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Fiber Bragg Gratings with reflectivity up to 25 dB have been photo-written in the core of a 2D all-solid Photonic Bandgap Fiber without modification of the guiding properties of the fiber. This result is obtained by combining an appropriate glass composition for the high index inclusions constituting the micro-structured cladding and a photosensitive low index core. Couplings of the fundamental core guided mode with cladding modes are investigated and compared to theoretical predictions.
©2009 Optical Society of America
1. Introduction
Among the wide family of Photonic Crystal Fibers (PCFs), 2D Solid-Core Photonic BandGap Fibers (SC-PBGFs), like Hollow-Core Photonic BandGap Fibers (HC-PBGFs), are one category of fiber that allows the light to be confined in a low index core (as compared to the effective index of the first cladding mode of the structure, generally noted nfsm) through Photonic BandGap (PBG) effect [1]. From a simple view, the guiding mechanism can be understood taking into account the resonant or anti-resonant couplings that exist between the high index inclusions of the cladding and the core modes of the fiber [2–3]. SC-PBGF geometries that have been reported up to now are of different kinds. The first ones are air/glass fibers for which high refractive index liquid has been inserted into the air holes [4]. This approach, even if it proved interesting tuning features [5], is not that easy to use as compared to an all-solid structure, especially if one needs to splice this fiber. The second category consists in all-glass fibers formed by the assembly of high refractive index glass surrounded by a low refractive index one [6–8]. These all-solid structures allow getting very low minimum losses (~dB/km) [9–10], compared to liquid filled air/glass PCFs (~dB/m). Note that alternative structures based on the addition of Interstitial Air Holes (IAHs) [11] or on hybrid geometries have also been reported [12]. In the case of all-glass SC-PBGFs, the most used structure is based on germanium-doped silica rods embedded in a silica background, the core consisting in a defect of the periodical structure (generally a missing high index rod). The main features of these fibers, in terms of guiding properties, are closely linked to the PBG effect and consist in: i) singular chromatic dispersion properties, ii) spectral filtering of specific wavelengths. Combined to a solid core that can be doped with luminescent ions, these properties have demonstrated potentialities for the realization of all-fiber subpicosecond soliton oscillator or Continuous Wave (CW) laser emission at unusual wavelength [13–15]. In addition to active core doping, the insertion of photosensitive species in the core of SC-PBGFs can, in theory, enable the realization of Fiber Bragg Grating (FBGs) in a PBGF, which is much more challenging in HC-PBGFs due to the limited overlap between core mode and silica. These FBGs could be helpful for the realization of efficient all-fiber laser based on SC-PBGF structure. In addition to this, photo-writing of FBGs in these fibers can help to probe the modal distribution of the fiber. Besides the realization of Long Period Gratings (LPGs) in liquid filled air/silica fibers [16–17] and all-solid SC-PBGFs [18], the realization of FBGs in SC-PBGF has led to poorly efficient gratings (the highest reflectivity reported up to now for the core guided mode is 1.1 dB for 1.3 cm-long FBG). In this paper, we show that it is possible to use conventional FBG writing process to realize efficient FBGs in the core of a SC-PBGF based on phosphosilicate high index inclusions. Different FBGs have then been realized and their spectral features have been analyzed and interpreted.
