1. Introduction

Fiber optic sensors are increasingly used in many applications owing to their extensive list of advantages including immunity to electromagnetic interference, small size, light weight, high sensitivity, multiplexing capabilities, etc [1,2]. Such sensors have proven to be very effective for measuring mechanical quantities such as strain, stress, pressure, transverse load and even shear stress [3–5]. Mechanical optical fiber sensors relying on fiber Bragg gratings (FBGs) or on polarization maintaining fibers (PMFs) typically exploit the photo-elastic effect for transducing a mechanical action into a measurable change of the Bragg wavelength or of the PMF beat length [6–8]. The photo-elastic effect is governed by the stress-optic law, which establishes the link between applied mechanical stress and the induced change in refractive index of the optical fiber material. The stress-optic law is formulated in Eq. (1).

The values of C for silica glass optical fibers that can be found in open literature are quite consistent. Variations of 13% around C = −3.37 × 10⁻¹² Pa⁻¹for bulk fused silica have been reported [9].

For polymer optical fibers (POFs), only a few references exist, which report significantly varying values of C. The values of C measured on bulk and thin film polymethyl methacrylate (PMMA) ranges from −1.08 × 10⁻¹⁰ Pa⁻¹ to 5.3 × 10⁻¹² Pa⁻¹ [10–15]. Moreover, in [14] the authors demonstrate that the photoelastic constant in PMMA also depends on the presence and concentration of dopants in the polymer. In [16] C has been measured in PMMA fibers and its value has been found to depend on the drawing conditions of the fiber. The values obtained are between 1.5 and 4.5 × 10⁻¹² Pa⁻¹. This may point at the necessity to measure C for every different type of POF. In addition, all the methods that have been proposed so far to determine C in glass and polymer fibers rely on the hypothesis that C is constant over the core cross-section of the fiber. Considering the fabrication method of optical fibers, either glass or polymer, that assumption is not necessarily straightforward. Fabricating an optical fiber typically involves manufacturing a preform including differently doped regions and hence different materials to form the core and cladding using specific synthesis or assembly techniques, and subsequently drawing the fiber from this preform at a particular drawing speed and tension, and in dedicated thermal conditions [16–19]. This may not only influence the value of C across the fiber but also change that value compared to bulk silica glass [20]or PMMA.

The purpose of this article is therefore to develop algorithms aiming to determine the radial profile of the photoelastic constant C(r) over the cross-section of glass and polymer optical fibers. We build on earlier work that we have reported in [21], in which we have described a first method to determine the radial distribution of C in silica glass optical fibers. The main conclusion of that earlier work was that the average value of C in silica glass fibers is slightly higher than that of bulk silica, in line with the findings described in [22]. However, we could not firmly conclude that C is constant or not over the fiber cross-section. Furthermore, we did not consider POF in our studies so far.

Our paper is structured as follows: Section 2 describes the measurement set-up and two algorithms that we have developed to determine C(r).Section 3 discusses the experimental results and compares the photoelastic constants measured in a Ge-doped glass fiber, in two fluorine-doped depressed clad fibers with increasing core diameter and in two PMMA fibers. Section 4 closes our paper with a summary of our findings.

2. Description of the measurement method

2.1. Retardance measurement

In previous work [21] we have already given a detailed description of the measurement method that we use to determine the radial distribution of the photoelastic constant C(r) in an optical fiber. We recall the experimental set-up here for sake of completeness. Figure 1 shows the Cartesian axes system, in which we take the z-axis along the fiber axis. We subject the fiber to a known axial stress${\sigma }_z$which is assumed constant across the fiber, and we illuminate the fiber with monochromatic light at 633nm that is linearly polarized at 45° in the yz-plane. The wave vector is perpendicular to the fiber axis and taken parallel with the x-axis, as also illustrated in Fig. 1. Since we immerse the fiber in index matching fluid during the measurement, the direction of the wave vector does not change at the boundaries of the optical fiber.

