1. Introduction

Fiber Bragg gratings (FBGs) have been extensively studied as a key element in the established and emerging fields of optical communications. FBGs have opened the way to all-fiber devices in communications. Their associated spectral properties directly influence the systems such as wavelength division multiplexing (WDM) and have led to developments in add drop filters, gain flattened fiber amplifiers, and high quality fiber lasers. In addition to the field of optical communication, FBGs have also been widely used as optical sensors owing to their fiber based, wavelength-encoded characteristics [1]. There are many advantages of FBG sensors over conventional electrical strain gauges, such as the immunity to electromagnetic interference, flexibility, and even the durability against high radiation environments. Especially, their lightweight, non-obstructive characteristics make it easy to embed them into materials for the development of smart structure technology [2]. Up to now, for most applications of FBGs, the research efforts have concentrated on the features of the spectral behavior of the gratings. However, with the increasing bit rate used in WDM systems, the polarization characterization of various fiber optic components has more impact on the quality of the transmission in optical communications. For example, the polarization dependent loss (PDL) can lead to pulse distortion, which will cause the degradation in terms of high speed communication systems performance. It is thus important to characterize the PDL properties of FBG, especially their dependence on the wavelength.

The PDL of FBG is generally caused by the birefringence of optical fiber, which is typically induced during the process of laser beam exposure onto optical fiber. There are three main contributions to photo-induced birefringence of UV induced FBG: orientation of the state of polarisation of the UV writing beam, asymmetric UV-induced index change profile and modification of the glass stress profiles from the asymmetric glass densification in the photosensitive regions of the exposed regions. Besides that, birefringence can also be generated by transverse load applied on the gratings. Actually this stress induced birefringence is hardly perceived in the amplitude spectral response of the grating due to its low sensitivity. However, it will lead to significant PDL values within the grating, which can provide more effective information and therefore lead to the potential development of new types of FBG-based optical sensors [3–5].

In brief, it becomes more and more important to characterize the PDL properties of FBG for optical communications as well as sensing purposes. Some research activities have been devoted to the study of the FBG PDL properties and a lot of studies have been reported [6–8]. However, to the best of our knowledge, no work has been reported concerning the PDL characterization of locally pressed FBG in the literature up to now. Since the FBG is always under local transverse load in some practical applications, especially for a FBG embedded in composite materials [9]. Establishing the PDL response of the FBG under local load is thus helpful for evaluating and avoiding the effects of polarization in optical communication. It is also beneficial in distributed FBG transverse load sensing, with the aim of developing a sensor that tracks the PDL variation as a function of the applied local load. Therefore the PDL characterization of locally pressed FBG needs additional investigation.

In this paper, the particulars of the relationship of local transverse load to FBG’s PDL response are completely analyzed and presented. The wavelength dependency of PDL evolution on the local transverse load is then numerically simulated by utilizing a modified transfer matrix method. Through numerical simulations, it is shown that the PDL curves can be strongly affected by the load magnitude the position of the load and the loaded length of the grating. Experiments are then carried out and good agreements between experimental results and numerical simulations have been obtained.

2.Theoretical model

When the FBG is subjected to a local transverse load, the difference between the effective refractive indices of the two orthogonal modes of the fiber within the loaded region will beproduced, the effect of which is equivalent to creating a phase shift and thus introduce a spectral hole within the bandwidth of the FBG [10]. Here, for simplicity the direction of the transverse load is assumed as y (fast axis), another direction perpendicular to y-axis is x direction (slow axis), z is along the fiber axial direction, as shown in Fig. 1 . Purely compression load on a glass cylinder could be modeled as a line force since both optical fiber and compression platform are hard media. Since the length of a FBG is much longer than the diameter of the fiber, it is reasonable to assume the loading situations to be contained in a single plane. In our experimental condition, the test FBG is fixed at both ends and thus the FBG is under a loading state of plane strain (ε_z=0). The stress state of the loaded section of grating can be found from the plane strain elasticity solution for stress along the central axis in a disk given by [11,12]:

 

Fig. 1 Schematic diagram of a FBG subjected to local transverse load.