2. All-solid photonic bandgap fiber based on phosphosilicate inclusions
In the case of all-solid SC-PBGFs, it is necessary to consider that the material constituting the high index inclusions of the cladding (generally, germanosilicate) is not transparent to UV beams commonly used for photo-writing FBGs (193 nm, 244 nm or 248 nm). Moreover, to ensure guiding through PBG effect, it is important to maintain the refractive index of the core close to that of silica, which forbids the use of high germanium concentration in the core and hence limits its photosensitivity. Therefore, in such structures, most of the writing beam will be absorbed by the high index germanosilicate inclusions, limiting the available UV power in the fiber core, and the FBG will be mainly realized in the high index inclusions [19–20]. Hence, it will be difficult to obtain strong resonance for the core guided mode due to its weak overlap with the FBG. Moreover, the transmission of the fiber will be modified due to the refractive index increase of the high index inclusions. An illustration of the result obtained in this situation is given by Jin et al. in [19]. In order to overcome the problem of strong absorption of the UV beam while keeping working with an all-solid structure, we chose to use phosphosilicate inclusions instead of germanosilicate inclusions to realize the micro-structured cladding of our SC-PBGF. We then combined this cladding to a photosensitive germanium/fluorine co-doped core with refractive index close to pure silica. This choice is based on the fact that phosphorous-doped silica presents a very small absorption around 240 nm and is therefore poorly photosensitive as compared to germanium-doped silica [21]. Combined to a germanium-doped silica core, it is hence possible, using a CW beam at 244 nm, to realize the FBG in the core of the SC-PBGF enabling to obtain efficient Bragg resonance. However, it has to be noted that phosphorous-doping of silica is more difficult to realize than germanium doping, especially if one wants to obtain large doped sections (as is generally needed for SC-PBGFs) and well controlled refractive index profile. This is mainly due to the low melting temperature of phosphorous oxide as compared to silica. Moreover, the maximum refractive index difference with silica is more limited (generally less than 15.10-3) as compared to what is accessible with germanium doping.
Fig. 1. (a) Refractive index profile of one of the phosphorous-doped silica core preforms used to realize the high index inclusions of the micro-structured cladding of the SC-PBGF (blue) and of the germanium/fluorine co-doped silica preform used to realize the photosensitive core of the fiber (red). (b) Optical image of the SC-PBGF.
The high-index inclusions have been obtained from large core phosphorous-doped silica preforms realized by Modified Chemical Vapor Deposition (MCVD). The phosphorous oxide contain is around 6 mol.%. The preforms used for this realization present a step-like index profile with a net central dip. The maximum refractive index difference compared to silica is roughly 9.10-3. The refractive index profile of one of the preforms used to realize the high index inclusions is presented in Fig. 1(a). The germanium/fluorine co-doped core has also been realized by MCVD. In order to limit the refractive index increase induced by germanium doping, fluorine (0.9 wt.%) has been added to germanium (3 wt.%) leading to a photosensitive preform presenting a net index increase smaller than 6.10-4 above the background pure silica. The refractive index profile of the corresponding preform is also presented on Fig. 1(a). All the preforms have then been chemically etched and then drawn into capillaries in order to realize the SC-PBGF by the stack and draw method. The final fiber presents 6 rings of high index inclusions with diameter, d, around 8.9 µm and a pitch, Λ, around 13.1 µm. The diameter of the photosensitive region is around 7.6 µm. An optical image of the fiber is given in Fig. 1(b).
3. Guiding properties
The design of the fiber described hereinbefore is based on the results of two different numerical methods: i) a Plane Wave Expansion (PWE) method that allows the bandgap diagram to be calculated, and ii) a Finite Element Method (FEM) for the determination of the Confinement Losses (CL) of the core guided modes. The PWE method used herein is similar to the one described in [22]. In both cases, no material dispersion has been taken into account.
In Fig. 2 are shown both the Density Of States (DOS) obtained from the PWE method and the effective index of the fundamental core guided mode (red curve) calculated using a 9×9 supercell. These calculations have been performed taking into account the refractive index profile of the phosphosilicate preform described previously and the d/Λ parameter equal to 0.7. Note that in the case of the supercell calculation, the core defect includes the photosensitive area. Figure 2 shows that bandgap #3, which was preferentially used in our previous work as it leads to a good compromise between low CL (for reasonable outer fiber diameter) and relatively low bend loss sensitivity, is no more opened below the silica refractive index (green line). This is due to the ring-like structure of the phosphosilicate inclusions (Fig. 1(a)). As a consequence, bandgap #2 has been chosen for this study, as it represents a good compromise in terms of CL, spectral bandwidth and associated pitch for a study around 1.55 µm (a larger pitch leading to stronger bend loss sensitivity). The CL of the fundamental core mode is estimated to have a minimum in the order of 0.2 dB/m around 1.5 µm. At this wavelength, a second core mode is theoretically predicted with CL larger than 600 dB/m, suggesting a single-mode behavior of the fiber.
