Introduction

Polarization dependent effects are becoming a major topic for modern optical communication and fiber sensing [1]. The fiber birefringence is one of the basic polarization effects which contribute to the well-known PMD, meanwhile it is an important physical parameter for many fiber-based sensing applications such as pressure sensors, temperature sensors, and etc.. Several methods have been developed to analyze the stress-induced birefringence especially the intrinsic birefringence in single mode fibers (SMFs), for example, a vector perturbation model proposed in [1], a modified coupled-mode theory described in [2], an optical tomographic technique applied in [3], an analogy between the pressure-induced strain and the thermal strain employed in [4], and etc.. Among them the stress distribution is the key issue in calculating the birefringence. In Ref. 3 and Ref. 5 the 2-D stress profile was measured with an optical tomographic technique while Ref. 6 introduced the finite element method for stress analysis, which are general methods for analysis of stress distribution in optical waveguides. To investigate the stress distribution of a round fiber under external forces, we think there should be a simple method when taking account of the circular symmetry of the fiber. In this paper, we propose a convenient approach to analyze the stress distribution in SMFs, by which we easily obtain all the stress components and the stress patterns in the core. Then we compute the stress-induced birefringence and perform a pressure vector sensing based on ~2 km SMF with the validity analyzed.

1. Pressure stress distribution in SMFs

Consider a round SMF with the core and cladding under the lateral pressure P as shown in Fig. 1, whose right part is the cross section of the fiber in the coordinate system, where a and R are the radii of the core and cladding. Since the residual stresses are relatively very small to the pressure stress in the SMF, we omit their effects on stress distribution.

 

Fig. 1. (a) The schematic graph and (b) enlarged cross section of SMF under lateral pressure.

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In the case that pressed length (5 mm in our experiment) is much larger than the size (0.125 mm) of fiber cross-section, the stress distribution in pressed fiber can be treated as a plane stress problem which means it only relates to the cross section while independent of the fiber length. Thus the stress function on A(r, θ) and the stress tensor components in the fiber are described by [7]

The integral formula solution of biharmonic equation (1) for a circle plane with radius R is [8]

in which μ(φ)=A_Q = ΣM⃗_i, $v\left(\phi \right)={\left(\frac{\partial A}{\partial n}\right)}_Q=\sum {\overrightarrow{F}}_i$ . Here Q is an arbitrary point on the circle boundary, n is the normal of the integration boundary, φ is the integration variable, while F⃗_i and M⃗_i are respectively a force element existing in the integration interval ( $\stackrel{︵}{\mathrm{BQ}}$ in Fig. 1b) and its moment with respect to point Q [7]. By simple calculation we get

Inserting expressions (4) into Eq. (3), and compute the integration we obtain

Since most optical field just exists in the core, we focus on its stress distribution which mainly determines the birefringence of the fiber under press. Considering r̃ = r/R≪1(r≤a) we take it into account till the second power (r̃ ²) and neglect the higher order terms, then substituting Eq. (5) into Eq. (2) we get the non-zero stress components

Transforming these components to the rectangular coordinates shown in Fig. 1(b) we have

Given parameters 2R = 125 µm, a = 4.25 µm, P = 200 N/m which means a force of 1 N acts on a 5-mm-length bare fiber, by Eq. (7) we obtain the stress patterns in the fiber core which are plotted in Fig. 2. Comparing the three parts we find the normal stresses σ_x and σ_y are nearly 100 times more than shearing stress τ_xy and the latter just fluctuates around zero, so the normal stresses bring main contributions to the birefringence in SMFs. In addition, the magnitude of σ_x is about three times as high as σ_y, therefore the normal stress along the direction of pressure P is the greatest stress component in the fiber.

 

Fig. 2. Stress distributions (a): σ_x, (b): σ_y, and (c): τ_xy in the core of the fiber under lateral pressure.

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2. Stress-induced birefringence vector with relation to lateral pressure

Based on above stress tensor components in Eq. (6), we can get the stress-induced perturbation in relative permittivity tensor as [3]

where Δε_r = 2n ₁(C ₁ σ_r+C ₂ σ_θ), Δε_θ = 2n ₁(C ₁ σ_θ+C ₂ σ_r), Δε_z = 2n ₁ C ₂(σ_r+σ_θ), Δε_rθ = 2n ₁(C ₁-C ₂)τ_rθ, and Δε_θz = Δε_zr = 0. The parameters C₁ and C₂ are photoelastic constants for pure silica.

Since the elastic modulus of the fiber is very large, its circular symmetry can not be broken by lateral pressure. And the stress-induced perturbation of relative permittivity is much smaller than the difference of the core and cladding refractive indices, so we can assume the eigen modes (two orthogonal solutions of the wave equation) do not vary and the perturbed electromagnetic fields are represented in terms of their linear combination. Then the coupled equations relating two eigen modes can be described as [1]

where A ₁ and A ₂ are the field amplitudes for two eigen modes which are only dependent on the fiber axis coordinate z.