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Here F is the diametric load, l is the length of optical fiber under load, b is the radius of the optical fiber, and v is the Poisson’s ratio. The refractive index changes within the loaded zone in response to the applied load are derived from photoelastic theory described by Eqs. (2) and (3):

Where E is the Young’s modulus, p₁₁ and p₁₂ are the strain-optic coefficients; n₀ is the average effective refractive index.

The modified transfer matrix method [13] is used to simulate the PDL spectrum of the FBG subjected to local transverse load. According to this method, the grating is divided into m uniform subgratings and the load applied on each subgrating can be treated as uniform. Based on the transfer matrix method, a 2x2 matrix is identiðed for each subgrating, and then multiplying all of these matrices together could obtain a single 2x2 matrix which describes the whole grating. Define R_i and S_i to be the field amplitudes after traversing the section i. And then the propagation through each uniform section can be described by a matrix T_i defined such that:

Here ${\widehat{\sigma }}_{x(y)}={\delta }_{x(y)}+{\sigma }_{x(y)}$,${\delta }_{x(y)}$and${\sigma }_{x(y)}$are respectively detuning, “dc” coupling coefficient corresponding to the x and y modes, and k is “ac” coupling coefficient, △z is the length of each subgrating and because the FBG is under plane strain state (ε_z = 0), so △z = L/m. Once all of the matrices for the individual sections are known, the output amplitude is:

where F = F^(m)·F^(m-1)·…·F⁽ⁱ⁾·…·F⁽¹⁾, R₀ and S₀ describe boundary conditions and R₀ = 1, S₀ = 0. The refractive index of the subgrating within the loaded zone should be substituted by (2) and (3) to modify the transfer matrix, leading to the simulation of the spectrum response of the FBG. The amplitude and power transmission coefficients of the x and y modes t_x(y) = S_m/R_m and T_x(y) = |t_x(y)|² can be derived from Eq. (6). PDL is defined as the maximum change in the transmitted power when the input state of polarization is varied over all polarization states:

where |t_max|²and |t_min|² denote the maximum and minimum power transmitted through the component. In the case of FBG, the final expression of PDL for transmission is:

Generally, when FBG is subjected to transverse load, the stress induced birefringence will produce two Bragg wavelengths eventually leading to pulse broadening and peak splitting. But in most applications the changes in the spectrum of the FBG are hard to detect because of its low sensitivity. Unlike the spectrum, the PDL spectrum is very distinct even if the spectrum of the FBG is almost unchanged. Thus, the PDL spectrum would be very useful for sensing purpose.

3. Numerical simulation and experimental investigation

A numerical simulation based on the modified transfer matrix method described in section 2 was adopted to simulate the PDL spectrum of a FBG under the local transverse load. The simulated result is then analyzed to determine the relationship between the evolution of the PDL response and the load magnitude the position of the load and the loaded length of the grating. The major parameters for all the simulations in this section are as follows: the central wavelength of the FBG without perturbation is 1557.1nm, the FBG length L is 1cm, n₀ = 1.5, E = 74.52Gpa, v = 0.17, p₁₁ = 0.121 and p₁₂ = 0.270. A simple experiment was also carried out to verify the simulated model and results.

