We demonstrate that thousands of periodic nano-craters are fabricated on a subwavelength-diameter tapered optical fiber, an optical nanofiber, by irradiating with just a single femtosecond laser pulse. A key aspect of the fabrication is that the nanofiber itself acts as a cylindrical lens and focuses the femtosecond laser beam on its shadow surface. We also demonstrate that the periodic nano-crater array on the nanofiber shows polarization dependent fiber Bragg grating (FBG) characteristics. Such FBG structures on the nanofiber may act as a 1-D photonic crystal due to the strong transverse and longitudinal confinement of the field.
© 2013 Optical Society of America
In recent years, tapered optical fibers with subwavelength diameters, known as optical nanofibers, have opened a new paradigm in manipulating single/few-atom fluorescence [1–3]. A nanofiber is realized by adiabatically tapering a conventional single mode optical fiber which enables light in the fundamental mode to be coupled in and out of the nanofiber with more than 95% efficiency. Due to the subwavelength diameter the field in the guided mode is strongly confined in the transverse direction. As a result, the spontaneous emission of atoms can be strongly modified around the nanofiber and an appreciable amount of atomic fluorescence can be coupled to the guided modes [4, 5]. Also the strong evanescent tail of the guided field gives excellent access for trapping and probing few atoms around the nanofiber [6–9]. The light-matter coupling in such a nanofiber system can be substantially improved by fabricating fiber Bragg grating (FBG) structures on the nanofiber. Such FBG structures on the nanofiber can provide longitudinal confinement of the field in the nanofiber guided modes and will act as a 1-D photonic crystal (PhC) [10–12]. Due to the strong transverse confinement of the nanofiber guided modes, the coupling between the atom and the guided modes will be significantly enhanced and the strong-coupling regime can be realized, even for moderate finesse . Such PhC nanofibers, may provide a promising platform for realizing quantum-state manipulation in a fiber-based system and may open new avenues in quantum information technology .
Apart from the quantum optics applications, the tapered micro/nanofibers have also revolutionized fiber optics technologies for various photonics applications . Conventional FBGs  and FBG-based interferometers have played an important role in various applications like fiber lasers , FBG sensors  and all-optical switching [19–21]. The merging of the FBG technology with the tapered micro/nanofibers can significantly improve the device performance due to tight field confinement and stronger near-field interaction with the surrounding medium.
In recent years various techniques for fabricating micro/nanofiber Bragg gratings and cavities have been reported. We have demonstrated fabrication of nanofiber cavities by drilling periodic nano-grooves on the nanofibers using a focused ion beam (FIB) milling technique . Other groups have also demonstrated FIB fabrication of microfiber Bragg gratings for highly sensitive refractive index sensing [23, 24]. However, there are various technical limitations in the FIB fabrication such as contamination from the substrate or the ion beam itself and mechanical instability of the nanofiber due to charging up effects. In this context fabrication of nanofiber Bragg gratings using femtosecond laser induced ablation or refractive index modification might be a better approach . There are only a few works reporting the fabrication of microfiber Bragg gratings using femtosecond laser irradiation [26, 27]. However to the best of our knowledge femtosecond laser fabrication on submicron fibers has not been reported yet.
In this paper, we present an optical method for fabricating FBG structures on optical nanofibers using a femtosecond laser. We demonstrate that thousands of periodic nano-craters are fabricated on optical nanofibers by irradiating with just a single femtosecond laser pulse. A key aspect of the fabrication is the lensing effect of the nanofiber itself. We also demonstrate that the periodic nano-crater array on the nanofiber shows polarization dependent FBG characteristics. Such FBG structures on the nanofiber may act as a 1-D PhC due to the strong transverse and longitudinal confinement of the field.