 

Fig. 1 Illustration of an optical fiber transversely illuminated with a plane wave (left) and resulting retardance profile R(y) (right). b is the radius of the fiber. The z-axis is taken along the fiber length with a direction exiting the page.

Download Full Size | PDF

Axial load applied to the fiber induces birefringence in the fiber material due to which the two linearly polarized components along the y and z directions will experience a different phase shift. This phase shift can be observed as the projected retardance R(y) between these two orthogonal components.

The projected retardance R(y) is related to the axial stress by an inverse Abel transform [23–25], as given by Eq. (2):

The photo-elastic constant C(r) is the regression coefficient linking the inverse Abel transform of the projected retardance R(y) and the applied known axial stress${\sigma }_z$. r is the radial distance taken from the fiber’s center and b is the radius of the fiber.

We use a polarizing microscope and apply the Sénarmont compensation method to measure R(y). The set-up is illustrated in Fig. 2.

 

Fig. 2 Polarization microscope set-up to measure the full-field retardance profile using the Sénarmont compensation method. To obtain a controlled tensile stress, a predefined axial load is applied to the fiber using an external loading system.

Download Full Size | PDF

The optical fiber is immersed in index matching liquid to avoid diffraction and scattering of the light at the edges and is placed between a polarizer and quarter wave plate aligned at 45° with respect to the fiber’s axis. The birefringence of the fiber causes the orientation of the linearly polarized light to change at the output of the quarter wave plate. The analyzer is oriented perpendicularly to that direction, which results in extinction of the light exiting the analyzer. The angular range of rotation of the analyzer ${\theta }_{Atot}$is chosen to achieve extinction for every pixel in the field of view of the microscope. A CCD camera records an image for each position of the analyzer ${\theta }_A$in that range. For each pixel the recorded intensity is plotted as a function of ${\theta }_A$. A polynomial fit is performed on the intensity profile to determine the minimal intensity and the corresponding analyzer angle${\theta }_{A\min }$. The retardance in the pixel under consideration is determined with $R(y)=\frac{{\theta }_{A\min }}{180}\lambda$ with ${\theta }_{A\min }$in degrees. We used objective lenses with 40x and 20x magnifications and numerical apertures of 0.90 and 0.50 respectively, resulting in spatial resolutions of 0.43 µm and 0.77 µm. This is sufficient to obtain radial profiles taking into account the typical dimensions of the optical fibers.

2.2. Original and adapted inverse Abel transform algorithm

Once that we have measured R(y) we still have to determine its inverse Abel transform. Inverse Abel transforms can be calculated using Fourier theory [24,26,27]. The inverse Abel transform of the Fourier expansion of the retardance is computed with Eq. (3):

where$\rho =r/b$ is the normalized radius, $t=\sqrt{1-{\rho }^2}$ and $a_k$is the k^th Fourier coefficient of the Fourier series of the retardance.

In [21] we have developed such an algorithm to which we will refer as ‘Algo 1’. The inverse Abel transform requires the integration of the derivative of R(y),which implies that measurement noise on R(y)significantly impacts the final result. As we will show in Section 3, the measured retardance profiles for the PMMA fibers are much noisier than for the silica glass fibers, leading to increased variance in the results obtained for the polymer fibers.

We therefore modified the approach to compute the inverse Abel transform. We label this new algorithm with ‘Algo 2’. It still relies on Fourier theory, but the product $C(r).{\sigma }_z$is expanded in Fourier series instead of the measured retardance, as inspired by [28]. The forward Abel transform of $C(r).{\sigma }_z$results in the measured retardance. The main advantage of the forward Abel transform is that it does not require the integration of a derivative. Furthermore, ‘Algo 2’ should converge faster as there is a constant term in the Fourier expansion of$C(r).{\sigma }_z$. In ’Algo1’ we expand the retardance R(y). The inverse Abel transform of R(y) requires deriving R(y). Consequently, the constant term disappears and more Fourier coefficients are required to allow ‘Algo 1’ to converge. We thus expect a better and more robust behavior of ‘Algo 2’when dealing with noisy data. The expansion of $C(r).{\sigma }_z$is shown in Eq. (4).

where$\rho =x/b$ is the normalized radius and $t=\frac{\sqrt{r^2-y^2}}{b}$.