3.1 Effect of the transverse load magnitude

Firstly, the effect of the transverse load magnitude on the PDL evolution of the FBG is investigated using the above mentioned method. We consider the case when the center of the FBG is loaded transversely in a very small region (0.25mm). Figure 2(a) shows the simulated PDL spectra for various transverse load magnitudes in the range of 0 to 25N. As we can see, the PDL spectra of the FBG exhibit almost (quasi) periodic behavior (the period is about 10N) with respect to the magnitude of loads. In the first half period, as the load increases, the values of PDL in the whole wavelength range increase with increasing load. Instead, in the last half period, the opposite is true. The asymmetry between the two main lobes of PDL is also observed in Fig. 2(a). The causes can be explained as: For a uniform FBG without any perturations, the two main lobes of PDL are usually symmetrically distributed on either side of the resonant Bragg wavelength, which is close to the minimum PDL value. When the FBG is subjected to a local transverse load, the pressed section will cause a phase shift and introduce a transmission notch within the bandwidth of the FBG spectrum, which acts like a phase-shifted grating. The transmission notch will be located at the center of the bandwidth if the phase-shift is equal to pi. In this case, the PDL spectrum will be symmetrical. As the localtransverse load changes, the transmission notch will moves to longer or shorter wavelengths since the phase shift is not equal to pi. Then the PDL spectrum will be no longer symmetrical, which cause the observed asymmetry between the two main lobes of PDL. For the same reason, the characterization of quasi periodic variations in PDL is also predicable since the phase shift is periodic and strain dependent, which will lead to repetitive behaviors in amplitude spectrum and eventually in PDL spectrum. Figure 2(b) depicts the maximum PDL amplitudes and their corresponding wavelengths of the FBG with respect to the local transverse loads magnitudes. As seen in Fig. 2(b), the variation of the maximum PDL values is similar to an inverted power function curve with slowly decreasing amplitude. The wavelengths corresponding to the max PDL values increase gradually with the load increase in a period of about 10N, however, the peak envelope of the wavelengths decreases slowly with the load increase. The simulated results could be efficiently used for recovering the local transverse load magnitude based on detecting both the maximum PDL amplitudes and their corresponding wavelengths.

 

Fig. 2 (a)Full PDL response (b) Maximum PDL amplitudes and their corresponding wavelengths of the FBG for variety of local transverse loads magnitudes.

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3.2 Effect of the position of the load

The second step is to consider the cases when the local transverse loads are applied at different positions of the FBG. The FBG length is 1cm, the position is varied from 0 to 1cm with an interval of 0.25 mm. The load magnitude is assumed to be 5N and the region length of the FBG under load is still 0.25mm. On the basis of the above method, the relationship between the position of the load and the PDL is simulated, as shown in Fig. 3(a) . Figure 3(a) shows how the PDL spectrum of the FBG changes as load is applied at different positions along the whole length of the FBG. The PDL values increase with the position moving from the edge of the FBG to the internal region until they reach the maximum when the load is applied at the exact center position, as described in Fig. 3(b) more clearly. It is obvious that the variation curve of the maximum PDL values with respect to the different position of the load is similar to the effect of loads magnitudes in a period, being like an inverted power function. However, for the peak resonance wavelength, Fig. 3(b) shows the wavelengthscorresponding to the maximum PDL amplitudes contain a constant at 1557.1nm. It is readily observed that Fig. 3 could provide meaningful information for spatial measurement of thedistributed FBG sensor.

 

Fig. 3 (a)Full PDL response (b) Maximum PDL amplitudes and their corresponding wavelengths of the FBG for different positions of the load.

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3.3 Effect of the loaded length

Finally, to estimate the loaded length effects, the full PDL responses of the FBG and their characterization are computed. It is assumed that the transverse local load is 5N and is initially applied at the center position of the FBG. The loaded region then spread out along both sides of the FBG center with the increase of the loaded length. Figure 4(a) depicts the PDL responses versus the loaded length. The PDL spectrum contains three obvious resonance peaks and is very similar in form even though the loaded length is different. Their maximum PDL amplitudes decrease gradually with the increase of the loaded length, as shown in Fig. 4(b). Figure 4(b) also shows the peak resonance wavelength shifts slowly to longer wavelengths. Figure 4 could be helpful for evaluating the loaded length of the FBG by detecting its PDL characterizations.

 

Fig. 4 (a) Full PDL responses (b) Maximum PDL amplitudes and their corresponding wavelengths of the FBG versus loaded length.