2.1. Fabrication setup
A schematic diagram of the fabrication setup is shown in Fig. 1(a). For the fabrication we use a femtosecond laser having a central wavelength (λ) of 400 nm, derived from the second harmonic generation of a Ti-Sapphire regenerative amplifier system (Coherent Libra-HE). The femtosecond laser generates 120 fs pulses at a repetition rate of 1 kHz with maximum pulse energy of 1.3 mJ. A Talbot interferometer , consisting of a phase mask as beam splitter and two folding mirrors (M1 and M2), is built to create the two-beam interference pattern on the nanofiber. The phase mask used for the fabrication has a uniform pitch (ΛP) of 700 nm and was designed for zero order suppression at 400 nm wavelength. The phase mask splits the femtosecond laser beam into ± first orders at angles (θ = sin−1(λ/ΛP)) of ±34.85°. The first orders are then recombined by the folding mirrors. The folding mirrors are mounted on Z-stages and their position is carefully adjusted to symmetrically recombine the first orders at the nanofiber position, thereby creating an interference pattern with a period (ΛG = ΛP/2) of 350 nm. A cylindrical lens of focal length 100 mm is used to line-focus the femtosecond laser beam along the nanofiber. The typical 1/e2 beam size at the nanofiber position is 5.4 mm × 60 μm. The polarization of the femtosecond laser beam is perpendicular to the nanofiber axis. Tapered nanofibers with waist diameters (2a) of 450–650 nm are used for the fabrication. The tapered fiber is mounted on a metallic holder having a central hole of 6 cm length, so that the nanofiber is protected from contamination due to ablation of the substrate. The metallic holder containing the nanofiber is fixed on a fabrication bench equipped with XYZ and rotation stages. The Y-stage can also control the tilt angles. All the experiments are done under dust-free conditions to maintain the transmission of the nanofiber.
2.2. Alignment of the interferometer
The interferometer alignment is highly sensitive to the optical path length difference between the two first orders. The path length difference must be smaller than 36 μm as the spatial overlap is limited by the femtosecond laser pulse width [25, 28]. Hence, the positions of the folding mirrors are controlled by Z-stages and the mirror tilt angles are precisely controlled. The alignment of the interferometer is performed using the following procedure. The femtosecond laser is irradiated on a glass plate placed on the fabrication bench and the pulse energy is increased to see ablation on the glass plate. The femtosecond laser induced ablation can be identified by the white light generation, and the ablation pattern appears as a damage-line on the glass plate. The position of the glass plate is varied using the Y-stage to find the beam focus position by searching for the strongest ablation at minimum pulse energy. Then the position and the tilt angles of the folding mirrors are adjusted to maximize the overlap of the ± first orders at this focus position. It must be noted that apart from the maximum overlap of the ± first orders, the path length must be matched to achieve interference. The interference condition can be identified by an increase in the ablation strength. It is due to the fact that when the ± first orders interfere the local intensity will be doubled and stronger ablation can occur at the same pulse energy. Thus the best interference condition is achieved by adjusting the position and the tilt angles of the folding mirrors and searching for the strongest ablation at minimum pulse energy. Finally the tilt angle of the fabrication bench is adjusted to optimize the alignment. The visibility of the interference fringes is further confirmed by observing the periodic ablation pattern on the glass plate using a scanning electron microscope (SEM).
2.3. Fabrication method
After the alignment of the interferometer, the glass plate is replaced by the tapered nanofiber. The waist part (nanofiber region) of the tapered fiber is identified, by sending a probe laser (wavelength: 850 nm) through the tapered fiber and observing the scattering pattern using a CCD camera. The maximum scattering of the guided field can be observed at the nanofiber region due to the subwavelength diameter. After identifying the nanofiber region, the probe laser is switched off and the femtosecond laser beam with minimum pulse energy is irradiated on the nanofiber. The overlap of the femtosecond laser beam with the nanofiber is achieved by adjusting the X-stage and rotation-stage. The overlap is optimized by observing the scattering of femtosecond laser light into the nanofiber guided modes using a photodiode connected to one end of the tapered fiber. After the alignment, the femtosecond laser is irradiated on the nanofiber by setting the pulse energy, repetition rate and irradiation time for the fabrication. After fabrication the nanofiber samples are observed using the SEM.