Fig. 2. DOS diagram of the periodic structure based on a triangular array of high index inclusions presenting the refractive index profile of Fig. 1(a). The color-scale stands for non-zero DOS regions (high DOS in red) whereas white color stands for zero DOS regions. Fundamental core guided mode dispersion in the bandgap #2 is represented in red and glass line is represented in green. The top scale is given for a pitch, Λ, of 13.1 µm. The scalar LPlm modes of the isolated high index inclusions (see for example [20, 23]) from which each band evolves in the high index regime are also reported.
The guiding properties of the fiber have then been investigated experimentally using conventional transmission and cutback methods. A high power supercontinuum source has been injected into a standard single-mode fiber (SSMF) butt-coupled to the SC-PBGF. The output of the SC-PBGF has been either butt-coupled to SSMF connected to an Optical Spectrum Analyzer (OSA) or imaged on a CCD camera using a microscope objective. The transmission curve of a 1.5 m-long sample is reported on Fig. 3(a). One can observe the different transmission windows corresponding to bandgap #2 to bandgap #7. Note that, as supposed by the band diagram modeling, bandgaps #3 and #5 are not guiding. An attenuation curve obtained from a cutback realized on an 8 m-long sample is reported on Fig. 3(b). On this figure, the intensity profile of the core guided mode around 1550 nm is also reported in inset. For this measurement, the bending radius of the fiber has been fixed to 15.8 cm in order to prevent the measurement from being disturbed by the bending losses. The bandgap spectral width over which experimental losses are below 4 dB/m is around 200 nm and the minimum value is around 0.84 dB/m at 1546 nm.
Fig. 3. (a) Transmission curve measured on a 1.5 m-long fiber. (b) Attenuation curve centered on bandgap #2 and obtained by cutting back an 8 m-long sample. Optical image of the guided mode at 1550 nm is reported in inset.
4. Characteristics of the FBGs
As explained before, the originality of the design proposed for the realization of FBG in SC-PBGF is based on the fact that the phosphorous-doped silica used for the high index inclusions presents low absorption around 240 nm, leaving most of the UV writing beam available for the photosensitive core. The set-up we used for the realization of FBGs is based on an interferometer coupled to a CW 244 nm laser (see for example [24]). This set-up is very flexible and enables FBGs to be realized at very different wavelengths depending on the laser beam incident angle on the Lloyd mirror we used to create the interference pattern. The length of our FBGs is around 2 mm and the power density is estimated to 70 W/cm2. It has to be noticed that, in order to increase the photosensitivity response of our fiber, hydrogen loading procedure has been performed at 90°C under 140 atm H2 pressure during 3 days. FBGs have then been written at different wavelengths (1437 nm, 1526 nm, 1547 nm, 1571 nm and 1641 nm) in bandgap #2.
Figure 4(a) presents the reflected and transmitted power spectra of the FBG written around 1526 nm. This illustrates the main features observed for all the FBGs realized in this work. These curves have been obtained using a high power supercontinuum source and a Fiber Optical Circulator (FOC). The supercontinuum signal is sent to the first channel of the FOC and the second channel is connected to a SSMF pigtail butt-coupled to the SC-PBGF input. To analyze the reflectivity of the FBG, the third channel of the FOC is send to an OSA. To analyze the transmission of the FBG, the SC-PBGF output is butt-coupled to a SSMF connected to an OSA. As can be seen on Fig. 4(a), a strong resonance (noted R1) appears on the transmission spectrum with amplitude of 25 dB. This Bragg resonance is also observed on the reflectivity spectrum and is slightly asymmetric on the long wavelength side. Three other resonance peaks (noted R2, R3 and R4) are also observed, on the transmission spectrum, on the short wavelength side of resonance R1. The maximum amplitude of these resonances is around 3 dB. Figure 4(b) presents the transmission spectrum of a roughly 2 m-long sample, over the full width of bandgap #2, before and after photo-writing the FBG at 1437 nm (a high-resolution transmission spectrum of this FBG is reported in inset of Fig. 4(b). Similar features are observed for the other FBGs.
Fig. 4. (a) Reflected and transmitted power spectra of the FBG written around 1526 nm. (b) Transmission curves of bandgap #2 before and after photo-writing of the FBG at 1437 nm. Resolution was set to 1 nm. A spectrum of the FBG recorded at high resolution (0.05 nm) is reported in inset.
