## Stress distribution and induced birefringence analysis for pressure vector sensing based on single mode fibers

Optics Express, Vol. 16, Issue 6, pp. 3955-3960 (2008)

http://dx.doi.org/10.1364/OE.16.003955

Acrobat PDF (638 KB)

### Abstract

We propose a theoretical approach to analyze the pressure stress distribution in single mode fibers (SMFs) and achieve the analytical expression of stress function, from which we obtain the stress components with their patterns in the core and compute their induced birefringence. Then we perform a pressure vector sensing based on ~2 km SMF. Using Mueller matrix method we measure the birefringence vectors which are employed to compute the pressure magnitudes and their orientation. When rotating the pressure around the fiber, the corresponding birefringence vector rotates around a circle with double speed. Statistics show the average deviation of calculated pressure-magnitude to practical value is ~0.17 *N* and it is ~0.85° for orientation.

© 2008 Optical Society of America

## Introduction

1. D. Chowdhury and D. Wilcox, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Sel. Topics Quantum Electron. **6**, 227–232 (2000). [CrossRef]

1. D. Chowdhury and D. Wilcox, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Sel. Topics Quantum Electron. **6**, 227–232 (2000). [CrossRef]

2. J. Sakai and T. Kimura, “Birefringence and polarization characteristics of single mode optical fibers under electric deformations,” IEEE J. Quantum Electron. , **17**, 1041–1051 (1981). [CrossRef]

3. Y. Park, U. C. Paek, and D.Y. Kim, “Determination of stress-induced intrinsic birefringence in a single-mode fiber by measurement of the two-dimensional stress profile,” Opt. Lett. , **27**, 1291–1293 (2002). [CrossRef]

4. K.S. Chang, “Pressure-induced birefringence in a coated highly birefringent optical fiber,” J. Lightwave Technol. , **8**, 1850–1855 (1990). [CrossRef]

3. Y. Park, U. C. Paek, and D.Y. Kim, “Determination of stress-induced intrinsic birefringence in a single-mode fiber by measurement of the two-dimensional stress profile,” Opt. Lett. , **27**, 1291–1293 (2002). [CrossRef]

5. S. M. Pietralunga, M. Ferrario, M. Tacca, and M. Martinelli, “Local Birefringence in Unidirectionally Spun Fibers,” J. Lightwave Technol. , **24**, 4030–4038 (2006). [CrossRef]

6. K. Saitoh, M. Koshiba, and Y. Tsuji, “Stress analysis method for elastically anisotropic material based optical waveguides and its application to strain-induced optical waveguides,” J. Lightwave Technol. , **17**, 255–259 (1999). [CrossRef]

## 1. Pressure stress distribution in SMFs

*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.

*A*(

*r*,

*θ*) and the stress tensor components in the fiber are described by [7]

*μ*(

*φ*)=

*A*= Σ

_{Q}*M⃗*,

_{i}*Q*is an arbitrary point on the circle boundary,

*n*is the normal of the integration boundary,

*φ*is the integration variable, while

*F⃗*and

_{i}*M⃗*are respectively a force element existing in the integration interval (

_{i}*Q*[7]. By simple calculation we get

*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

*R*= 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*and

_{x}*σ*are nearly 100 times more than shearing stress

_{y}*τ*and the latter just fluctuates around zero, so the normal stresses bring main contributions to the birefringence in SMFs. In addition, the magnitude of

_{xy}*σ*is about three times as high as

_{x}*σ*, therefore the normal stress along the direction of pressure

_{y}*P*is the greatest stress component in the fiber.

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

3. Y. Park, U. C. Paek, and D.Y. Kim, “Determination of stress-induced intrinsic birefringence in a single-mode fiber by measurement of the two-dimensional stress profile,” Opt. Lett. , **27**, 1291–1293 (2002). [CrossRef]

*ε*= 2

_{r}*n*

_{1}(

*C*

_{1}

*σ*+

_{r}*C*

_{2}

*σ*), Δ

_{θ}*ε*= 2

_{θ}*n*

_{1}(

*C*

_{1}

*σ*+

_{θ}*C*

_{2}

*σ*), Δ

_{r}*ε*= 2

_{z}*n*

_{1}

*C*

_{2}(

*σ*+

_{r}*σ*), Δ

_{θ}*ε*= 2

_{rθ}*n*

_{1}(

*C*

_{1}-

*C*

_{2})

*τ*, and Δ

_{rθ}*ε*= Δ

_{θz}*ε*= 0. The parameters C

_{zr}_{1}and C

_{2}are photoelastic constants for pure silica.

1. D. Chowdhury and D. Wilcox, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Sel. Topics Quantum Electron. **6**, 227–232 (2000). [CrossRef]

*A*

_{1}and

*A*

_{2}are the field amplitudes for two eigen modes which are only dependent on the fiber axis coordinate

*z*.