2.4. Measurement of optical properties
The optical properties of the nanofiber samples are characterized by measuring the transmission and reflection spectra. A schematic of the experimental setup is shown in Fig. 1(b). A supercontinuum light source (SuperK Extreme, NKT Photonics) is launched into the tapered fiber sample and both transmission and reflection spectra are measured simultaneously. The transmission spectrum is measured using a Fourier transform spectrometer (FT-spectrometer). Whereas, the reflection spectrum is measured using an optical multi-channel analyzer (OMA). The resolutions of the FT-spectrometer (Nicolet 8700, Thermo Fisher Scientific) and OMA (QE65000, Ocean Optics) are 0.01 nm and 2 nm, respectively. The transmittance and reflectance values are calibrated using a CW Ti-Sapphire laser source (MBR-110, Coherent Inc.). An inline fiber polarizer is used to control the polarization of the input light. For confirmation, a part of the transmitted light is used for polarization measurements.
3. Results and discussions
3.1. Nano-crater formation on nanofibers
For the fabrication, we vary the femtosecond laser pulse energy, repetition rate and irradiation time (number of shots) to find the optimum conditions. It was found that for 1 kHz repetition rate, the nanofiber can be completely destroyed within 1 second, even at pulse energy of only 200 μJ. A repetition rate as low as 100 Hz can still seriously damage the nanofiber. However, we found that by reducing the number of shots to less than 20, we could realize controlled nanofabrication on the nanofiber. By systematically changing the number of shots and pulse energy, we found that single-shot irradiation is the best condition to realize clean nanofabrication. In the following, we discuss the fabrication results for multiple-shot and single-shot irradiation.
3.1.1. Multiple-shot fabrication
Figure 2(a) shows the SEM image of a typical sample fabricated by 20-shot irradiation with a pulse energy of 370 μJ. We have found that many ablated structures are formed on the nanofiber. We must mention that these ablated structures are formed, not on the irradiation side, but on the shadow side of the nanofiber. As one can see the ablation pattern is quite irregular. However, a crucial observation is that although the beam size along the X-axis (60 μm) is much larger than the nanofiber diameter, the ablation pattern is formed in a line along the fiber axis (Z-axis).
Figure 2(b) shows the SEM image of a typical sample fabricated by 3-shot irradiation with a pulse energy of 560 μJ. One can see that a rather regular and periodic ablation pattern is formed on the shadow side of the nanofiber. However, the ablated structures are elliptical and are elongated along the Z-axis. The typical size of an ablated structure is 90 nm × 210 nm. We would like to note that such ellipticity is observed throughout the ablation pattern, even for the weakest ablated structure.
3.1.2. Single-shot fabrication
Figure 3(a) shows the SEM image of a typical sample fabricated using single-shot irradiation with a pulse energy of 630 μJ. As one can clearly see, periodic nano-crater structures are formed on the shadow side of the nanofiber. Such periodic structures are quite systematically formed over a length of a few mm, consisting of thousands of such nano-craters. The inset shows the enlarged view of the sample. The shape of the nano-craters is almost circular and the diameter of a typical nano-crater is around 210 nm. The nano-craters are formed with a periodicity of 350 nm, which corresponds well to the interference fringe spacing. The depth of a typical nano-crater was measured by cutting the nanofiber at the nano-crater position and observing the cross-section using the SEM. The cross-sectional view is shown in Fig. 3(b). The nano-crater has a bowl-like shape and the depth is ∼120 nm.
3.1.3. Lensing effect of the nanofiber
It should be noted that the ablated structures shown in Fig. 2 and Fig. 3 are formed on the shadow side of the nanofiber. We could not observe any particular structure on the irradiation side. This suggests that the nanofiber itself acts as a cylindrical lens and focuses the femtosecond laser beam on its shadow surface [29,30]. Also as shown in Figs. 2(a) and 2(b), the ablation pattern is formed exactly in a line along the fiber axis (Z-axis), confirming the lensing effect of the nanofiber.
