## Influence of strain and pressure to the effective refractive index of the fundamental mode of hollow-core photonic bandgap fibers

Optics Express, Vol. 18, Issue 13, pp. 14041-14055 (2010)

http://dx.doi.org/10.1364/OE.18.014041

Acrobat PDF (1893 KB)

### Abstract

We investigate the phase sensitivity of the fundamental mode of hollow-core photonic bandgap fibers to strain and acoustic pressure. A theoretical model is constructed to analyze the effect of axial strain and acoustic pressure on the effective refractive index of the fundamental mode. Simulation shows that, for the commercial HC-1550-02 fiber, the contribution of mode-index variation to the overall phase sensitivities to axial strain and acoustic pressure are respectively ~-2% and ~-17%. The calculated normalized phase-sensitivities of the HC-1550-02 fiber to strain and acoustic pressure are respectively 1 ε^{−1} and −331.6 dB re μPa^{−1} without considering mode-index variation, and 0.9797 ε^{−1} and −333.1 dB re μPa^{−1} when mode-index variation is included in the calculation. The latter matches better with the experimentally measured results.

© 2010 OSA

## 1. Introduction

1. C. K. Kirkendall and A. Dandridge, “Overview of high performance fiber-optic sensing,” J. Phys. D **37**(18), R197–R216 (2004). [CrossRef]

2. V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Phase sensitivity to temperature of the fundamental mode in air-guiding photonic-bandgap fibers,” Opt. Express **13**(18), 6669–6684 (2005). [CrossRef] [PubMed]

4. M. Pang and W. Jin, “Detection of acoustic pressure with hollow-core photonic bandgap fiber,” Opt. Express **17**(13), 11088–11097 (2009). [CrossRef] [PubMed]

*L,*the effective index of fundamental mode is

*n*, and the operating light wavelength in vacuum is

_{eff}*λ*, the phase (

*φ*) accumulated by the fundamental mode travelling through this section of HC-PBF is given by Eq. (1).

*NR = 20log|S|*) can be increased by as much as 35dB (

*S*= ~60 times) over the conventional single mode fiber [4

4. M. Pang and W. Jin, “Detection of acoustic pressure with hollow-core photonic bandgap fiber,” Opt. Express **17**(13), 11088–11097 (2009). [CrossRef] [PubMed]

1. C. K. Kirkendall and A. Dandridge, “Overview of high performance fiber-optic sensing,” J. Phys. D **37**(18), R197–R216 (2004). [CrossRef]

4. M. Pang and W. Jin, “Detection of acoustic pressure with hollow-core photonic bandgap fiber,” Opt. Express **17**(13), 11088–11097 (2009). [CrossRef] [PubMed]

8. A. Cucnotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. **14**(11), 1530–1532 (2002). [CrossRef]

9. J. Ju, W. Jin, and M. S. Demokan, “Properties of a Highly Birefringent Photonic Crystal Fiber,” IEEE Photon. Technol. Lett. **15**(10), 1375–1377 (2003). [CrossRef]

## 2. Theoretical Modeling

### 2.1 Elasticity model of HC-PBF

**17**(13), 11088–11097 (2009). [CrossRef] [PubMed]

*a*; region 1 is the microstructure cladding (honeycomb cladding) with thickness of

*b-a*; region 2 is the silica outer-cladding with thickness of

*c-b*; and finally they are coated with a jacket region to the radius of

*d*.

*E*and Poisson’s ratio

*v*. The materials of silica outer-cladding and jacket regions are homogeneous, and their Young’s modulus and Poisson’s ratios are

*E*,

_{2}*E*, and

_{3}*ν*,

_{2}*ν*respectively. The microstructure cladding of the HC-PBF is an exception in that it is not a homogeneous material but it behaves mechanically like a honeycomb [5,6

_{3}6. R. M. Christensen, “Mechanics of cellular and other low-density materials,” Int. J. Solids Struct. **37**(1-2), 93–104 (2000). [CrossRef]

*η*. For a hexagonal pattern of air holes in silica, they are given by [5]:where

*E*and

_{0}*ν*are the Young’ modulus and Poisson’s ratio of the silica material;

_{1}*E*,

_{r}*E*,

_{θ}*E*are the effective Young’s modulus of the silica-air honeycomb in the three directions, and

_{z}*ν*,

_{r-θ}*ν*,

_{θ-r}*ν*,

_{z-θ}*ν*,

_{θ-z}*ν*, and

_{r-z}*ν*are the six Poisson’s ratio items of the honeycomb.