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3.4 Experimental investigation

Experiments were set up to verify the simulated model and results. As shown in Fig.5(a), the input light from a tunable laser source (1528-1563nm) was launched into the PDL multimeter (GPC PDL-101). The recorded PDL measurement data was read out via an USB interface and transferred to the PC. Figure 5(b) shows the amplificatory view of the compression arrangement, where the bottom compression area is an aluminum plate. The test FBG whichhas an unstrained Bragg wavelength of 1557.1 nm was written with a length of 1 cm. The balance reference fiber was placed parallel to the test FBG. The test FBG with the initial strain applied was fixed at both sides on the bottom surface of the plate by epoxy resin. Thisprocedure required careful processing to avoid shear stress which will cause the FBG to break easily. To validate the PDL behaviors observed numerically in sections 3.1-3.3, two different arrangements for the application of local transverse load to the FBG had been used in our experiment. For the situations in section 3.1 and 3.2, a copper pillar about 3mm in diameter was attached to an optical bench at the pivot point and the load was applied to the opposite end, as shown in Fig. 5(c). Since both the FBG and the copper pillar are hard mediacylinders, the loads assumed to be applied in a small region is reasonable, which is in accordance with our simulation conditions. The load was then increased gradually from 0N to 40N with a step of 2.5N and was applied at the center of the FBG. It should be noted that only half the load was actually applied on the FBG because the balance fiber shared the other half. The experimental results of the effect of the transverse load magnitude on the PDL of the FBG was shown in Fig. 6(a) . Figure 6(b) indicated that the the envelope of the Fig. 6(a)(maximum PDL amplitudes magnitudes) showed a periodic change as a function of increasing load, which was well consistent with the theoretical predictions comparing to Fig. 2. To study theeffect of the position of the load, the copper pillar was moved precisely in a step of 1.5mm by use of a micrometer. The actual load on the FBG was arranged at 5N to meet the simulation condition in section 3.2. The experimental results was shown in Fig. 7(a) . Figure 7(b) showed the maximum PDL amplitudes obtained from the experiment with respect to the different positions of the load. Both Fig. 7(a) and Fig. 7(b) were in good agreement with the simulated evolutions (refer Fig. 3). Finally, for the situations in section 3.3, to verify the validity of the theoretical model used for the simulation of PDL behavior for different loaded length of FBG, two pieces of copper block with different width were used as compression device for the application of local transverse load. We arbitrarily consider two different integrated loading conditions: 1) load magnitude: 5N, position of the load: 1/3 of the FBG, loaded length: 6mm (which means the top copper block have a width of 6mm). 2) load magnitude: 2.5N, position of the load: 1/4 of the FBG, loaded length: 4mm. The experimental result also coincided well with the theoretical predictions of the PDL response spectrum, as shown in Fig. 8 , demonstrating the validity of the theoretical model in our study once again. This weak disagreement may be caused by the following factors: a). the power from the laser source used for PDL measurement is not highly stable. b). the fiber cables and connectors used in the measurement also having small PDL values cause the error. c). the effect of unexpected shear stress on the FBG.

 

Fig. 5 Experimental setup (a). Optical configuration (b). Amplificatory figure of the mechanical configuration.

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Fig. 6 Experimental PDL spectra of the FBG for variety of local transverse loads magnitudes: (a)Full PDL response (b) Maximum PDL amplitudes magnitudes.

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Fig. 7 Experimental PDL spectra of the FBG for different positions of the load: (a)Full PDL response (b) Maximum PDL amplitudes.

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Fig. 8 The experimental result and simulated evolutions of the PDL response, the load parameters are: (a).F = 5N, Position = 1/3, Loaded length = 6mm ;(b). F = 2.5N, Position = 1/4, Loaded length = 4mm.

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4. Conclusion

Locally pressed FBGs open new possibilities for novel devices and applications in optical communication systems such as polarization delay line、WDM add/drop filter and so on. It is thus important to characterize the FBG polarization properties. Moreover, their polarization properties can play an important role for retiring the local loading information in distributed fiber sensing. In this work, the theoretical analysis of spectral characterization of PDL of locally pressed FBG has been presented. Experimental investigations were also carried out to verify the feasibility of the theoretical model. Good agreement was obtained between experimental result and theoretical predictions.