. Y. Kim, B. H. Lee, Y. Chung, U. C. Paek, and W. T. Han, “Residual stress relaxation in the core of optical fiber by CO(2) laser irradiation,” Opt. Lett. 26(21), 1657–1659 (2001). [CrossRef]
21. J. M. Lázaro, B. T. Kuhlmey, J. C. Knight, J. M. Lopez-Higuera, and B. J. Eggleton, “Ultrasensitive UV-tunable grating in all-solid photonic bandgap fibers,” Opt. Commun. 282(12), 2358–2361 (2009). [CrossRef]