5. Discussion
The non-continuous transmission spectrum (Fig. 3(a)) and the shape of the core guided mode in bandgap #2 with clear resonances in the high index inclusions of the first ring (inset of Fig. 3(b)) are two evidences that the guiding mechanism associated to our SC-PBGF is correlated to PBG effect. Hence, the photosensitive region in the core and its associated refractive index difference as compared to silica does not impact significantly on the guiding properties of the fiber. Concerning losses, it appears that the experimental minimum background losses are more than 4 times higher than the theoretical one. This indicates that the design geometry (number and opto-geometrical parameters of the high index inclusions that fix CL level) is not the only limiting loss factor of our structure and that geometry defects and contaminations induced by the process contribute to the losses. These defects can be attributed to the phosphosilicate nature of the high index inclusions that presents a significantly smaller glass working temperature as compared to the pure silica background. In combination with geometry defects induced by the chemical etching of the preforms, this explains the non-circular shape of the high index inclusions and the geometry defects of the cladding. However, it has been shown that CL in SC-PBGFs can be reduced using an air-clad [9] or IAHs [11] and we are confident that these approaches could be applied to the present structure. Finally, it has to be noticed that the fiber can be considered as single-mode as no high order core guided mode could be observed even on samples as short as roughly 1 cm.
Concerning FBGs, the main resonance observed (noted R1) can clearly be attributed to the core guided mode as it appears both on the transmission and reflected power spectra. The reflectivity observed (generally more than 20 dB and up to 25 dB for the FBG written at 1526 nm) for this Bragg resonance is the highest reported up to now for a PBGF. As can be seen on Fig. 4(b), this high reflectivity is obtained without modification of the transmission properties of the fiber. This is explained by the fact that the UV beam has not modified the refractive index of the phosphosilicate inclusions, which hence possess the same resonance features before and after UV writing. It has to be noted that the increase of the core guided mode mean effective index, induced by photo-writing, is estimated to 6.3.10-4. This value has been obtained by analyzing the Bragg wavelength shift during photo-writing.
As for any kind of fibers, FBGs written in PBGFs enable to observe couplings of the core guided mode with cladding modes of smaller effective index, leading to resonances at shorter wavelengths than the Bragg resonance. However, as discussed by Jin et al., FBGs written in PBGFs also offer the singular possibility of couplings between the core guided mode and cladding modes of higher effective index, leading to resonances appearing now at longer wavelengths than the Bragg resonance [19]. Effectively, looking at Fig. 2, one can imagine that phase matching conditions can exist between the core guided mode and the counter propagating cladding modes located below (respectively above) the bottom (respectively top) limit of bandgap #2. As shown by Jin et al., these phase-matching conditions can even be extended to the cladding modes delimiting bandgap #1 [20].
Fig. 5. (a) Zoom of Fig. 2 in the wavelength range of interest. (b) Representation of the phase-matching condition defined by Eq. 1 for the FBG illustrated on Fig. 4(a). Black curve illustrates the coupling of the core guided mode with counter-propagating core guided modes (corresponding to resonance R1 on Fig. 4(a)) whereas pink, red, blue, green and orange curves illustrate the phase-matching conditions between core guided mode and cladding modes located in, respectively, high DOS regions 5, 1, 2, 3 and 4. All the spectral dependences of the effective indices have been extracted from Fig. 2 including, for the core guided mode, the mean refractive index increase induced by UV beam.
To take this effect into account, we have considered possible phase-matching conditions with cladding modes located in high DOS regions of Fig. 2, for which a zoom is given on Fig. 5(a). Above-mentioned phase-matching conditions are illustrated on Fig. 5(b) and can be summarized by Eq. (1).
where ΛFBG is the spatial periodicity of the FBG, λ is the wavelength, neff1 and neff2 are the effective indices of the two modes considered (core guided mode and counter-propagating core guided mode OR core guided mode and counter-propagating cladding mode). Note that, in the case of the core guided mode, the mean effective index increase induced by UV must be taken into account. The Bragg resonance R1 has been considered, together with the effective index calculated by FEM, to determine ΛFBG (526.36 nm in the example of the FBG written at 1526 nm).