The uniqueness of the present method is that the lensing effect of the nanofiber makes it robust to any mechanical instabilities in the transverse direction (X-axis). However instabilities along the Z-axis can seriously affect the fabrication for multiple-shot irradiation, as the intensity pattern on the nanofiber may differ for each shot. This is evident from the observed irregular ablation pattern for 20-shot irradiation (Fig. 2(a)), where the periodicity is completely washed out due to instabilities along Z-axis. Even for 3-shot irradiation (Fig. 2(b)) one can see that the ablated structures are elongated along the fiber axis. However such instability does not affect the fabrication in single-shot irradiation as the irradiation time is only 120 fs (i.e. pulse length). As a result periodic nanostructures with well defined shape (shown in Fig. 3(a)) and periodicity are fabricated, without taking any special care to suppress mechanical vibrations. Moreover, such a single-shot irradiation technique makes the fabrication method highly reproducible. The lensing effect of the nanofiber may explain the shape of the nano-craters. Although the field distribution along the Z-axis is limited by the interference pattern, the circular shape of the nano-craters is mainly due to focusing of the femtosecond laser along the X-axis.
3.2. Diameter distribution of nano-craters fabricated using single-shot irradiation
The diameter distributions of the nano-craters for two nanofiber samples, fabricated using single-shot irradiation, are plotted in Figs. 4(a) and 4(b), along with the corresponding nanofiber diameter. For Sample 1 the nano-craters are fabricated in the tapered region, and the diameter of the nanofiber varies from 500 nm to 680 nm. A periodic array of nano-craters is formed in a region exceeding 2 mm along the nanofiber axis. The diameter profile of the nano-craters shows a peak-like behavior with diameter ranging from 95 nm to 290 nm. On the other hand, for Sample 2 the fabrication is done near to the waist region, and the diameter of the nanofiber is almost constant around 490 nm. The nano-craters are formed in a region of 0.8 mm along the nanofiber. The diameter of the nano-craters varies from 95 nm to 220 nm.
The peak-like profile of the nano-crater array may be understood from the intensity profile of the femtosecond laser. However, the length of the nano-crater array is smaller than the fabrication beam size of 5.4 mm. This may be due to a fact that the femtosecond laser induced ablation involves a multi-photon ionization process [25, 31]. Hence the ablation process scales as a higher power of the intensity and has some threshold condition. In order to understand the diameter profile of the nano-craters, one must also consider the nanofiber diameter as the lensing effect of the nanofiber depends on the size parameter (2πa/λ). In the case of Sample 2, the reduction in the length of the nano-crater array may be understood from the decrease in the lensing effect of the nanofiber due to thinner diameter [29, 30].
3.3. Optical properties of nanofibers with periodic nano-craters
Optical properties of the nanofiber samples, fabricated using single-shot irradiation, were characterized by measuring the transmission and reflection spectra. The transmission spectrum of Sample 1 in Fig. 5(a), shows a strong and broad dip centered around 825 nm with a width of 37 nm FWHM. In the wavelength region from 815 nm to 836 nm there is almost no transmission, indicating the stop band. High reflectivity of 99% was measured in this stop band region using the CW Ti-Sapphire laser.
The transmission and reflection spectra of Sample 2 are shown in Fig. 5(b). The transmission and reflection spectra were measured for two orthogonal linear polarizations, X-polarization and Y-polarization. The transmission spectrum for the X-polarization shows a dip centered around 823 nm with a width of 10 nm FWHM. The wavelength region from 819.5 nm to 824 nm denotes the stop band. Note that the transmission spectrum for the Y-polarization is blue shifted having a dip at 817.5 nm with a width of 6.8 nm FWHM. The inset shows the transmission spectrum for the Y-polarization on an expanded scale. The spectrum is asymmetric and shows several sharp peaks in the red side. The reflection spectra almost match to the transmission spectra except that the fine features are washed out due to the lower resolution of the OMA used for the reflection measurements. For the X-polarization, we observed 99% reflection in the stop band region, whereas for the Y-polarization the peak reflectivity was measured to be 80%.
