Abstract

In this study, we propose a photoresist-free, laser-assisted wet chemical etching process used to control the grating depth of a long-period fiber grating (LPFG) termed laser-assisted etching LPFG (LLPFG). This process can considerably reduce production time, while the photoresist-free laser etching allows tiny long-period notches to be etched on the fiber surface, distinguishing the etching rate of the process from that of standard wet etching processes. The LLPFG, which has a period of 610 µm, was scanned using a KrF excimer laser. The results showed a resonant-attenuation wavelength of 1551 nm through a fiber diameter of 60 µm and a grating depth of 26 µm and to being a green process due to the photoresist-free etching.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Optical fiber sensors have attracted considerable attention due to their use in a variety of industrial applications, such as chemical and biological sensing applications [15]. Recently, there has been great interest in the use of fiber grating refractive index refractive index (RI) sensors because of their high sensitivity, efficiency, and convenience. In particular, much attention has been paid to long-period fiber gratings (LPFGs) due to the way in which light at the resonant wavelength of an LPFG is coupled from the guided mode to the cladding modes [6], which induces some loss dips depending on the difference between the effective indexes of the guided and cladding modes. Therefore, LPFGs can be directly applied as codirectional coupling mechanisms [7], allowing them to be used as band-rejection filters, sensors, wavelength selective filters, and beam shapers. In addition, in-series LPFG pairs can be used to form all-fiber Mach-Zender interferometers that are useful as wavelength-selective filters for wavelength-division multiplexing (WDM) applications, while parallel LPFGs can act as wavelength selective couplers or wavelength multiplexers [8].

Many methods are used for the fabrication of LPFGs, including the wet and/or dry etching of sandwiched LPFGs (SLPFGs) involving exposure to inductively coupled plasma (ICP), a laser excimer, or a CO2 laser on the surface of the fiber to achieve a periodical perturbation of the refractive index in the core, the cladding, or both along the fiber optics. To compare our results with those of some previously published work, we would first note that SLPFG has previously been fabricated using a thick photoresist patterned with a wet etching process [9], with a periodic photoresist layer being applied on the optical fiber to form a periodically varied structure. In another study, dry etching achieved via exposure to an ICP was used to fabricate a notched LPFG (NLPFG) made of optical fiber by etching the cladding layer of the fiber to produce the periodic grating structure with an exact period, vertical sidewalls, and smooth etched surfaces [10]. Meanwhile, CO2 laser irradiation based on the thermal shock effect of focused high-frequency pulses at several kilohertz was previously used in another study in order to solve the problem of cross-sensitivity between the bend and other measurements [11]. Still another study found that acousto-optic interaction enables electrically tunable notches in each of the photonic bandgap fiber (PBGF) transmission bands, where both the center frequency and depth of the resonances can be varied [12]. In short, these and various other approaches have been applied to innovate the fabrication of LPFGs.

The conventional process for fabricating LPFGs utilizes a UV excimer laser to obtain periodic refractive index variations in the core of a photosensitive fiber, such as germanium (Ge)-doped or hydrogen-soaked fiber [13,14]. Several reports have discussed using irradiation with UV light to cause an increase in the refractive index of Ge-doped fibers through the formation of Ge-related glass defects; in fact, the production of this photorefractive effect through the use of UV light is limited only to Ge-doped fibers. However, the inscription of manufactured LPFGs using facile and simple processes is rarely reported. At the same time, LPFGs fabricated using UV-light irradiation have a problem with stability as they age because the index change relaxes even at temperatures below 100 °C [15]. Another fabrication method, dry etching through the use of an ICP or a laser has good precision, but it is time-consuming and expensive in comparison to laser-assisted wet chemical etching. In addition, dry etching methods have the disadvantage of unpredictable attenuation losses when used for mass production, an issue which restricts its applicability.

Laser-assisted wet chemical etching relies on the use of a laser to provide ablation on the surface of the fiber and the use of a solution etchant for thinning the cladding of the fiber. The acid solution used has the capacity to interact with glass fiber. When laser beam ablation is also applied, the rate of interaction of the acid solution and the glass fiber will be increased. The laser beam is applied through a metal mask, with specific periods, incident on the fiber surface. Due to this laser ablation, the structure (O-Si-O compound) of the glass fiber will break. Moreover, the depth of the broken structure of the fiber depends on the intensity of the laser. Relatedly, the key point of the laser-assisted wet chemical technique is that it allows for the control of the periodical laser ablation area formed by the laser (that is, the grating cross-section), as well as the un-ablated areas (that is, the cladding cross-section). The interaction of the laser and the acid solution will change the cross-section of the grating area and the cladding cross-section, and as a result, the diameter of the grating and cladding sections will change.

In this study, we propose a novel sensor fabricated by means of a laser-assisted wet chemical etching process that was used to manufacture an LPFG termed laser-assisted etching LPFG (LLPFG). In this process, the laser etching causes tiny long-period notches to be etched on the fiber surface, distinguishing the etching rate of the process from that of standard wet etching processes while also allowing the process to be performed without photoresist or Ge-doped fiber. This in turn means that single-mode fiber can be used in the fabrication procedure, making the process both green and cost-competitive compared to other fabrication processes.