Looking at Fig. 5(b), it appears that resonances R2 and R3 fall in the region of coupling of the core guided mode with cladding supermodes located in the high DOS regions 1 to 3. As R2 and R3 do not appear as two discrete resonances on the transmission spectrum but more as a broad resonance, one can imagine that a quasi-continuous coupling of the core guided mode with cladding modes located in the high DOS regions 1 to 3 is observed in the 1523.6–1525.2 nm wavelength range. In the case of resonance R4, a coupling of the core guided mode with cladding modes located in the high DOS region 4 is suspected according to the fairly good agreement observed on the phase-matching diagram between resonance position and phase matching curve. Concerning the long wavelength side, it has to be noticed that the FBGs realized generally present an asymmetric shape for the resonance R1 (see transmission curve on Fig. 4(a)). As FBGs realized on the same set-up in a SSMF do not present such asymmetry, it can be supposed that this asymmetry could be linked to a coupling of the core guided mode with cladding modes located in the high DOS region 5. This assumption is rather in good agreement with the phase-matching conditions shown in pink on Fig. 5(b) between the core mode and this high DOS region 5.
Fig. 6. Optical images of the light reflected by the FBG written around 1526 nm. From left to right, the wavelength of the light collected is around 1524.8nm (a), 1526.1 nm (b) and 1526.7 nm (c). Position of high index inclusions is reported in clear yellow.
To validate further this hypothesis, the set-up used for the characterization of FBGs has been modified: collimated light from a tunable laser source was injected into the PBGF core thanks to a microscope objective. A semi-transparent mirror was inserted, at an angle of 45°, before the microscope objective in order to collect on a CCD camera the light reflected by the FBG. As illustrated by Fig. 6, both cladding modes and core guided modes can be collected, which proves the existence of coupling of the co-propagating core guided mode with both counter propagating core guided mode (Fig. 6(b)) and counter propagating cladding modes (Figs. 6(a) and 6(c)). The cladding modes observed at short wavelengths (Fig. 6(a)) fall in the wavelength range of resonance R2 and confirm a coupling with radiative cladding modes (effective index smaller than the background index) of the high DOS regions 1 to 3. As expected for these modes, the light spreads significantly out of the high index inclusions. On the opposite, the cladding modes observed at long wavelengths (Fig. 6(c)) correspond to guided cladding LP11 supermodes located in the high DOS region 5 for which coupling with core guided mode is predicted by the pink curve of Fig. 5(b). For these cladding supermodes, light is much more localized in the high index inclusions (effective index higher than background index) with LP11-like modes in each high index inclusions, as predicted by PWE method (see Fig. 2 and [20]). This result confirms that the coupling with the high DOS region 5 appearing very close to the Bragg resonance R1 could explain the asymmetry of this resonance peak.
6. Conclusion
By an adapted choice of glass compositions used for the realization of a 2D SC-PBGF, it has been shown the possibility to realize, for the first time to our knowledge, efficient FBGs in a PBGF. The reflectivity obtained for the core guided mode is at least 20 dB with limited coupling to cladding supermodes. We believe that these results could open new perspectives in the realization of all-fiber lasers based on SC-PBGFs.
Acknowledgments
The authors would like to thank Karen Delplace for providing technical support. This work was supported in part by the “Conseil Régional Nord Pas de Calais”, the “Fonds Européen de Développement Economique des Régions” and the “Agence Nationale de la Recherche” (ANR-05-BLAN-0080).