The observation of high reflectivity values in the stop band region, clearly demonstrates that the periodic nano-craters on the nanofiber induce strong modulation of refractive index and act as a FBG. The observed spectral width for Sample 1 is 3.7 times broader than that of Sample 2. This might be due to the nanofiber diameter variation for Sample 1. The diameter of the nanofiber determines the effective index (neff) and the Bragg resonance (λR = 2neffΛG) [10, 22]. For thinner diameter of the nanofiber the effective index is smaller and results in a blue shift in the Bragg resonance. So for Sample 1, the diameter variation along the nano-crater array may induce a chirp in the Bragg resonance resulting in a broader spectrum. In the case of Sample 2, the observed blue shift for the Y-polarization may also be explained using the above argument. Due to the formation of nano-craters, the effective diameter along the Y-axis is reduced, resulting in such blue shift. Similar characteristics were also reported for nanofiber Bragg gratings fabricated using an FIB milling technique .
The peak reflectivity value for the Y-polarization is 80% whereas the transmittance value at this wavelength is 12.5%. This suggests that, for the Y-polarization, there is a few percent scattering loss due to the nano-craters. However, for the X-polarization, the scattering loss is negligible as is evident from the high transmittance values away from the stop-band region. The observed low scattering loss and high reflectivity might be due to the peak-like profile of the nano-crater array as shown in Fig. 4(b). The gradual increase of the nano-crater diameter induces an adiabatic index modulation and reduces the scattering into the radiation modes.
The sharp peaks observed in the transmission spectrum for the Y-polarization, can be understood from the peak-like profile of the nano-crater array, as discussed for apodized FBGs in Ref. . As shown in Fig. 4(b) the diameter of the nano-craters is reduced towards the wings of the nano-crater array, resulting in relatively higher effective index at the wings compared to the central part. As a result in these wing regions the Bragg resonance may occur at relatively longer wavelengths compared to the central part. Therefore the nano-crater array acts as a Fabry-Perot (FP) cavity for longer wavelengths. As shown in the inset of Fig. 5(b), the spacing between the side peaks is reducing towards the red side. From the average value of the spacing between the side peaks we estimate an effective cavity length of 0.28 mm which is reasonable given the observed profile of the nano-crater array shown in Fig. 4(b). Also for the X-polarization, similar side peaks are faintly observed at the band edge of the spectrum in the red side. Such FP resonances may be used for various applications. However for certain applications such resonances might be undesirable and may be removed by using a flat-top intensity distribution for the fabrication beam.
In conclusion we have demonstrated the formation of periodic nano-craters on a sub-wavelength diameter silica fiber using a femtosecond laser ablation technique. The ablation is achieved by irradiating only a single femtosecond laser pulse. Thanks to the lensing effect of the nanofiber, thousands of circular nano-crater structures are formed on the shadow surface of the nanofiber. Such a fabrication method may open new prospects in nanophotonics and nanofabrication technologies. We have demonstrated that the periodic nano-crater array on the nanofiber shows polarization dependent FBG characteristics. Such FBG structures on the nanofiber may act as a 1-D PhC due to the strong transverse and longitudinal confinement of the field. We expect that the PhC nanofiber system can become a promising workbench for quantum non-linear optics and will open new avenues in quantum information technology, by combination with laser-cooled atoms or solid-state quantum emitters. Also the PhC nanofibers may open up exciting new applications in lasing, optical switching and chemical/biological sensing.
We are thankful to Mark Sadgrove and Makoto Morinaga for fruitful discussions. Also we wish to thank Hitachi High-Tech Corp., Japan, for helping in SEM measurements with SU8040. This work was supported by the Japan Science and Technology Agency (JST) as one of the Strategic Innovation projects.