2. Theory

The photoelectric effect in a material coupling the mechanical strain to the optical index refraction can be calculated as shown in Eq. (1).

$$\Delta {\left( {\frac{1}{{{n^2}}}} \right)_{ij}} = {P_{ijk}}{S_{kl}}$$
In the above equation, $\Delta {\left( {\frac{1}{{{n^2}}}} \right)_{ij}}$ is the change in the optical impermeability tensor and ${S_{kl}}\textrm{and}{P_{ijk}}$ are the strain tensor and coefficient strain-optics tensor, respectively. When a tensile force is applied to long-period grating (LPG) under the condition of mechanical equilibrium, the longitudinal forces are the same for the etched and unetched regions. Here, the photoelastic coefficient is supposed to be constant [16], while the etched and unetched regions experience different loading strains which are inversely proportional to the cross-sectional areas of the corresponding region [17]. These strains in turn change the refractive index according to Eq. (2).
$$\begin{aligned}\Delta {\textrm{n}_{strain}}(r ) &= {\delta }{n_e}(r )- {\delta }{n_u}(r ) \quad \quad {`}0 \le r \le {r_e}\\ &= - \frac{1}{2}{P_e}{[{{n_e}^{(0 )}(r )} ]^3}\left[ {1 - \frac{{{A_a}}}{{{A_u}}}} \right]s \end{aligned}$$
In the above equation, ${P_e}$, ${n_e}^{(0 )}(r )$, ${A_u}$, and ${A_a}$ are the photoelastic coefficient, grating refractive index, cross-section of the unetched area, and cross-section of the grating, respectively. Laser-assisted wet chemical etching techniques can be used to produce LPG with a smooth-edged profile, as shown in Fig. 1. Relatedly, by adjusting the effects of a smooth edge in the refractive index profile, we expected to be able to cause changes in the transmission spectrum.

 figure: Fig. 1.

Fig. 1. Wet etching caused a smooth edge depending on the time of ablation and time of etching.

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Hence, we expected another modulation of the refraction index due to the shape of the grating required for a smooth edge in the LLPFG. The refraction index modulation for a periodic structure obeys Eq. (3) [18].

$${\delta \textrm{n}}(\textrm{z} )= \overline {\Delta \textrm{n}} \left\{ {1 + \frac{{\gamma }}{2}\left( {{\textrm{e}^{ - \textrm{i}\left[ {\left( {\frac{{2{\pi \textrm{N}}}}{{\Lambda }}} \right)\textrm{Z} + {\varphi}(\textrm{Z} )} \right]}} + \textrm{cc}} \right)} \right\}$$
In the above equation, $\Delta n$, $\gamma $, ${e^{ - \textrm{i}\left[ {\left( {2\frac{{\pi N}}{{\Lambda }}} \right)Z + \varphi (Z )} \right]}}$, $\varphi (\textrm{z} )$, $\Lambda $, and N indicate the refractive index change averaged over a single period of the grating, the visibility of the fringes, the exponent term along with the complex conjugate that describe the periodic modulation in complex notation, an arbitrary spatially varying phase change, and the harmonic order of perturbation, respectively.

Depending on the laser exposure time and the etching time, the slope and the depth of the grating will be changed. Therefore, the profile of the LPFG will be changed periodically so that it causes an extra modulation refraction index due to the smooth edge of the LLPFG.

$$\Delta {n_{total}} = \Delta {\textrm{n}_{strain}}(r )+ \Delta {\textrm{n}_{SE - LPG}}(r )$$
The resonance wavelength of an LPFG is determined by the phase-matching condition. The coupling transmission spectrum of an LPFG appears with power losses at specific wavelengths ${\lambda ^m}$ described by Eq. (5), where ${n_{eff}}^{co}$ is the core effective refractive index, ${n_{eff}}^{cl,m}$ is an effective index of the $\textrm{m}$th order of the cladding mode, and Λ is the grating period [19]:
$${\lambda ^m} = ({{n_{eff}}^{co} - {n_{eff}}^{cl,m}} )\Lambda $$
Because the core-mode field is highly confined in the core region, the periodic index changes in Eq. (4) have almost no effect on the core mode. According to Eq. (5), if the period of the fiber grating is unchanged and the cladding thickness is decreased, the effective refractive index of the fiber cladding will also be decreased. This phenomenon causes the resonance dip of the LLPFG to be generated in the longer wavelength region as the diameter of the optical fiber becomes smaller, and vice versa. However, the confinement of cladding modes is due to the cladding-surrounding interface, and the structure of an LPFG indeed influences the coupling behaviors through the tails of the cladding-mode fields. Because we were mainly concerned with coupling with the core mode through the strain-induced index perturbation in Eq. (4), the interaction region was mainly concentrated near the core mode area. Thus, as regards coupling descriptions, we can use some average cladding modes as the unperturbed fields with respect to the strain-induced $\Delta {\textrm{n}_{total}}(r )$. Let us define this averaged cladding mode as ${\bar{e}_{cl}}$ . In that case, the averaged, or empirical, value of the coupling coefficient $\mathrm{\bar{\kappa }}$ between the core mode and cladding mode is defined as follows:
$$\mathrm{\bar{\kappa }} = \frac{{\omega {\varepsilon _0}}}{4}\int_{ - \infty }^{ + \infty } {2{n_e}^{(0 )}\Delta {n_{total}} {e_{co}}^\ast (r ).{{\bar{e}}_{cl}}(r )dA} $$
Here, the mode fields are assumed to be normalized to carry unit power. Thus, as an empirical formula, we use this averaged coupling coefficient to express the transmission loss of the corrugated LPFG for the core mode using
$$\textrm{T} \cong co{s^2}({\mathrm{\bar{\kappa}}l} )= co{s^2}\left[ {\alpha \left( {\frac{{{r_u}^2}}{{{r_e}^2}} - 1} \right)sl} \right]$$
In the second expression of Eq. (7), we introduce another empirical coefficient. There, $\alpha $ represents the overall dependence on the overlapping integrals and photoelastic constants. The transmittance of an LLPFG can be expressed with the coupling coefficient between the core and the cladding and the grating length. The length of the LLPFG is determined by the overlap integral of the core and cladding mode and by the amplitude of the periodic modulation of the mode propagation constants. Because the deformation of the LLPFG under loading is very small (< 0.2%), the grating period lengthening produced by the mechanical load has a negligible effect on the grating wavelength in Eq. (5). When loading is applied, the value of $\mathrm{\bar{\kappa }}$ in Eq. (7) will change according to the elastic-optic effect. Therefore, the transmittance can be tuned by changing the external loading. We also emphasize the dependence of transmission on measurable quantities, specifically, the applied tensile strain s, the total length l, and the difference in the fiber-cladding areas. We call the resulting LPFG laser-assisted etching long-period fiber grating, or LLFPG, an example of which is also depicted in Fig. 1.