2. J. Sakai and T. Kimura, “Birefringence and polarization characteristics of single mode optical fibers under electric deformations,” IEEE J. Quantum Electron. , **17**, 1041–1051 (1981). [CrossRef]

*α*

_{12}=

*α*

_{21}= 0 and the theoretical birefringence

*C*=

*C*

_{1}-

*C*

_{2}and it is 3.184 × 10

^{−12}m

^{2}/N for fused silica [9

9. A. J. Barlow and D. N. Payne, “The stress-optic effect in optical fibers,” IEEE J. Quantum Electron. , **19**, 834–839 (1983). [CrossRef]

*R*,

*a*and

*C*into Eq. (10) we get the birefringence

*B*= 0.2779

*P*rad/m for wavelength λ = 1550nm.

*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

*B̂*

_{0}corresponds to the pressing azimuth at 0°. While the magnitude of the pressure is

## 3. Experiment and analysis for pressure vector sensing

*M*of the fiber under different pressure.

*m*denotes the component of matrix

_{ij}*M*. Then the birefringence vector is calculated by

*Tr*(

*M*) and

_{B}*m*(

^{B}_{ij}*i*,

*j*= 1,2,3,4) are the trace and components of matrix

*M*.

_{B}*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]

*. 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.*

^{T}## 4. Conclusions

## Acknowledgments

## References and links

1. | D. Chowdhury and D. Wilcox, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Sel. Topics Quantum Electron. |

2. | J. Sakai and T. Kimura, “Birefringence and polarization characteristics of single mode optical fibers under electric deformations,” IEEE J. Quantum Electron. , |

3. | Y. Park, U. C. Paek, and D.Y. Kim, “Determination of stress-induced intrinsic birefringence in a single-mode fiber by measurement of the two-dimensional stress profile,” Opt. Lett. , |

4. | K.S. Chang, “Pressure-induced birefringence in a coated highly birefringent optical fiber,” J. Lightwave Technol. , |

5. | S. M. Pietralunga, M. Ferrario, M. Tacca, and M. Martinelli, “Local Birefringence in Unidirectionally Spun Fibers,” J. Lightwave Technol. , |

6. | K. Saitoh, M. Koshiba, and Y. Tsuji, “Stress analysis method for elastically anisotropic material based optical waveguides and its application to strain-induced optical waveguides,” J. Lightwave Technol. , |

7. | S. Timoshenko and J. N. Goodier, |

8. | D.H. Yu, |

9. | A. J. Barlow and D. N. Payne, “The stress-optic effect in optical fibers,” IEEE J. Quantum Electron. , |

10. | S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A |

**OCIS Codes**

(060.2310) Fiber optics and optical communications : Fiber optics

(060.2330) Fiber optics and optical communications : Fiber optics communications

(060.2370) Fiber optics and optical communications : Fiber optics sensors

(260.1440) Physical optics : Birefringence

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: January 29, 2008

Revised Manuscript: March 5, 2008

Manuscript Accepted: March 5, 2008

Published: March 10, 2008

**Citation**

Zhengyong Li, Chongqing Wu, Hui Dong, P. Shum, C. Y. Tian, and S. Zhao, "Stress distribution and induced birefringence analysis for pressure vector sensing based on single mode fibers," Opt. Express **16**, 3955-3960 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-3955

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### References

- D. Chowdhury and D. Wilcox, "Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation," IEEE J. Sel. Topics Quantum Electron. 6, 227-232 (2000). [CrossRef]
- J. Sakai and T. Kimura, "Birefringence and polarization characteristics of single mode optical fibers under electric deformations," IEEE J. Quantum Electron. 17, 1041-1051 (1981). [CrossRef]
- Y. Park, U. C. Paek, and D.Y. Kim, "Determination of stress-induced intrinsic birefringence in a single-mode fiber by measurement of the two-dimensional stress profile," Opt. Lett. 27, 1291-1293 (2002). [CrossRef]
- K.S. Chang, "Pressure-induced birefringence in a coated highly birefringent optical fiber," J. Lightwave Technol. 8, 1850-1855 (1990). [CrossRef]
- S. M. Pietralunga, M. Ferrario, M. Tacca, and M. Martinelli, "Local Birefringence in Unidirectionally Spun Fibers," J. Lightwave Technol. 24, 4030-4038 (2006). [CrossRef]
- K. Saitoh, M. Koshiba, and Y. Tsuji, "Stress analysis method for elastically anisotropic material based optical waveguides and its application to strain-induced optical waveguides," J. Lightwave Technol. 17, 255-259 (1999). [CrossRef]
- S. Timoshenko and J. N. Goodier, Theory of elasticity (McGraw-Hill, 1970), Chap. 4.
- D.H. Yu, Mathematical theory of natural boundary element method (Science Press, Beijing, 1993), Chap. 3.
- A. J. Barlow and D. N. Payne, "The stress-optic effect in optical fibers," IEEE J. Quantum Electron. 19, 834-839 (1983). [CrossRef]
- S. Y. Lu and R. A. Chipman, "Interpretation of Mueller matrices based on polar decomposition," J. Opt. Soc. Am. A 13, 1106-1113 (1996). [CrossRef]

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