When an optical fiber is dipped in an etching solution, the internal stress profile in the fiber varies with the fiber diameter. We observed a physical contraction as much as 0.2% in the fiber axial dimension when the fiber was reduced from its original diameter to ~6 µm through analysis using high resolution microscope images of the grating period of an etched FBG at different fiber diameters. This axial contraction is related to the varying axial stress profile in the fiber when the fiber diameter is reduced. On top of that, the refractive index of fiber core increases with reducing fiber diameter due to stress-optic effect. The calculated index increment is as much as 1.8 × 10−3 at the center of fiber core after the diameter is reduced down to ~6 µm. In comparison with the conventional model that assumes constant grating period and neglects the variation in stress-induced index change in fiber core, our proposed model indicates a discrepancy as much as 3nm in Bragg wavelength at a fiber diameter of ~6 µm.
© 2013 OSA
Ge-doping holds the key to the fundamental operating principle in optical fibers where it acts as a refractive index riser to the fiber core which has a higher refractive index than the cladding to achieve total internal refraction (TIR), so that the light power is confined within the fiber core to enable low optical transmission loss. For some specialty fibers, i.e. high numerical aperture (NA) fibers with low bending loss and high photosensitivity for Fiber Bragg Grating (FBG) fabrication , the concentration of Ge-dopant in the fiber core is higher [2, 3]. High Ge-dopant concentration in the fiber core also leads to other effects to the fiber, for instance, high coupling loss with other fibers due to large NA mismatch, high Rayleigh coefficient in the fiber  and thermoelastic stresses in the core arise from thermal expansion coefficient mismatch between core glass and cladding glass . These stresses were frozen into the fiber due to rapid cooling of fibers during the fiber drawing process. Strong thermoelastic stresses are likely to be found in heavily doped fibers .
Many research works have been carried out to further study the stress in glass fibers. Theoretical models were built to explain these stresses, which results in the introduction of stress-optic effects and birefringence in the fibers [7, 8]. Most attentions have been given to the analysis of stress distribution and birefringence . It is well known that the fiber stress profile and birefringence are governed by the fiber cross sectional geometry, optical characteristics and physical parameters. These include the shape and dimension of the core, position and size of the stress-producing components, refractive index and thermal expansion profiles of the fibers . Numerical modeling for stress profile by means of finite element method is suitable for most types of fibers including Hi-Bi fibers, microstructure fibers with complex lattices and non-axisymetry fibers; however, it is computationally taxing and time consuming. On the other hand, analytical method is a more accurate, faster and less computationally demanding approach, which is preferable for fibers with simple cross-section geometries. Most theoretical model for stress component is based on the assumption that the cylindrical fiber is infinitely long. For fibers with a finite length, some study indicates that a fraction of the axial stress is released at the close proximity of the cleaved fiber end . It was shown that the axial stress relaxation takes place within 40µm from the fiber end, the axial stress profile is unaffected beyond 40µm from the fiber end. The same stress model for infinite length fiber is applicable for any position beyond this point.
Considering the gravity of stress-optic effect to fibers, post-treatment processess on the fibers are suggested for modification of stress and refractive index profile in the fibers. Most of them are intended for internal stresses reduction. One of the techniques is by annealing the fiber at a temperature near to the transition temperature of the fiber glass, which is followed by a slow cooling process of the fiber to prevent recreation of stresses . CO2 laser irradiation technique shares similar principle, where residual stress relaxation in the fiber is achieved through thermal absorption of CO2 laser power and subsequent heating of the silica glass fiber . Hydrogen loading is also known to reduce axial stress in the fiber. In , several different fibers with different dopants, including Sn-Ge, B-Ge and SMF-28, are hydrogenated and investigated. B-Ge codoped core fibers exhibit the smallest stress reduction percentage, which is in the range from 22% to 27%. The birefringence property of the Hi-Bi optical fibers is the product of the core geometry and the fiber internal stress effect. Between them, stress-induced birefringence plays a dominant role, as compared to the birefringence induced by the geometry of the fiber core [10, 15]. A detailed study of the characteristics of Bragg grating in Hi-Bi fiber under the influence of reducing fiber diameter has been carried out by Abe et. al. . In their finding, the spacing between the two Bragg wavelengths in the reflection spectrum decreases when the fiber diameter is reduced via Hydrofluoric (HF) acid etching technique. Indications of reduction of birefringence and internal stress are observed in the Hi-Bi fibers.