3. Methods

A single-mode fiber (Corning SMF28) was chosen for the fabrication of the LLPFGs through the presented laser-assisted wet chemical etching technique. The LLFPGs were 3 cm in length and had a period of 610 µm. Reduction of the cladding diameter was performed by wet etching of the fused silica cladding while it was also exposed to a laser excimer. Figure 2 shows how the cladding removal was performed by immersing the optical fiber in the etching solution. Each step of the fabrication and detection was conducted at room temperature (30°C) and also with optical table systems to prevent vibrations. Moreover, the sensor head can be easily broken by transverse stress. Therefore, the fiber was inserted into a hole in the glass tube and both ends of the fiber were fixed into the glass tube by UV-type resin to avoid environmental impacts in further applications. As shown in Fig. 2(a), reducing the diameter of the optical fiber by chemical etching required placing the bare fiber into the buffered oxide etch solution (BOE), which contained hydrofluoric acid (HF) and ammonium fluoride (NH3F), with a ratio of BOE to water of 1:6 being used for the chemical reaction between the SiO2 and HF etchant according to Eq. (7) [20].

$$\textrm{SiO}+\textrm{6HF} \to \textrm{2}{\textrm{H}^{+}} + \textrm{Si}{\textrm{F}^{\textrm{6}-}} + 2{\textrm{H}_2}\textrm{O}$$
The etch rate of the SiO2 in the HF solution was controlled by the duration of immersion, with durations of 0, 60, 90, 110, 120, 130, 140, and 150 min at a temperature of 30 °C variously being applied, after which the fiber was rinsed in deionized water. The reaction stopped at the Si surface and left the surface hydrogen passivated. That provided the possibility of removing the native oxide easily and reliably without leaving a highly reactive and unstable surface. Figure 2(b) shows the proposed laser excimer scheme that was used. Specifically, two metal masks were pressed over both sides of the 125 um fiber to present a three-layer structure on the platform, and a controlled Z-axis platform was used for precision positioning. The power of the laser excimer (KrF excimer laser 248 nm) used in this setup was 12 mj through the periscope and focal lens focused on the surface of the fiber, while the speed of the motorized linear stage was 0.2 mm/s and the frequency was 100 Hz.

 figure: Fig. 2.

Fig. 2. (a) The process of chemical etching. (b) KrF excimer laser setup used to assistant chemical etching form period structure.

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4. Result and discussion

The metal mask period and laser energy were fixed at 610 µm and 12 mj, respectively. In the first step of the fabrication process, that is, at the first shooting of the laser, partial ablation on the fiber surface while the cladding diameter remained unchanged at 125 µm was observed. The depth of the etching was determined by the duration of time for which the fibers were immersed inside the acid, as shown in Table 1. We studied a range of possible cases to find an optimum point for transmission, with the main factors, that is, the depth of the grating and the diameter of the cladding of the LLPFG, being changed accordingly.

Tables Icon

Table 1. Variations in the grating depth and diameter corresponding to the immersion duration for the LLPFG

Figure 3 and Fig. 4 show how the diameter and depth of the LLPFG decreased depending on the wet etching time periods and also show the corresponding procedure for each step. A simple LLPFG was produced after immersion into the acid plate for 60 min. A 24 µm depth in the grating areas was produced due to the reaction with the acid solution, yielding the LLPFG shown in Fig. 3. The experimental results further indicated that the 24 µm depth after 60 min of immersion was accompanied by a diameter of the LLPFG cladding of 95 µm. The experimental results indicated that not only did the etched section diameter change even as the cladding diameter was changed. Rather, they further indicated that when the duration of the immersion was extended to 100 min, the depth of the grating was 26 µm and the diameter of the LLPFG was 75 µm. As such, the process can be repeated as many times as necessary in order to reach a specific cladding diameter, as shown in Table 1. Nonetheless, in our experiments, we eventually stopped at a 50 µm diameter of the LLPFG, at which point the fiber core was 10 µm. It is necessary to keep control over the diameter reduction in order to reach an optimum point. Therefore, the processing in this experiment was done for specific time periods. Overall, the results clearly show that 5 µm of the diameter of the cladding of the LLPFG was removed for every addition 10 min of immersion, which allowed us to obtain high transmission and shift the wavelength according to the cladding thickness. Then, for each structure of LLPFG, the spectrum and wavelength shift under tensile strain were investigated through the observation of the transmission spectrum.

 figure: Fig. 3.

Fig. 3. A micro-scan photo of LLPFG effect of the incident laser beam on the grating depth during wet etching.

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 figure: Fig. 4.

Fig. 4. The grating depth of LLPFG depends on the wet etching immersion duration.

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The manufacturing process is supported by laser-assisted etching in order to generate periodically exposed areas on the surface of the fiber through metal masks, with the etching speed of the exposed area being increased in comparison with the unexposed area. In this situation, when the fiber is immersed in an acid solution, the ablated area of the fiber is removed more rapidly in comparison to the un-ablated area; therefore, a period on the surface of the fiber is produced by the different etching speeds, resulting in the LLPFG.

Figure 5 shows the transmission spectra under strain as the diameter of the LLPFG was decreased from 95 to 55 µm and the depth of the grating was increased from 24 to 26 µm, respectively. As shown in Figs. 5(a) and 5(g), the LLPFGs with diameters of 95 and 50 µm did not have reliable and correct transmission spectrum responses under strain. Although the morphology formed an LPFG in the case of the fiber diameter at 95 µm, the depth of the grating structure was insufficient to produce a long-period effect. When the diameter of the fiber was 50 µm, meanwhile, substantial noise was observed in the transmission spectrum on account of the grating structure being too deep and close to the core layer (about 10 µm). As shown in Fig. 5(a), the spectra of these LLPFGs under strain showed some different resonant dip wavelengths from 1490 to 1600 nm. This happened due to the non-uniformity on the surface of the grating and the smooth slope edge between the etched and unetched areas of the LLPFG that can clearly be seen in Fig. 3. Proceeding with the fabrication by wet etching according to the steps in Table 1, after 90 to 140 min of immersion, the cladding diameter would be reduced from 80 to 55 µm. As shown in Figs. 5(b)–5(f), there was a big dip in the spectra under specific strain for each of the resulting LLPFGs. Figure 5(e) shows a resonant-attenuation wavelength of 1551 nm and a maximum resonance-attenuation of -26.489 dB when the diameter of the fiber was 60 µm and the depth of the grating was 26 µm. For the LLPFG with a diameter of 50 µm, because the diameter of the fiber core was 10 µm, the grating depth extended into the core area so that all of the covers of the grating section were totally removed. As shown in Fig. 5(g), in this situation, the transmission loss was very large (about −60 dB), and noise appeared in the LLPFG transmission spectrum. As shown in Fig. 6, when the fiber diameter was between 80 and 55 µm, the smaller the diameter of the fiber, the more the wavelength of the resonance would drift toward a longer wavelength, increasing from 1512 to 1553 nm overall.

 figure: Fig. 5.