To compute the amplitude of the k Fourier coefficients a_k we apply the least square criterion [29] and we evaluate the expression given in Eq. (7).

To compare the performance of the two algorithms we take R(y) equal to a half an ellipse E(y). Assuming that tensile stress and C are constant across the fiber, the shape of the retardance profile should indeed be elliptical. The analytical expression of the inverse Abel transform of the semi-ellipse is a constant value dependent of the semi-long and semi-short axis of the ellipse (see Eq. (8)).

where$A=\max (abs(R(y)))=44.98nm$is the semi-short axis and $B=62.50\mu m$ the semi-long axis of the ellipse E(y). To simulate real measurements we add Gaussian noise to E(y). The standard deviation of the Gaussian noise is fixed to 1.0 × 10⁻⁹ nm, which corresponds to the actual standard deviation of the retardance measurements on polymerfibers (see Section 3). Figure 3 compares the result of the inverse Abel transform of E(y) computed with both algorithms. We have chosen the amount of Fourier coefficients k₁ = 10 and k₂ = 50. ‘Algo 2’ returns a profile close to the analytical profile for the small amount of Fourier coefficients. ‘Algo 1’ requires a larger amount of Fourier coefficients to get close to the shape of the analytical inverse Abel transform. This goes at the expense of having a very noisy profile.

 

Fig. 3 The inverse Abel transform of the noisy ellipse for k₁ = 10 (left) and k₂ = 50 (right) computed with ‘Algo 1’ and ‘Algo 2’. The standard deviation of the Gaussian noise on top of the signal is$1.0\times {10}^{-9}$nm. The results are compared to the analytical value of the inverse transform of E(y).

Download Full Size | PDF

To quantify the performance of both algorithms we consider the root-mean-square error (RMSE) between the analytical expression of the inverse Abel transform and the computed inverse Abel transform of the noisy ellipse for a range of Fourier coefficients from k = 1 to k = 200. We calculate the average error as a function of k. The details of the methodology to determine the RMSE are described in a previous publication [21]. In that publication we demonstrate that a high number of Fourier components leads to a better result. But when the standard deviation of the measured signal increases, the influence of the noise on the inverse Abel transform becomes dominant. Increasing the number of Fourier coefficients adds substantial oscillations to the resulting profile.

Figure 4 evidences that ‘Algo 2’ is more robust than ‘Algo 1’ for computing inverse Abel transforms of profiles with a high variance. Aminimal error of 12% with ‘Algo 1’ requires about 40 coefficients, whilst ‘Algo 2’ achieves much lower errors with a small value of k. For an increasing amount of Fourier coefficients both algorithms tend to behave in the same manner with an increasing RMSE.

 

Fig. 4 RMSE between the analytical value of the inverse Abel transform of the ellipse and the computed Abel transform of the noisy ellipse as a function of the amount of Fourier coefficients k. The standard deviation of the Gaussian noise added to the signal is$\sigma =1.0\times {10}^{-9}$nm.

Download Full Size | PDF

3. Experimental set-up and results

3.1 Important parameters

To evaluate and compare both algorithms based on real measurement data we have experimented with commercially available step index singlemode and multimode silica glass fibers with increasing core diameter. We have also characterized commercially available PMMA optical fibers. The main specifications of those fibers are summarized in Table 1.

Table 1. Main characteristics of the optical fibers used to measure the retardance and determine the radial profile of the photoelastic coefficient [30–32].