In this work, we investigate the internal stresses in etched optical fiber manufactured from hydrogenated B-Ge codoped fiber and their impacts to its physical properties. When the fiber is etched in a Buffered Oxide Etchant (BOE) solution, the fiber diameter is reduced and the reducing cross-sectional area of the cladding alters the stress distribution in the fiber. To counter-balance the axial stresses between core region and cladding region, the fiber experiences a physical contraction in its axial dimension. The analytical models for the stress distribution profiles and axial contraction of an etched optical fiber are presented in this work. In the fabrication of uniform etched optical fiber containing FBG, homogeneous etching is desired to prevent fiber surface irregularities which may result to phase perturbations in the etched FBGs and alteration in the spectral characteristics . In the experiment, a reduction of grating period was observed which indicates an axial contraction in the etched FBG. We used a Zeiss Axioplan differential interference contrast (DIC) microscope [18, 19] to obtain clear images of the grating structure with a refractive index change of 1.7 × 10−4 and the microscope images were captured by a high resolution CCD camera. From the images, the measured grating periods of an etched FBG at different fiber diameters were obtained and we observed a trend of reduction in the grating period with reducing fiber diameter was observed. This observation is in agreement with the analytical model where an axial contraction of 0.2% is achieved as the fiber diameter is reduced from 125 µm to 6 µm. In relation with the internal stress profile, the core refractive index of the etched FBG varies with fiber diameter due to the stress-optic effect. Lastly, a new model comprises of the axial contraction effect and stress-induced index change is constructed and compared with the output Bragg wavelength of the etched FBG.
2. Analytical model
For a single mode fiber, the stress components in the polar coordinate are derived from the following relations 
For a non-step index profileEq. (4) is the Airy stress function. As the core is the only stress-producing component, the solution for an axisymetry fiber is given by F(r) = b0r2 + a0.
Satisfying the boundary conditions of at r = b.
The following functions for stress components are obtained6, 20],
3. The axial stress and contraction in etched optical fiber
In the equilibrium state, the sum of axial stress over the etched fiber cross-sectional area is zero.
The fiber equilibrium state is disrupted when the cladding recedes in an etching solution. The internal tensile stress is reduced and hence the fiber core experiences a contraction ε in the axial dimension. The axial stress profile for an etched FBG, with the radius ξ can be expressed as:Eq. (13) into Eq. (12),Eqs. (16) and (17) that the axial stress distribution profile varies with the changing fiber radius, ξ.
Figures 1(a) -1(c) shows the stress profile in radial and axial components, σr, σθ and σr of a step index fiber at different fiber diameters. In the figure, tensile stress is represented as positive and compression stress is denoted as negative. From the stress profiles, all the three stresses in the core region are of tensile and the stress magnitude is lower in fibers with smaller diameter.
To liaise with the two polarization modes in single mode fiber, the stress components are transformed into Cartesian coordinate as introduced below :Eqs. (18) and (19) to Eq. (20), the stress-induced index change can be represented as:
And the average of index change over the θ-axis from 0 to 2π is givenEq. (10) into Eq. (22) and related to the etched FBG, the stress induced index change in the fiber core is given by
The stresses in the core region are paramount to the analysis of stress-optic effect as a large fraction of light is confined within this region. The simulated results in Fig. 2(a) illustrate the relationship between the axial stress at the core center and the fiber diameter based on the relations in Eq. (16) and (17). The fiber initial diameter is 125 µm and the frozen axial stress at the center of the fiber core is 14 kg/mm2. The results indicate that the magnitude of the stress decays as the fiber diameter decreases during the etching process. The rate of decay becomes more rapidly when fiber diameter reduces to 20 µm. Depending on the index parameter γ, the axial stress decays faster with higher index parameter. Figure 2(b) illustrates the stress-induced index change calculated from Eq. (23). Based on the parameters presented in the caption of Fig. 1, the computed magnitude of the fiber core index change is in the order of 10−3 which is the same with that of the Ge-dopant constituent.
The actual stress model of an etched FBG with three-dimensional structure is complex. However for FBGs written with low UV fluence exposure, UV-induced stresses in the fiber can be neglected. It is reasonable to assume that the stress model for an etched optical fiber developed here can be applied for etched FBGs written with low UV fluence exposure.