_{z-r}*A*,

_{i}*C*, and

_{i}*D*and

_{i}(i = 1,2*3)*are constants. Using Hooke’s law, we obtain the strain tensors of different layers shown as:

*A*,

_{i}*C*, and

_{i}*D*can be determined. Hence the stress, strain and displacement distributions of each layer of the HC-PBF can be obtained. We here are interested in the stress, strain and displacement of the microstructure region, i.e.,

_{i}*σ*,

^{1}_{r}*σ*,

^{1}_{θ}*σ*,

^{1}_{z}*ε*,

^{1}_{r}*ε*,

^{1}_{θ}*ε*, and

^{1}_{z}*u*, from which the field and effective refractive index variations of the fundamental mode may be evaluated by following the procedure as outlined in the next section.

^{1}_{r}### 2.2 Change of microstructure cladding profile

*r*,

*θ*), as shown in Fig. 2 (1), the in-profile stresses acting on this cell may be written as

*σ*and

^{1}_{r}*σ*, as shown in Fig. 2 (2) and Fig. 2 (3) .

^{1}_{θ}*σ*may be decomposed into three components: the

^{1}_{r}*x*- and

*y*-directional stresses

*σ*and

^{1}_{r}|_{x}*σ*, and the shear

^{1}_{r}|_{y}*τ*, which are respectively given by [11

^{1}_{r}|_{xy}11. Y. L. Su, Y. Q. Wang, Z. G. Zhao, and Y. L. Kang, *Mechanics of Materials* (Tianjin Univ. Press, China 2001). [PubMed]

*σ*is the sum of the deformations under loads

^{1}_{r}*σ*,

^{1}_{r}|_{x}*σ*and

^{1}_{r}|_{y}*τ*as shown in Fig. 3, and the deformations of the cell walls (1~6) due to each of the load components are calculated respectively as follows [5]:

^{1}_{r}|_{xy}#### 2.2.1 Deformation due to *σ*^{1}_{r}|_{x}

^{1}

_{r}|

_{x}

*σ*deforms in a linear-elastic way, the cell walls 1 and 4 keep straight and the walls 2, 3, 5 and 6 bend by following a similar pattern. In the new set of coordinates {

^{1}_{r}|_{x}*x’, y’*}, the moment of the wall 2 may be written as:where

*W = (√3/2)elσ*is the length of the cell wall and

^{1}_{r}|_{x}, l*e*is unit depth in

*z*-direction. The subscript “

*x*” means the moment is caused by

*σ*. Then the deformation of cell wall 2 along the

^{1}_{r}|_{x}*y’*direction as the function of

*x’*can then be obtained by double integration of Eq. (10) with respective to

*x*’ and with proper boundary conditions and expressed aswhere

*I = et*,

^{3}/12*t*is the thickness of the wall. The subscripts “

*y’, x*” means the displacement is along

*y’*-direction and is caused by

*σ*.

^{1}_{r}|_{x}#### 2.2.2 Deformation due to *σ*^{1}_{r}|_{y}

^{1}

_{r}|

_{y}

*σ*deforms in a linear-elastic way, the cell walls 1 and 4 keep straight and walls 2, 3, 5 and 6 bend. In the new set of coordinates {

^{1}_{r}|_{y}*x’, y’}*, the moment of the wall 2 can be written as:where

*P = (3/2)elσ*. The subscript “

^{1}_{r}|_{y}*y*” means the moment is caused by

*σ*. The deformation of cell wall 2 as the function of

^{1}_{r}|_{y}*x’*can be obtained from Eq. (12) and expressed as Eq. (13).

*y’, y”*means the deformation is along

*y’*-direction and is caused by

*σ*

^{1}_{r}|_{y}_{.}

#### 2.2.3 Deformation due to *τ*^{1}_{r}|_{xy}

^{1}

_{r}|

_{xy}

*τ*is shown in Fig. 3(3). According to knowledge of cellular solids [5], when the honeycomb cell is sheared, all of cell walls bend, and walls 2, 3, 5, 6 will rotate compared with their original positions. The angle of rotation (

^{1}_{r}|_{xy}*ΔΦ*) can be written as:where

*F = √3leτ*. Because of symmetry, the cell walls 4 and 1, 5 and 2, 3 and 6 have similar bending patterns respectively. In their own new set of coordinates, the moments of walls 1, 2 and 6 can be written as:

^{1}_{r}|_{xy}*y’*-direction as the function of

*x’*can be calculated from Eq. (15) and expressed as Eq. (16). The subscripts

*“y’, τ*means the deformation is along

_{i}”*y’*-direction and is caused by the shear

*τ*acting on the “

^{1}_{r}|_{xy}*i*” wall. It should be emphasized that the coordinates {

^{th}*x’*,

*y’*} for different walls are different and the deformation for a particular wall is expressed in the coordinates of that wall.