Fig. 5. Transmission spectrum of LLPFG under loading while after etching processing reach a specific diameter (a) 95 µm; (b) 80 µm; (c)70 µm; (d) 65 µm; (e) 60 µm; (f) 55µm; (g) 50 µm.

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 figure: Fig. 6.

Fig. 6. Transmission spectra of LLPFG after etching processing reach a specific diameter of 80, 70, 65, 60 and 55µm.

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Figure 7 illustrates in detail the dependence of the transmission losses on force applied for the LLPFGs with diameters of 80 to 55 µm. The transmission losses dipped for all the LLPFGs with increasing force, stopping at a specific force for each LLPFG. Based on Eq. (7), these transmission spectra of the LLPFGs under force loading were consistent with our theoretical expectations. As shown in Fig. 7, for all of the LLPFGs, there was a turning point at a specific force, such that after this point, the transmission losses were decreased with increasing force. As the loading decreased, the LLPFGs with cladding diameters ranging from 80 to 55 $\mu\textrm{ m}$ tended to exhibit high transmission dips.

 figure: Fig. 7.

Fig. 7. The dependency of transmission to strain for LLPFG with diameters 80 to 55 µm.

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The highest transmission dips were observed for the LLPFG with a cladding diameter of 55 µm. For the LLPFGs with cladding diameters of 80 to 55 µm, the cladding diameter was the only parameter that was changed during the process of etching, while the grating depth, periods of grating, and other effective factors remained approximately the same. Therefore, the role of the cladding diameter and also the cladding to etched area ratio can be made dominant depending on the applications for which the LLPFG is used. Theoretically, such applications can be justified by the ratio of the cladding to the etched cross-section. When a force loading is applied on the LLPFG, a strain field is created as a periodically rectangular shape according to the cross-section area. In addition, the effects of strain on the cladding cross-section and the grating cross-section are not the same. Rather, the effect of strain on the grating cross-section is greater than that on the cladding cross-section. This means that the grating cross-section is subjected to greater deformation under loading. For LLPFGs with diameters of 75 to 55 µm, as indicated in Eq. (2), when the grating depth is fixed with a specific diameter, the refraction modulation for the given LLPFG will increase with increasing strain. For an LLPFG with a diameter of 80 µm, the transmission spectrum is effectively a function of the refraction modulation caused by the cladding and grating cross-section under loading, with the specific effect on the transmission effect depending on how much of both areas are placed under the loading. This means that the diameter and grating depth can be considered the decisive factors in the fabrication and design of LLPFGs for different applications.

5. Conclusion

In this work, a laser-assisted wet etching technique for fabricating LPFG structures with corrugated smooth edges on one side of a fiber surface is proposed, with the resulting LPFG being termed laser-assisted etching long-period fiber grating (LLPFG). The behavior of LLPFG under strain and the decisive factors in determining that behavior were experimentally optimized by utilizing the energy of a KrF excimer laser and the duration of the chemical etching to control the period and depth of the ablation zone. By controlling the duration of etching and the energy of the incident laser, we were able to obtain specific diameters. The depth and edge of the LLPFG could be controlled depending on the etched notch of the laser and the duration of the etching process, which in turn enabled quick estimation of the cladding diameter needed to achieve tensile loading. The LLPFG with a period of 610 µm showed a resonant-attenuation wavelength of 1551nm and a maximum resonance-attenuation of −26.489dB at a fiber diameter of 60 µm and grating depth of 26 µm, and the sensitivity of the LLPFG would be increased when the fiber diameter was decreased. Those results show that the proposed process, in addition to being a green process due to the photoresist-free etching, is a more economical manufacturing process for making loss-tunable filters used in force transducer applications.

Funding

Ministry of Science and Technology, Taiwan (MOST 107-2221-E-992-043-MY3).

Disclosures

The authors have declared no conflict of interest.

Author Contribution statement

Seyed Reza Hosseini Largani and Hsin-yi Wen wrote the manuscript with support from Jing-luen Chen, who carried out the experiment and produced the Figs and table. Chia-chin Chiang conceived the original idea for the study.

References

1. L. Rindorf, J. B. Jensen, M. Dufva, L. H. Pedersen, P. E. Høiby, and O. Bang, “Photonic crystal fiber long-period gratings for biochemical sensing,” Opt. Express 14(18), 8224–8231 (2006). [CrossRef]  

2. V. Bhatia, “Applications of long-period gratings to single and multi-parameter sensing,” Opt. Express 4(11), 457–466 (1999). [CrossRef]  

3. V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996). [CrossRef]  

4. N. T. Pham, S. L. Lee, S. Park, Y. W. Lee, and H. W. Kang, “Real-time temperature monitoring with fiber Bragg grating sensor during diffuser-assisted laser-induced interstitial thermotherapy,” J. Biomed. Opt. 22(4), 045008 (2017). [CrossRef]  

5. S. L. Woon, C. H. Pua, H. S. Lin, and F. A. Rahman, “Application of Erbium-Doped Fibre Laser Dynamics in Pipeline Leak Detection and Location Estimation,” in 2018 IEEE 7th International Conference on Photonics (ICP), 2018), 1–3.