4. Fabrication of etched FBGs
Type-I FBGs with a length of 2.0 cm were fabricated through grating inscription in B/Ge co-doped photosensitivity fiber (Fibercore Ltd: PS1250/1500). The FBG has an initial fiber diameter of 125.3μm, core-cladding index difference of 0.5% and 10 mol% of GeO2 and 14-18 mol% of B2O3. Before the inscription process, the fibers were hydrogenated for 10 days under a pressure of 2000psi at room temperature. The grating structure was inscribed into the fiber core by a continuous-wave UV laser at 244nm with an average output power of 4mW through a phasemask (Lasiris, ± 1st order and period 1072.6nm). The obtained Bragg reflection is at 1552 nm with 99.97% reflectivity.
The transmission and reflection spectra were measured with an optical spectrum analyzer (OSA from Ando AQ6331) with the resolution of 0.05μm. Transmission loss of up to −35dB is observed at the Bragg wavelength after the fiber was exposed to 244nm laser irradiation for a total fluence of 18J/cm2. It is believed that the change of refractive index is dominated by change of color-center, in which the change of stress profile and amplitude can be ignored for the grating exposed by such small UV fluence. The inscribed FBGs were left at room temperature for two weeks to allow out-diffusion of hydrogen from the fiber. Then, the transmission dip of around −25dB were measured from these gratings. The center wavelength of the grating is red-shifted during grating inscription, indicating Type-I grating was formed.
The etched FBGs with uniform fiber diameter are obtained by immersing them in BOE solution. The etching rate was controlled by the volume ratio of NH4F solution (40% in water) and HF solution (48% in water). In this work, a volume ratio of 6:1 was adopted. After etching is completed, the etched FBG tip was rinsed with de-ionized water to remove the residual etchant.
5. Determination of grating period through image processing
The axial contraction in etched FBG can be physically observed from fibers photo-imprinted with the Bragg gratings. Figure 3(a) shows a DIC microscope image of an unetched FBG with clear visibility of alternate dark and bright regions along the fiber. The black rectangular box in the Fig. 3(a) marks the position of the sample, a position where the grating structure is clearly visible in the image. When the etched FBG is observed under the microscope, the center wavelength of FBG was monitored using OSA to ensure that no additional strain is applied to the FBG, in which may alter the axial dimension of the thinned FBG. The solid curve in Fig. 3(b) shows a sample of the intensity profile taken from microscope image in Fig. 3(a) with a sample size of 10. The profile is low-pass filtered with a moving average filter size of 15. The peaks of the fringes can be easily located from the smoothened profile as indicated by the red circles in Fig. 3(b). The grating period is estimated by calculating the average peak-to-peak spacing.
The image noise was kept at a minimum level to ensure the high accuracy of the period estimation. The period is predicted from the data set with smallest standard deviation as shown in Fig. 3(c). This can be achieved through a proper control of the intensity of illumination on the fiber to optimize visibility of the fringes in the grating structure. To further reduce the noise, a moving average filtering is applied on the intensity profile in the horizontal axis (Refer to dotted curve in Fig. 3(b)).
Figure 3(c) depicts standard deviation produced from the intensity profile of microscope image in Fig. 3(a). Both Fig. 3(a) and 3(c) are arranged in such a way that the vertical coordinates of both figures are synchronous. The standard deviation is found to be the lowest at the fiber core with a clear visibility of refractive index (RI) perturbation. The calculated average grating period is 1075.2 ± 0.6 nm. The same measurement is performed on the phasemask and the measured period is 1072.6 ± 0.1 nm, which is exactly the same as what is provided by the manufacturer. The observed RI perturbation of the grating is of second-order reflection and thus the period of the grating is equivalent to that of phasemask . The difference between the grating period in the fiber and phasemask can be attributed to various factors occurring during writing process, for instance the strain applied on the fiber, orientation of the fiber, phasemask and lenses and geometric properties of the UV laser source.