References and links
1. P. St. J. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24, 4729–4749 (2006). [CrossRef]
2. N. M. Litchinister, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. Martijn de Sterke, “Resonances in microstructured optical waveguides,” Opt. Express 11, 1243–1251 (2003). [CrossRef]
3. G Renversez, P. Boyer, and A. Sagrini, “Antiresonant reflecting optical waveguide microstructured fibres revisited: a new analysis based on leaky mode coupling,” Opt. Express 14, 5682–5687 (2006). [CrossRef] [PubMed]
4. J. Jasapara, T. H. Her, R. Bise, R. Windeler, and D. J. DiGiovani, “Group-velocity dispersion measurements in a photonic bandgap fiber,” J. Opt. Soc. Am. B 20, 1611–1615 (2003). [CrossRef]
5. T. P. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, “Optical devices based on liquid crystal photonic bandgap fibres,” Opt. Express 11, 2589–2596 (2003). [CrossRef] [PubMed]
6. A. Argyros, T. A. Birks, S. G. Leon-Saval, C. B. Cordeiro, F. Luan, and P. St. J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express 13, 309–314 (2005). [CrossRef] [PubMed]
7. F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. J. Russell, “All solid photonic bandgap fiber,” Opt. Lett. 29, 2369–2371 (2004). [CrossRef] [PubMed]
8. G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm,” Opt. Express 13, 8452–8459 (2005). [CrossRef] [PubMed]
9. A. Bétourné, V. Pureur, G. Bouwmans, Y. Quiquempois, L. Bigot, M. Perrin, and M. Douay, “Solid photonic bandgap fiber assisted by an extra air-clad structure for low-loss operation around 1.5µm,” Opt. Express 15, 316–324 (2006). [CrossRef]
10. G. Ren, P. Shum, L. Zhang, X. Yu, W. Tong, and J. Luo, “Low loss all solid photonic bandgap fiber,” Opt. Lett. 32, 1023–1025 (2007). [CrossRef] [PubMed]
11. A. Bétourné, G. Bouwmans, Y. Quiquempois, M. Perrin, and M. Douay, “Improvements of solid-core photonic bandgap fibers by means of interstitial air holes,” Opt. Lett. 32, 1719–1721 (2007). [CrossRef] [PubMed]
12. A. Cerqueira, F. Luan, C. M. B. Cordeiro, A. K. George, and J.C. Knight, “Hybrid photonic crystal fiber,” Opt. Express 14, 926–931 (2006). [CrossRef]
13. A. Isomaki and O. G. Okhotnikov, “Femtosecond soliton mode-locked laser based on ytterbium-doped photonic bandgap fiber,” Opt. Express 14, 9238–9243 (2006). [CrossRef] [PubMed]
14. A. Wang, A. K. George, and J. C. Knight, “Three-level neodymium fiber laser incorporating photonic bandgap fiber,” Opt. Lett. 31, 1388–1390 (2006). [CrossRef] [PubMed]
15. V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008). [CrossRef]
16. 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, 3007 (2007). [CrossRef]
17. D. Noordegraaf, L. Scolari, J. Laegsgaard, L. Rindorf, and T. T. Alkeskjold, “Electrically and mechanically induced long period gratings in liquid crystal photonic bandgap fibers,” Opt. Express 15, 7901–7912 (2007). [CrossRef] [PubMed]
18. L. Jin, Z. Wang, Y. Liu, G. Kai, and X. Dong, “Ultraviolet-inscribed long period gratings in all-solid photonic bandgap fibers,” Opt. Express 16, 21119–21131 (2008). [CrossRef] [PubMed]
19. L. Jin, Z. Wang, Q. Fang, B. Liu, Y. Liu, G. Kai, X. Dong, and B. O. Guan, “Bragg grating resonances in all-solid bandgap fibers,” Opt. Lett. 32, 2717–2719 (2007). [CrossRef] [PubMed]
20. L. Jin, Z. Wang, Q. Fang, Y. Liu, B. Liu, G. Kai, and X. Dong, “Spectral characteristics and bend response of Bragg gratings inscribed in all-solid bandgap fibers,” Opt. Express 1515555–15565 (2007). [CrossRef] [PubMed]
21. B. Malo, J. Albert, F. Bilodeau, T. Kitagawa, D. C. Johnson, K. O. Hill, K. Hattori, Y. Hibino, and S. Gujrathi, “Photosensitivity in phosphorus-doped silica glass and optical waveguides,” Appl. Phys. Lett. 65394–396 (1994). [CrossRef]
22. G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptative curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystal,” Phys. Rev. B 71195108- (2005). [CrossRef]
23. T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express 14, 9483–9490 (2006). [CrossRef] [PubMed]
24. G. Meltz, W. W. Morey, and W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989). [CrossRef] [PubMed]