6. S.-Y. Tan, Y.-T. Yong, S.-C. Lee, and F. Abd Rahman, “Review on an arc-induced long-period fiber grating and its sensor applications,” J. Electromagnet Wave. 29(6), 703–726 (2015). [CrossRef]  

7. J. Jung and Y. W. Lee, “Polarization-Independent Wavelength-Tunable First-Order Fiber Comb Filter,” IEEE Photonics J. 11(1), 1–15 (2019). [CrossRef]  

8. F. Y. Chan and K. Yasumoto, “Design of wavelength tunable long-period grating couplers based on asymmetric nonlinear dual-core fibers,” Opt. Lett. 32(23), 3376–3378 (2007). [CrossRef]  

9. C.-C. Chiang, T.-C. Cheng, H.-J. Chang, and L. Tsai, “Sandwiched long-period fiber grating filter based on periodic SU8-thick photoresist technique,” Opt. Lett. 34(23), 3677–3679 (2009). [CrossRef]  

10. C.-C. Chiang and L. Tsai, “Perfectly notched long-period fiber grating filter based on ICP dry etching technique,” Opt. Lett. 37(2), 193–195 (2012). [CrossRef]  

11. Y.-J. Rao, Y.-P. Wang, Z.-L. Ran, and T. Zhu, “Novel fiber-optic sensors based on long-period fiber gratings written by high-frequency CO 2 laser pulses,” J. Lightwave Technol. 21(5), 1320–1327 (2003). [CrossRef]  

12. D.-I. Yeom, P. Steinvurzel, B. Eggleton, S. Do Lim, and B. Y. Kim, “Tunable acoustic gratings in solid-core photonic bandgap fiber,” Opt. Express 15(6), 3513–3518 (2007). [CrossRef]  

13. C.-C. Chiang and C.-C. Tseng, “Characterization of notched long-period fiber gratings: effects of periods, cladding thicknesses, and etching depths,” Appl. Opt. 53(20), 4398–4404 (2014). [CrossRef]  

14. K. P. Lor, Q. Liu, and K. S. Chiang, “UV-written long-period gratings on polymer waveguides,” IEEE Photonics Technol. Lett. 17(3), 594–596 (2005). [CrossRef]  

15. S. R. Baker, H. N. Rourke, V. Baker, and D. Goodchild, “Thermal decay of fiber Bragg gratings written in boron and germanium codoped silica fiber,” J. Lightwave Technol. 15(8), 1470–1477 (1997). [CrossRef]  

16. A. Yariv and P. Yeh, Optical Waves. Crysta. (Wiley New York, 1984), Vol. 5.

17. C.-Y. Lin, L. A. Wang, and G.-W. Chern, “Corrugated long-period fiber gratings as strain, torsion, and bending sensors,” J. Lightwave Technol. 19(8), 1159–1168 (2001). [CrossRef]  

18. R. Kashyap, Fiber bragg gratings (Academic press, 2009).

19. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14(1), 58–65 (1996). [CrossRef]  

20. S. Verhaverbeke, I. Teerlinck, C. Vinckier, G. Stevens, R. Cartuyvels, and M. Heyns, “The etching mechanisms of SiO2 in hydrofluoric acid,” J. Electrochem. Soc. 141(10), 2852–2857 (1994). [CrossRef]  

References

  • View by:

  1. L. Rindorf, J. B. Jensen, M. Dufva, L. H. Pedersen, P. E. Høiby, and O. Bang, “Photonic crystal fiber long-period gratings for biochemical sensing,” Opt. Express 14(18), 8224–8231 (2006).
    [Crossref]
  2. V. Bhatia, “Applications of long-period gratings to single and multi-parameter sensing,” Opt. Express 4(11), 457–466 (1999).
    [Crossref]
  3. V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996).
    [Crossref]
  4. N. T. Pham, S. L. Lee, S. Park, Y. W. Lee, and H. W. Kang, “Real-time temperature monitoring with fiber Bragg grating sensor during diffuser-assisted laser-induced interstitial thermotherapy,” J. Biomed. Opt. 22(4), 045008 (2017).
    [Crossref]
  5. S. L. Woon, C. H. Pua, H. S. Lin, and F. A. Rahman, “Application of Erbium-Doped Fibre Laser Dynamics in Pipeline Leak Detection and Location Estimation,” in 2018 IEEE 7th International Conference on Photonics (ICP), 2018), 1–3.
  6. S.-Y. Tan, Y.-T. Yong, S.-C. Lee, and F. Abd Rahman, “Review on an arc-induced long-period fiber grating and its sensor applications,” J. Electromagnet Wave. 29(6), 703–726 (2015).
    [Crossref]
  7. J. Jung and Y. W. Lee, “Polarization-Independent Wavelength-Tunable First-Order Fiber Comb Filter,” IEEE Photonics J. 11(1), 1–15 (2019).
    [Crossref]
  8. F. Y. Chan and K. Yasumoto, “Design of wavelength tunable long-period grating couplers based on asymmetric nonlinear dual-core fibers,” Opt. Lett. 32(23), 3376–3378 (2007).
    [Crossref]
  9. C.-C. Chiang, T.-C. Cheng, H.-J. Chang, and L. Tsai, “Sandwiched long-period fiber grating filter based on periodic SU8-thick photoresist technique,” Opt. Lett. 34(23), 3677–3679 (2009).
    [Crossref]
  10. C.-C. Chiang and L. Tsai, “Perfectly notched long-period fiber grating filter based on ICP dry etching technique,” Opt. Lett. 37(2), 193–195 (2012).
    [Crossref]
  11. Y.-J. Rao, Y.-P. Wang, Z.-L. Ran, and T. Zhu, “Novel fiber-optic sensors based on long-period fiber gratings written by high-frequency CO 2 laser pulses,” J. Lightwave Technol. 21(5), 1320–1327 (2003).
    [Crossref]
  12. D.-I. Yeom, P. Steinvurzel, B. Eggleton, S. Do Lim, and B. Y. Kim, “Tunable acoustic gratings in solid-core photonic bandgap fiber,” Opt. Express 15(6), 3513–3518 (2007).
    [Crossref]
  13. C.-C. Chiang and C.-C. Tseng, “Characterization of notched long-period fiber gratings: effects of periods, cladding thicknesses, and etching depths,” Appl. Opt. 53(20), 4398–4404 (2014).
    [Crossref]
  14. K. P. Lor, Q. Liu, and K. S. Chiang, “UV-written long-period gratings on polymer waveguides,” IEEE Photonics Technol. Lett. 17(3), 594–596 (2005).
    [Crossref]
  15. S. R. Baker, H. N. Rourke, V. Baker, and D. Goodchild, “Thermal decay of fiber Bragg gratings written in boron and germanium codoped silica fiber,” J. Lightwave Technol. 15(8), 1470–1477 (1997).
    [Crossref]
  16. A. Yariv and P. Yeh, Optical Waves. Crysta. (Wiley New York, 1984), Vol. 5.
  17. C.-Y. Lin, L. A. Wang, and G.-W. Chern, “Corrugated long-period fiber gratings as strain, torsion, and bending sensors,” J. Lightwave Technol. 19(8), 1159–1168 (2001).
    [Crossref]
  18. R. Kashyap, Fiber bragg gratings (Academic press, 2009).
  19. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14(1), 58–65 (1996).
    [Crossref]
  20. S. Verhaverbeke, I. Teerlinck, C. Vinckier, G. Stevens, R. Cartuyvels, and M. Heyns, “The etching mechanisms of SiO2 in hydrofluoric acid,” J. Electrochem. Soc. 141(10), 2852–2857 (1994).
    [Crossref]