View Table

We determine the retardance profile for tensile stress that varies from 82MPa to 185 MPa for the glass fibers and from 10MPa to 40 MPa for the polymer fibers. We reduce the tensile stress for the polymer fibers to minimize the possible deformation of the fiber. The polymer fibers were stored and measured in a room with stable temperature (20°C) and humidity (59%).

As described in Section 2, to determine R(y) for one specific pixel in the field of view, we apply a polynomial fit of the measured intensities to determine the minimum intensity in an accurate manner. Effective fitting requires determining the degree of the polynomial fit L and the number of measured data points taken into consideration. We determined the polynomial fit degree according to Akaike’s information criterion [33,34]and we took L = 4.

For all silica glass fibers and for the singlemode PMMA fiber a picture is taken at every 2° rotation of the analyzer. For the multimode PMMA fiber the retardance is larger and requires a larger range${\theta }_{Atot}$. A picture is therefore taken after every rotation of 4° of the analyzer. Furthermore, increasing the axial stress increases the retardance in the fiber and the range ${\theta }_{Atot}$has to be adapted accordingly to obtain extinction for every pixel. The same range ${\theta }_{Atot}$is used for every pixel; it implies asymmetric intensity profiles for some pixels in the field of view. This causes the polynomial fit to be inaccurate. To cope with this problem we considered a fixed amount of 13 datapoints around the measured minimum intensity for the measurements of R(y) and we determined the polynomial fit using only these datapoints. The estimated value of${\theta }_{A\min }$becomes more accurate as illustrated in Fig. 5.

 

Fig. 5 : Polynomial fitting of the measured intensity profile for one pixel. The graph on the left side is the result of the polynomial fit for all the datapoints in the range of the analyzer${\theta }_{Atot}$. The graph on the right side represents the polynomial fit for 13 datapoints around the measured minimum. The polynomial fit degree L is 4.

Download Full Size | PDF

Once the retardance R(y) is known we still have to compute the inverse Abel transform to determine C(r). From [21]and [27],and as described in Section 2.2,it is clear that the amount of Fourier coefficients k considered for the expansion of R(y) is the essential parameter that influences the final result. The choice of k depends on the noise level of the retardance R(y) and is different for ‘Algo1’ and ‘Algo2’.

3.2 Experimental results and discussion

We use two approaches to determine the photoelastic constant C. The first aims to determine the mean value of C by approximating the measured retardance profile with half an ellipse, whilst the second derives the radial distribution C(r) from the measured retardance without any simplifications.

To determine the mean value of the photoelastic constant the retardance is approximated with the shape of a semi ellipse E(y). Since the inverse Abel transform of E(y) is constant, as given by Eq. (8), the relation between the axial stress and the retardance becomes:

b is the fiber radius, R(y) the measured retardance and σ_z the axial stress. C is the regression coefficient that has to be determined. The results we obtain with this method are shown in Fig. 6 for fibers 1,3, 4 and 5. Note that the retardance profile of fiber 2 suggests that it is a dual-clad fiber with depressed inner cladding. Reference [35] indicates that the core is surrounded with a fluorine-doped cladding. A measurement of the refraction index of this fiber can be found for example in [36] from which we can also deduce that the fiber has a depressed cladding. As a consequence of the particular index profile, the retardance profile over the entire cross-section of fiber 2 is not elliptical and hence the elliptical approximation cannot be applied.

 

Fig. 6 C .σ_z as a function of the axial stress. The regression coefficient is the photoelastic constant C. Its values are indicated in the graph along the respective linear fit. Fibers 1 and 3 are silica fibers, fibers 4 and 5 are PMMA fibers.

Download Full Size | PDF

The elliptical approximation allows determining the mean value of the photoelastic constant. We obtain C values of −3.71 × 10⁻¹²Pa⁻¹and −3.74 × 10⁻¹² Pa⁻¹for the silica glass fibers 1 and 3. This value is slightly higher than the value measured on bulk fused silica (C = −3.52 × 10⁻¹²Pa⁻¹) determined in [37]. For the PMMA fibers, we obtain C = −8.56 × 10⁻¹³Pa⁻¹ for fiber 4 and C = 5.14 × 10⁻¹²Pa⁻¹ for fiber 5.