References and links
1. K. P. Nayak, P. N. Melentiev, M. Morinaga, F. L. Kien, V. I. Balykin, and K. Hakuta, “Optical nanofiber as an efficient tool for manipulating and probing atomic Fluorescence,” Opt. Express 15, 5431–5438 (2007). [CrossRef] [PubMed]
2. K. P. Nayak and K. Hakuta, “Single atoms on an optical nanofibre,” New J. Phys. 10, 053003 (2008). [CrossRef]
4. F. L. Kien, S. Dutta Gupta, V. I. Balykin, and K. Hakuta, “Spontaneous emission of a cesium atom near a nanofiber: Efficient coupling of light to guided modes,” Phys. Rev. A 72, 032509 (2005). [CrossRef]
5. R. Yalla, F. L. Kien, M. Morinaga, and K. Hakuta, “Efficient channeling of fluorescence photons from single quantum dots into guided modes of optical nanofiber,” Phys. Rev. Lett. 109, 063602 (2012). [CrossRef] [PubMed]
6. F. L. Kien, V. I. Balykin, and K. Hakuta, “Scattering of an evanescent light field by a single cesium atom near a nanofiber,” Phys. Rev. A 73, 013819 (2006). [CrossRef]
7. F. L. Kien, V. I. Balykin, and K. Hakuta, “Atom trap and waveguide using a two-color evanescent light field around a subwavelength-diameter optical fiber,” Phys. Rev. A 70, 063403 (2004). [CrossRef]
8. E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010). [CrossRef] [PubMed]
9. A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012). [CrossRef] [PubMed]
10. F. L. Kien, K. P. Nayak, and K. Hakuta, “Nanofibers with Bragg gratings from equidistant holes,” J. Modern Opt. 59, 274–286 (2012). [CrossRef]
11. J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997). [CrossRef]
13. F. L. Kien and K. Hakuta, “Cavity-enhanced channeling of emission from an atom into a nanofiber,” Phys. Rev. A 80, 053826 (2009). [CrossRef]
15. L. Tong, F. Zi, X. Guo, and J. Lou, “Optical microfibers and nanofibers: A tutorial,” Opt. Commun. 285, 4641–4647 (2012). [CrossRef]
16. K. O. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276(1997). [CrossRef]
17. C. R. Giles, “Lightwave applications of fiber Bragg gratings,” J. Lightwave Technol. 15, 1391–1404 (1997). [CrossRef]
18. A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997). [CrossRef]
19. Zhi-Gang Zang and Wen-xuan Yang, “Theoretical and experimental investigation of all-optical switching based on cascaded LPFGs separated by an erbium-doped fiber,” J. Appl. Phys. 109, 103106 (2011). [CrossRef]
20. Zhi-Gang Zang and Yu-Jun Zhang, “Low-switching power (<45 mW) optical bistability based on optical nonlinearity of ytterbium-doped fiber with a fiber Bragg grating pair,” Modern J. Opt. 59, 161–165 (2012). [CrossRef]
21. Zhi-Gang Zang, “Numerical analysis of optical bistability based on Fiber Bragg Grating cavity containing a high nonlinearity doped-fiber,” Opt. Commun. 285, 521–526 (2012). [CrossRef]
22. K. P. Nayak, F. L. Kien, Y. Kawai, K. Hakuta, K. Nakajima, H. T. Miyazaki, and Y. Sugimoto, “Cavity formation on an optical nanofiber using focused ion beam milling technique,” Opt. Express 19, 14040–14050 (2011). [CrossRef] [PubMed]
25. S. J. Mihailov, D. Grobnic, C. W. Smelser, P. Lu, R. B. Walker, and H. Ding, “Induced Bragg gratings in optical fibers and waveguides using an ultrafast infrared laser and a phase mask,” Laser Chem. 2008, 416251 (2008). [CrossRef]
26. D. Grobnic, S. J. Mihailov, H. Ding, and C. W. Smelser, “Bragg grating evanescent field sensor made in biconical tapered fiber with femtosecond IR radiation,” IEEE Photon. Tech. Lett. 18, 160–162 (2006). [CrossRef]
28. M. Becker, J. Bergmann, S. Brückner, M. Franke, E. Lindner, M. W. Rothhardt, and H. Bartelt, “Fiber Bragg grating inscription combining DUV sub-picosecond laser pulses and two-beam interferometry,” Opt. Express 16, 19169–19178 (2008). [CrossRef]
29. H. C. van de Hulst, Light Scattering by Small Particles (Dover Publication Inc., 1981) pp 297–328.
30. F. L. Kien and K. Hakuta, “Microtraps for atoms outside a fiber illuminated perpendicular to its axis: Numerical results,” Phys. Rev. A 80, 013415 (2009). [CrossRef]
32. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997). [CrossRef]