2019 (1)

J. Jung and Y. W. Lee, “Polarization-Independent Wavelength-Tunable First-Order Fiber Comb Filter,” IEEE Photonics J. 11(1), 1–15 (2019).
[Crossref]

2017 (1)

N. T. Pham, S. L. Lee, S. Park, Y. W. Lee, and H. W. Kang, “Real-time temperature monitoring with fiber Bragg grating sensor during diffuser-assisted laser-induced interstitial thermotherapy,” J. Biomed. Opt. 22(4), 045008 (2017).
[Crossref]

2015 (1)

S.-Y. Tan, Y.-T. Yong, S.-C. Lee, and F. Abd Rahman, “Review on an arc-induced long-period fiber grating and its sensor applications,” J. Electromagnet Wave. 29(6), 703–726 (2015).
[Crossref]

2014 (1)

2012 (1)

2009 (1)

2007 (2)

2006 (1)

2005 (1)

K. P. Lor, Q. Liu, and K. S. Chiang, “UV-written long-period gratings on polymer waveguides,” IEEE Photonics Technol. Lett. 17(3), 594–596 (2005).
[Crossref]

2003 (1)

2001 (1)

1999 (1)

1997 (1)

S. R. Baker, H. N. Rourke, V. Baker, and D. Goodchild, “Thermal decay of fiber Bragg gratings written in boron and germanium codoped silica fiber,” J. Lightwave Technol. 15(8), 1470–1477 (1997).
[Crossref]

1996 (2)

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14(1), 58–65 (1996).
[Crossref]

V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996).
[Crossref]

1994 (1)

S. Verhaverbeke, I. Teerlinck, C. Vinckier, G. Stevens, R. Cartuyvels, and M. Heyns, “The etching mechanisms of SiO2 in hydrofluoric acid,” J. Electrochem. Soc. 141(10), 2852–2857 (1994).
[Crossref]

Abd Rahman, F.

S.-Y. Tan, Y.-T. Yong, S.-C. Lee, and F. Abd Rahman, “Review on an arc-induced long-period fiber grating and its sensor applications,” J. Electromagnet Wave. 29(6), 703–726 (2015).
[Crossref]

Baker, S. R.

S. R. Baker, H. N. Rourke, V. Baker, and D. Goodchild, “Thermal decay of fiber Bragg gratings written in boron and germanium codoped silica fiber,” J. Lightwave Technol. 15(8), 1470–1477 (1997).
[Crossref]

Baker, V.

S. R. Baker, H. N. Rourke, V. Baker, and D. Goodchild, “Thermal decay of fiber Bragg gratings written in boron and germanium codoped silica fiber,” J. Lightwave Technol. 15(8), 1470–1477 (1997).
[Crossref]

Bang, O.

Bhatia, V.

V. Bhatia, “Applications of long-period gratings to single and multi-parameter sensing,” Opt. Express 4(11), 457–466 (1999).
[Crossref]

V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996).
[Crossref]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14(1), 58–65 (1996).
[Crossref]

Cartuyvels, R.

S. Verhaverbeke, I. Teerlinck, C. Vinckier, G. Stevens, R. Cartuyvels, and M. Heyns, “The etching mechanisms of SiO2 in hydrofluoric acid,” J. Electrochem. Soc. 141(10), 2852–2857 (1994).
[Crossref]

Chan, F. Y.

Chang, H.-J.

Cheng, T.-C.

Chern, G.-W.

Chiang, C.-C.

Chiang, K. S.

K. P. Lor, Q. Liu, and K. S. Chiang, “UV-written long-period gratings on polymer waveguides,” IEEE Photonics Technol. Lett. 17(3), 594–596 (2005).
[Crossref]

Do Lim, S.

Dufva, M.

Eggleton, B.

Erdogan, T.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14(1), 58–65 (1996).
[Crossref]

Goodchild, D.

S. R. Baker, H. N. Rourke, V. Baker, and D. Goodchild, “Thermal decay of fiber Bragg gratings written in boron and germanium codoped silica fiber,” J. Lightwave Technol. 15(8), 1470–1477 (1997).
[Crossref]

Heyns, M.

S. Verhaverbeke, I. Teerlinck, C. Vinckier, G. Stevens, R. Cartuyvels, and M. Heyns, “The etching mechanisms of SiO2 in hydrofluoric acid,” J. Electrochem. Soc. 141(10), 2852–2857 (1994).
[Crossref]

Høiby, P. E.

Jensen, J. B.

Judkins, J. B.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14(1), 58–65 (1996).
[Crossref]

Jung, J.