The second method calculates the radial distribution of C(r).We first compute the inverse Abel transform of each measured retardance profile corresponding to a specific tensile stress. We then apply ‘Algo 1’ and ‘Algo2’ to determine the inverse Abel transform of the retardance profiles. This allows obtaining the relationship$f(r)=C(r).{\sigma }_z$. Working out the linear regression of $f(r)$for every radial point finally yields the radial distribution of the photoelastic constant C(r). To determine the amount of Fourier coefficients for our calculations, we selected that value of k that minimizes the variance of the resulting $C(r).{\sigma }_z$profile in the central parts of the fiber cross-section (core/cladding). We did not take into account the transitions and discontinuities at interfaces and edges. Figure 7 shows the radial profiles of the photoelastic constant for the three different glass optical fibers. The standard deviation of the retardance measured for glass fiber 1 is small (σ_glass< 0.5 × 10⁻⁹). The results are similar and confirm that both algorithms behave equivalently for profiles with a low noise level.

 

Fig. 7 Radial distribution of the photoelastic constant C(r) in one fiber section. The profiles are respectively computed with ‘Algo 1’ and ‘Algo 2’. (a) C(r) in fiber 1, (b) C(r) in fiber 2 and (c)C(r) in fiber 3. The respective value of k is mentioned in the graph.

Download Full Size | PDF

The difference in refractive index between core and cladding for glass fibers 2 and 3 is four times larger compared to fiber 1.This increases the scattering at the boundaries between the two materials and increases the noise on the measured retardance R(y). The Ge-doped core of fiber 1 does not influence the profile of C(r) and we can conclude from Fig. 7(a) that C(r) is constant throughout the cross-section of this fiber. On the other hand, Fig. 7(b) and 7(c) evidence that the value of C(r) in the fluorine-doped cladding portion differs significantly from the value of C(r) in the undoped core of fiber 2 and 3 and from the outer cladding of fiber 2.

In agreement with the conclusions of Section 2, the first algorithm requires a higher number of Fourier coefficients k to achieve a correct profile at the expense of increasing the impact of the measurement noise. For the second algorithm less Fourier coefficients are needed to achieve a reliable and stable result. We also compute the average value of C(r) in the stable parts of the radial profiles and we obtain respectively −3.83 × 10⁻¹²Pa⁻¹, −3.75 × 10⁻¹²Pa⁻¹and −3.80 × 10⁻¹²Pa⁻¹ for glass fiber 1 and for the undoped portions of fibers 2 and 3. These figures are comparable with the mean values of the photoelastic constant that we have determined with the elliptical approximation. The mean value of C(r) equals −2.75 × 10⁻¹² Pa⁻¹ in the fluorinated trenches of the cladding. To the best of our knowledge this is the first demonstration that one cannot assume the photoelastic coefficient to be constant throughout the fiber section for certain types of fibers. Our results show that the fluorine doping decreases the absolute value of C with about 27% compared to pure silica or Ge-doped silica glass fibers.

The noise level in our experimental results with PMMA fibers is higher compared to silica glass fibers [38]. The fabrication tolerance for the core and cladding diameter is about 10%, ten times higher than for glass fibers [33]. Additionally the tolerance of the cladding refractive index is around 3%. This induces mismatches between the refractive index of the cladding and the refractive index of the index matching liquid surrounding the fiber. Moreover the difference in refractive index between core and cladding is 20 times larger for fiber 5 compared to fiber 1. These mismatches cause additional scattering and diffraction at the edges and in the fiber. These factors contribute to a decreased accuracy of the measured intensity profiles and the retardance R(y) for polymer fibers.