Figures 4(a) -4(c) show the microscope images of the core regions of an etched FBG at different fiber diameters, namely 125 µm, 18 µm and 6 µm respectively. These fiber diameters were obtained through BOE etching. Microscope images were recorded using an objective lens with 50 × magnification factor and 0.85 numerical aperture, blue light illumination (wavelength 450 nm) and a 10MP high resolution digital CCD camera with 24-bit RGB pixel and SNR of 40.5dB. The parallel fringes in the grating structure can be clearly seen at the core region (with estimated index modulation amplitude of 1.3 × 10−4) for every microscope images and the fringes become more visible at smaller fiber diameter. The estimated grating periods for the grating structure in Fig. 4(a)-4(c) are 1076.4 ± 0.5 nm, 1075.0 ± 0.6 nm and 1.0743 ± 0.3 nm respectively. From a fiber diameter of 125 µm to 6 µm, the fiber undergoes 0.2% contraction in axial direction. The contraction of a single grating period is too small to be detected with good accuracy from the microscope images. Therefore, a bulk grating period of 109 fringes was measured. The sample standard deviation is calculated based on 10 samples of bulk grating (10 distinct samples from a microscope image, particularly in the region of grating structure), each sample comprises of 100-110 fringes. Sample standard deviation as low as 0.3nm can be obtained from clear microscope images with good fringe visibility. The sample standard deviation for the microscope image of phasemask is even lower, 0.1 nm.
The contraction of a single grating period is deduced via dividing the bulk grating period with the total number of fringes detected. Figure 5 shows six sample images taken from each microscope images which are enclosed by the white boxes in Fig. 4. Each sample has a dimension of 282 pixels × 51 pixels which is equivalent to 9.01µm x 1.63µm. For every set of two sample images (i) and (ii), they are axially spaced by a fixed distance of 103µm in their corresponding microscope images. The white sinusoidal profile in each sampled image represents the photo-induced index perturbations in the grating structure. The white vertical lines in each sample image mark the peak positions of the fringes. All sampled images (i) are arranged in such a way that the intensity profiles are in phase and the spatial spacing between images (i) and (ii) are maintained. Throughout the length from images (i) to (ii), the contraction in every grating period is cumulatively summed and the phase difference in the sinusoidal profiles in images (ii) can be seen. This result clearly shows the trend of axial contraction in etched FBG.
The period difference, ΛEtched - Λ0, between the grating period of an etched FBG and un-etched FBG, is given by:
Figure 6(a) provides a schematic illustration of an FBG before (top) and after (bottom) etching. The etched FBG experiences an axial contraction due to reduction in total compressive stress contributed by the cladding. The grating period suffers the same contraction as shown in Fig. 6(b). The analytical curve (solid) calculated based on Eqs. (15) and (24) appears to be in good agreement with the measured data from the microscope images. It can be noticed that the axial contraction of the etched FBG increases with reducing fiber diameter. However, the rate of change of axial contract plummets after 20µm. Consistent with the theoretical curve presented in Fig. 1 and Fig. 2(a), there is very little change in the axial dimension until the fiber diameter approaches 20 µm where the axial stress at the core center starts to drop and contraction increases abruptly.
Figure 7 shows the response of the Bragg wavelength in fiber with different diameters. The fiber was immersed in a BOE solution with a concentration of 16.7%, the diameter of the fiber was etched at a rate of 0.25µm/min. The blue shift of Bragg wavelength shown in Fig. 7 is attributed to the decrease of effective index and axial contraction of etched FBG. The relationship between effective index neff of an etched FBG with varying fiber diameter was calculated based on Variational Calculation . Taking into account of the stress-induced core index change from Eq. (23) based on step index profile and the contraction effect in etched FBG from the analytical expression Eq. (24), the Bragg wavelength of this model is given byFig. 7. The experimental result (as indicated as “circles” in Fig. 7) is in agreement with the calculated curve. In comparison with the original model (refer to dotted curve in Fig. 7) which assumes constant grating period and no stress-induced index change in the etched FBG, , the proposed model has larger blue shift in the Bragg wavelength and the wavelength difference can be as much as 3nm when the fiber diameter is reduced to 6µm.
The axial contraction in etched optical fiber is analytically presented and experimentally investigated based on the high resolution images of grating structures of an etched FBG. The grating periods of the etched FBGs were determined by image processing of the microscope images taken by a DIC microscope coupled with a 10MP digital camera. The same technique was applied on the microscope image of a FBG phasemask and the measured grating period is in agreement with the specification given by the manufacturer. The observations in this work has revolutionized the conventional ideas about the blue shifting in Bragg wavelength in etched FBG, that it was solely attributed to the decreasing effective index of a fiber of reducing fiber diameter in an etching solution. Apart from the effective index change, both axial contraction and core index change in an etched FBG contribute to the Bragg wavelength shift.
We would also like to thank the University of Malaya for providing the HIR Grant University of Malaya HIR Grant (UM.C/ 625/ 1/ HIR/ 071) and UMRG Grant (UM.TNC2/ RC/ AET/ 261/ 1/ 1/ RP019-2012C) for funding this project.
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