J. Jung and Y. W. Lee, “Polarization-Independent Wavelength-Tunable First-Order Fiber Comb Filter,” IEEE Photonics J. 11(1), 1–15 (2019).
[Crossref]

Kang, H. W.

N. T. Pham, S. L. Lee, S. Park, Y. W. Lee, and H. W. Kang, “Real-time temperature monitoring with fiber Bragg grating sensor during diffuser-assisted laser-induced interstitial thermotherapy,” J. Biomed. Opt. 22(4), 045008 (2017).
[Crossref]

Kashyap, R.

R. Kashyap, Fiber bragg gratings (Academic press, 2009).

Kim, B. Y.

Lee, S. L.

N. T. Pham, S. L. Lee, S. Park, Y. W. Lee, and H. W. Kang, “Real-time temperature monitoring with fiber Bragg grating sensor during diffuser-assisted laser-induced interstitial thermotherapy,” J. Biomed. Opt. 22(4), 045008 (2017).
[Crossref]

Lee, S.-C.

S.-Y. Tan, Y.-T. Yong, S.-C. Lee, and F. Abd Rahman, “Review on an arc-induced long-period fiber grating and its sensor applications,” J. Electromagnet Wave. 29(6), 703–726 (2015).
[Crossref]

Lee, Y. W.

J. Jung and Y. W. Lee, “Polarization-Independent Wavelength-Tunable First-Order Fiber Comb Filter,” IEEE Photonics J. 11(1), 1–15 (2019).
[Crossref]

N. T. Pham, S. L. Lee, S. Park, Y. W. Lee, and H. W. Kang, “Real-time temperature monitoring with fiber Bragg grating sensor during diffuser-assisted laser-induced interstitial thermotherapy,” J. Biomed. Opt. 22(4), 045008 (2017).
[Crossref]

Lemaire, P. J.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14(1), 58–65 (1996).
[Crossref]

Lin, C.-Y.

Lin, H. S.

S. L. Woon, C. H. Pua, H. S. Lin, and F. A. Rahman, “Application of Erbium-Doped Fibre Laser Dynamics in Pipeline Leak Detection and Location Estimation,” in 2018 IEEE 7th International Conference on Photonics (ICP), 2018), 1–3.

Liu, Q.

K. P. Lor, Q. Liu, and K. S. Chiang, “UV-written long-period gratings on polymer waveguides,” IEEE Photonics Technol. Lett. 17(3), 594–596 (2005).
[Crossref]

Lor, K. P.

K. P. Lor, Q. Liu, and K. S. Chiang, “UV-written long-period gratings on polymer waveguides,” IEEE Photonics Technol. Lett. 17(3), 594–596 (2005).
[Crossref]

Park, S.

N. T. Pham, S. L. Lee, S. Park, Y. W. Lee, and H. W. Kang, “Real-time temperature monitoring with fiber Bragg grating sensor during diffuser-assisted laser-induced interstitial thermotherapy,” J. Biomed. Opt. 22(4), 045008 (2017).
[Crossref]

Pedersen, L. H.

Pham, N. T.

N. T. Pham, S. L. Lee, S. Park, Y. W. Lee, and H. W. Kang, “Real-time temperature monitoring with fiber Bragg grating sensor during diffuser-assisted laser-induced interstitial thermotherapy,” J. Biomed. Opt. 22(4), 045008 (2017).
[Crossref]

Pua, C. H.

S. L. Woon, C. H. Pua, H. S. Lin, and F. A. Rahman, “Application of Erbium-Doped Fibre Laser Dynamics in Pipeline Leak Detection and Location Estimation,” in 2018 IEEE 7th International Conference on Photonics (ICP), 2018), 1–3.

Rahman, F. A.

S. L. Woon, C. H. Pua, H. S. Lin, and F. A. Rahman, “Application of Erbium-Doped Fibre Laser Dynamics in Pipeline Leak Detection and Location Estimation,” in 2018 IEEE 7th International Conference on Photonics (ICP), 2018), 1–3.

Ran, Z.-L.

Rao, Y.-J.

Rindorf, L.

Rourke, H. N.

S. R. Baker, H. N. Rourke, V. Baker, and D. Goodchild, “Thermal decay of fiber Bragg gratings written in boron and germanium codoped silica fiber,” J. Lightwave Technol. 15(8), 1470–1477 (1997).
[Crossref]

Sipe, J. E.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14(1), 58–65 (1996).
[Crossref]

Steinvurzel, P.

Stevens, G.

S. Verhaverbeke, I. Teerlinck, C. Vinckier, G. Stevens, R. Cartuyvels, and M. Heyns, “The etching mechanisms of SiO2 in hydrofluoric acid,” J. Electrochem. Soc. 141(10), 2852–2857 (1994).
[Crossref]

Tan, S.-Y.

S.-Y. Tan, Y.-T. Yong, S.-C. Lee, and F. Abd Rahman, “Review on an arc-induced long-period fiber grating and its sensor applications,” J. Electromagnet Wave. 29(6), 703–726 (2015).
[Crossref]

Teerlinck, I.

S. Verhaverbeke, I. Teerlinck, C. Vinckier, G. Stevens, R. Cartuyvels, and M. Heyns, “The etching mechanisms of SiO2 in hydrofluoric acid,” J. Electrochem. Soc. 141(10), 2852–2857 (1994).
[Crossref]

Tsai, L.

Tseng, C.-C.

Vengsarkar, A. M.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14(1), 58–65 (1996).
[Crossref]

V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996).
[Crossref]

Verhaverbeke, S.

S. Verhaverbeke, I. Teerlinck, C. Vinckier, G. Stevens, R. Cartuyvels, and M. Heyns, “The etching mechanisms of SiO2 in hydrofluoric acid,” J. Electrochem. Soc. 141(10), 2852–2857 (1994).
[Crossref]

Vinckier, C.

S. Verhaverbeke, I. Teerlinck, C. Vinckier, G. Stevens, R. Cartuyvels, and M. Heyns, “The etching mechanisms of SiO2 in hydrofluoric acid,” J. Electrochem. Soc. 141(10), 2852–2857 (1994).
[Crossref]

Wang, L. A.