The radial distribution of C(r) in fibers 4 and 5is depicted in Fig. 8. The profile we obtain with ‘Algo 1’ is very noisy as the number of Fourier coefficients has to be large to achieve a stable profile. But a larger k increases the influence of measurement noise on the inverse Abel transform. On the other hand the radial distribution computed with ‘Algo 2’ is rather regular and smooth except at the edges of the fiber. Fiber 4 has a small core and the rather sharp transitions of the retardance profile between core and cladding increases the impact of the numerical artefacts of the inverse Abel transform at these locations. The average value of C(r) in the stable parts equals respectively −8.8 × 10⁻¹³Pa⁻¹ and 5.5 × 10⁻¹² Pa⁻¹for the PMMA fibers 4 and 5, close to the values that we found with the elliptical approximation. From the profile of C(r) we can conclude that the photoelastic constant is constant over the fiber section in these polymer fibers. The results on the PMMA fibers also tend to prove that the second algorithm based on the Fourier expansion of $C(r).{\sigma }_z$ is very robust when one has to compute the inverse Abel transform of a noisy profile. Unlike the values of the photoelastic constant of the silica glass fibers that are all similar in either the undoped or fluorine-doped portions, the values of the coefficients of the two PMMA fibers differ significantly; moreover, their signs are opposite. This finding can be explained by the strong dependence of the photoelastic constant of a polymer fiber on the fabrication and production process [16]. On the other hand, we obtain identical results for different lengths of fiber taken from the same batch. C(r) should therefore be measured for each type of fiber, but measuring one fiber length of the whole batch seems sufficient to determine the photoelastic coefficient of the polymer fiber of that specific batch.

 

Fig. 8 Radial distribution of the photoelastic constant C(r) in one POF fiber section. The profile is respectively computed with ‘Algo 1’ and ‘Algo 2’. (a) C(r) in fiber 4.(b) C(r) in fiber 5.

Download Full Size | PDF

4. Conclusion

We have compared two algorithms to determine the radial distribution of the photoelastic coefficient C(r).

First, we have shown that the parameters required to carry out the polynomial fit of the intensity profiles to determine the retardance need to be chosen carefully. The polynomial fitting order has been fixed to L = 4 and the number of datapoints taken into consideration has to be determined in order to obtain a well-fitted profile.

Then we analyzed the algorithms to perform the inverse Abel transform of the retardance. Both algorithms are based on Fourier series. The first algorithm decomposes the measured retardance in his Fourier coefficients and afterwards computes the inverse Abel transform. The second algorithm, on the other hand, starts from the expansion of the desired profile in Fourier series and then computes the forward Abel transform. The obtained function is subsequently compared to the measured retardance based on the least squares criterion. The most important conclusion here is that the second algorithm is more robust to compute the inverse Abel transform when dealing with noisy measurements.

Finally, we have determined C(r) of different silica glass and PMMA fibers using both algorithms. The mean value of C determined with the elliptical approximation and the average value of C(r) in the stable parts of the profiles are equivalent. For silica glass fibers the mean value of C computed for the three fibers under test equals −3.78 × 10⁻¹² Pa⁻¹ in the undoped and Ge-doped portions of the cross-sections. The average value of C in the fluorine-doped part of the cladding region is 27% smaller. For PMMA fibers the photoelastic coefficients of the two fibers under test are entirely different. The singlemode POF has a mean C equal to-8.8 × 10⁻¹³Pa⁻¹, while for the multimode POF C equals 5.5 × 10⁻¹² Pa⁻¹. These results lead us to conclude that, in contrast to glass fibers, C cannot be approximated by any ‘standard’ value and that it has to be measured for every POF.

With respect to the radial distribution of C(r), we conclude that the photoelastic coefficient can be considered constant throughout the fiber section for silica fibers that only conclude a Ge-doped core and for PMMA fibers. This is no longer the case in silica glass fibers containing a fluorine-doped depressed cladding.