Our fiber parameters are given as n1 = 1.449, Δ=n1-n2 = 0.005, v = 2.072, u = 1.552, and w = 1.373. By solving the simultaneous equations (6), (8) and (9) with the benefit of the computational method described in [2] we obtain α ₁₂=α ₂₁ = 0 and the theoretical birefringence

In Eq. (10) the stress-optic coefficient C = C ₁-C ₂ and it is 3.184 × 10⁻¹² m²/N for fused silica [9]. Inserting above values of R, a and C into Eq. (10) we get the birefringence B = 0.2779P rad/m for wavelength λ = 1550nm.

In the case that pressure P squeezes the fiber at different orientation it should be written as a vector P⃗ = PP̂. Correspondingly, we note the birefringence in the vector format B⃗ = BB̂. Concerning the relationship between vectors P⃗ and B⃗, we find when P⃗ rotates around the fiber for angle θ, B⃗ will rotates around a circle for double θ on the Poincaré sphere, which we will prove later experimentally. Since the birefringence vector can be measured by Mueller matrix method (MMM), the orientation angle θ will be determined by

in which B̂ ₀ corresponds to the pressing azimuth at 0°. While the magnitude of the pressure is

3. Experiment and analysis for pressure vector sensing

Based on Eqs. (11) and (12) we can realize the pressure vector sensing. Our experimental setup is shown schematically in Fig. 3, where the right part is the pressure vector generator (PVG) made of a force gauge and a rotatable PZT-based squeezing device. We use a computer-controlled measure system to obtain the Mueller matrix M of the fiber under different pressure.

 

Fig. 3. Experimental setup for pressure vector measurement based on Mueller matrix method.

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The polarization state generator (PSG) produces four kinds of linear independent states of polarization (SOPs) S⃗ ₁, S⃗ ₂, S⃗ ₃ and S⃗ ₄ measured by Polarimeter (IPM5300, Thorlabs) 1 which form a matrix S _in=[S⃗ ₁ S⃗ ₂ S⃗ ₃ S⃗ ₄], while the corresponding output SOPs obtained by Polarimeter 2 form another matrix S _out. Since output and input SOPs are related by S _out=MS _in, we get the Mueller matrix M = S _out S ⁻¹ _in. Actually besides the birefringence our system contains a polarization dependent loss (PDL) so matrix M should include both birefringence and PDL terms M_B and M_D. However the pressure P mainly relates to the birefringence and its matrix M_B. We use the decomposing method described in [10] to obtain this pressure dependent M_B.

here m_ij denotes the component of matrix M. Then the birefringence vector is calculated by

where Tr(M_B) and m^B_ij(i, j = 1,2,3,4) are the trace and components of matrix M_B.

To get the initial value B̂ ₀, recalling the coordinate system in Fig. 1(b), squeezing the fiber with P = 1 N as shown in the figure while measuring the corresponding birefringence we obtain B ₀ = 0.2839 and its unit vector B̂ ₀=[-0.6218,0.4995,-0.6045]^T. Then we change the magnitude of pressure and measure the birefringence while recording the practical pressure values to analyze the error, using Eq. (12) we get the pressure and plot them in Fig. 4(a). By statistics we find the average error between calculated and practical pressure is ~0.17 N.

Furthermore, we rotate the device in Fig. 3 to press the fiber at different orientation determined by geometrical method. Then we calculate birefringence vectors by Eq. (15) and their pressure azimuthes by Eq. (11). Some results are shown in Fig. 4(b), which demonstrates when pressure rotating around the fiber the corresponding birefringence vector rotates around a circle with double speed.

 

Fig. 4. (a) Measurement and calculation of pressure magnitude, (b) birefringence vectors under different pressure orientations.

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To know the validity of our analysis for pressure vector orientation, we compare the calculated and practical (geometrical) azimuthes. Statistical analysis shows the average deviation is ~0.85°. Our experimental results are partly shown in Fig. 5.

 

Fig. 5. Comparison for calculated and practical azimuthes of lateral pressureure.

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4. Conclusions

In summary, based on stress function equation we propose a theoretical method to analyze the pressure stress distribution on the SMF profile, and obtain the analytical expression of the stress function with the stress patterns in the core. The stress-induced birefringence is calculated and applied to realize the pressure vector sensing. We demonstrate the birefringence vector rotates around a circle with double speed as the pressure rotating around the fiber. Experimental results confirm our analysis and the average errors of pressure magnitude and direction are ~0.17 N and ~0.85°. Our conclusions are applicable to high-speed optical communications and to fiber-based pressure sensing.