Wang, Y.-P.

Woon, S. L.

S. L. Woon, C. H. Pua, H. S. Lin, and F. A. Rahman, “Application of Erbium-Doped Fibre Laser Dynamics in Pipeline Leak Detection and Location Estimation,” in 2018 IEEE 7th International Conference on Photonics (ICP), 2018), 1–3.

Yariv, A.

A. Yariv and P. Yeh, Optical Waves. Crysta. (Wiley New York, 1984), Vol. 5.

Yasumoto, K.

Yeh, P.

A. Yariv and P. Yeh, Optical Waves. Crysta. (Wiley New York, 1984), Vol. 5.

Yeom, D.-I.

Yong, Y.-T.

S.-Y. Tan, Y.-T. Yong, S.-C. Lee, and F. Abd Rahman, “Review on an arc-induced long-period fiber grating and its sensor applications,” J. Electromagnet Wave. 29(6), 703–726 (2015).
[Crossref]

Zhu, T.

Appl. Opt. (1)

IEEE Photonics J. (1)

J. Jung and Y. W. Lee, “Polarization-Independent Wavelength-Tunable First-Order Fiber Comb Filter,” IEEE Photonics J. 11(1), 1–15 (2019).
[Crossref]

IEEE Photonics Technol. Lett. (1)

K. P. Lor, Q. Liu, and K. S. Chiang, “UV-written long-period gratings on polymer waveguides,” IEEE Photonics Technol. Lett. 17(3), 594–596 (2005).
[Crossref]

J. Biomed. Opt. (1)

N. T. Pham, S. L. Lee, S. Park, Y. W. Lee, and H. W. Kang, “Real-time temperature monitoring with fiber Bragg grating sensor during diffuser-assisted laser-induced interstitial thermotherapy,” J. Biomed. Opt. 22(4), 045008 (2017).
[Crossref]

J. Electrochem. Soc. (1)

S. Verhaverbeke, I. Teerlinck, C. Vinckier, G. Stevens, R. Cartuyvels, and M. Heyns, “The etching mechanisms of SiO2 in hydrofluoric acid,” J. Electrochem. Soc. 141(10), 2852–2857 (1994).
[Crossref]

J. Electromagnet Wave. (1)

S.-Y. Tan, Y.-T. Yong, S.-C. Lee, and F. Abd Rahman, “Review on an arc-induced long-period fiber grating and its sensor applications,” J. Electromagnet Wave. 29(6), 703–726 (2015).
[Crossref]

J. Lightwave Technol. (4)

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14(1), 58–65 (1996).
[Crossref]

S. R. Baker, H. N. Rourke, V. Baker, and D. Goodchild, “Thermal decay of fiber Bragg gratings written in boron and germanium codoped silica fiber,” J. Lightwave Technol. 15(8), 1470–1477 (1997).
[Crossref]

Y.-J. Rao, Y.-P. Wang, Z.-L. Ran, and T. Zhu, “Novel fiber-optic sensors based on long-period fiber gratings written by high-frequency CO 2 laser pulses,” J. Lightwave Technol. 21(5), 1320–1327 (2003).
[Crossref]

C.-Y. Lin, L. A. Wang, and G.-W. Chern, “Corrugated long-period fiber gratings as strain, torsion, and bending sensors,” J. Lightwave Technol. 19(8), 1159–1168 (2001).
[Crossref]

Opt. Express (3)

Opt. Lett. (4)

Other (3)

S. L. Woon, C. H. Pua, H. S. Lin, and F. A. Rahman, “Application of Erbium-Doped Fibre Laser Dynamics in Pipeline Leak Detection and Location Estimation,” in 2018 IEEE 7th International Conference on Photonics (ICP), 2018), 1–3.

R. Kashyap, Fiber bragg gratings (Academic press, 2009).

A. Yariv and P. Yeh, Optical Waves. Crysta. (Wiley New York, 1984), Vol. 5.

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Figures (7)

Fig. 1.
Fig. 1. Wet etching caused a smooth edge depending on the time of ablation and time of etching.
Fig. 2.
Fig. 2. (a) The process of chemical etching. (b) KrF excimer laser setup used to assistant chemical etching form period structure.
Fig. 3.
Fig. 3. A micro-scan photo of LLPFG effect of the incident laser beam on the grating depth during wet etching.
Fig. 4.
Fig. 4. The grating depth of LLPFG depends on the wet etching immersion duration.
Fig. 5.
Fig. 5. Transmission spectrum of LLPFG under loading while after etching processing reach a specific diameter (a) 95 µm; (b) 80 µm; (c)70 µm; (d) 65 µm; (e) 60 µm; (f) 55µm; (g) 50 µm.
Fig. 6.
Fig. 6. Transmission spectra of LLPFG after etching processing reach a specific diameter of 80, 70, 65, 60 and 55µm.
Fig. 7.
Fig. 7. The dependency of transmission to strain for LLPFG with diameters 80 to 55 µm.

Tables (1)

Tables Icon

Table 1. Variations in the grating depth and diameter corresponding to the immersion duration for the LLPFG

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

Δ ( 1 n 2 ) i j = P i j k S k l
Δ n s t r a i n ( r ) = δ n e ( r ) δ n u ( r ) 0 r r e = 1 2 P e [ n e ( 0 ) ( r ) ] 3 [ 1 A a A u ] s
δ n ( z ) = Δ n ¯ { 1 + γ 2 ( e i [ ( 2 π N Λ ) Z + φ ( Z ) ] + cc ) }
Δ n t o t a l = Δ n s t r a i n ( r ) + Δ n S E L P G ( r )
λ m = ( n e f f c o n e f f c l , m ) Λ
κ ¯ = ω ε 0 4 + 2 n e ( 0 ) Δ n t o t a l e c o ( r ) . e ¯ c l ( r ) d A
T c o s 2 ( κ ¯ l ) = c o s 2 [ α ( r u 2 r e 2 1 ) s l ]
SiO + 6HF 2 H + + Si F 6 + 2 H 2 O

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