*σ*. The deformation of the cells due to

^{1}_{r}*σ*can be obtained by following a similar process. The total deformation is the sum of deformations due to

^{1}_{θ}*σ*and

^{1}_{r}*σ*. By using Eqs. (6) and (8), we can then plot the new transverse profile of the deformed fiber. For example, Fig. 4 shows the transverse profile of the HC-PBF when it is subjected to axial straining of

^{1}_{θ}*ε*0.3. In this Figure, the black pattern shows the deformed profile, while the red one shows the original profile of the HC-PBF. In the calculation, the following fiber parameters are used:

^{1}_{z}=*Pitch = 3.8μm*, air-filling ratio

*η*= 94%,

*a*= 5.45μm,

*b*= 35μm and

*c*= 60μm. All the cladding cells, except the ones in the innermost ring, are regarded as ideal honeycomb hexagons as shown in Fig. 2. For the 12 cells near the core surround, 6 of them keep the original hexagonal shape while the other 6 are modified with the two sides of the hexagons are replaced by a straight line that connects the two corners of the original hexagons [12

12. K. Saitoh, N. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express **12**(3), 394–400 (2004). [CrossRef] [PubMed]

*r = a + u*. The strain and displacement of the core surround was calculated directly from Eqs. (5), (6), (7) and (8); while Eqs. (11), (13), (14) and (16) are also needed to obtain the deformations of the cell walls.

_{r}^{1}|_{r = a}*y’*-axis) and equal to the strain along the center-line of the wall (i.e., along

*x’*-axis). The walls that form the core-surround are treated in the same way as the walls of a uniform hexagon in the air-silica cladding. The strain-optic effect appears as a change in the optical indicatrix of the silica material [13

13. C. D. Butter and G. B. Hocker, “Fiber optics strain gauge,” Appl. Opt. **17**(18), 2867–2869 (1978). [CrossRef] [PubMed]

*p*is the strain-optic tensor, the subscripts are in the standard contracted notation, and

_{ij}*n*is the original reflection index of the material. Silica material is a homogeneous isotropic medium and has only two numerical values

_{0}*p*and

_{11}*p*, thus the changes in the optical indices in the

_{12}*r*,

*θ*and

*z*directions can be written as Eq. (18).

8. A. Cucnotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. **14**(11), 1530–1532 (2002). [CrossRef]

9. J. Ju, W. Jin, and M. S. Demokan, “Properties of a Highly Birefringent Photonic Crystal Fiber,” IEEE Photon. Technol. Lett. **15**(10), 1375–1377 (2003). [CrossRef]

### 2.3 Predictions from the model

14. Crystal Fiber website, http://www.nktphotonics.com/

10. J. D. Shephard, P. J. Roberts, J. D. C. Jones, J. C. Knight, and D. P. Hand, “Measuring Beam Quality of Hollow Core Photonic Crystal Fibers,” J. Lightwave Technol. **24**(10), 3761–3769 (2006). [CrossRef]

*(dn*; 2) the contribution of the fiber material index modification

_{eff}/dX)_{D}*(dn*, as expressed in Eq. (19). Importing the model discussed in Section 2.1 and 2.2 into the finite element solver [8

_{eff}/dX)_{N}8. A. Cucnotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. **14**(11), 1530–1532 (2002). [CrossRef]

9. J. Ju, W. Jin, and M. S. Demokan, “Properties of a Highly Birefringent Photonic Crystal Fiber,” IEEE Photon. Technol. Lett. **15**(10), 1375–1377 (2003). [CrossRef]

*n*due to geometry deformation, material index, and their combination as the functions of the axial strain and acoustic pressure can be obtained. The results are shown respectively in Figs. 6 (1) and 6 (2).

_{eff}*(dn*,

_{eff}/ε)_{D}*(dn*, and

_{eff}/ε)_{N}*dn*are calculated to be −0.0039 ε

_{eff}/ε,^{−1}and −0.0162 ε

^{−1}, and −0.0202 ε

^{−1}, respectively. Obviously the material index contribution is dominant and

*(dn*is ~4 times larger than

_{eff}/ε)_{N}*(dn*. This is because the fiber’s in-profile deformation due to axial strain is considerably smaller than the axial deformation. Figure 7 (1) shows the calculated

_{eff}/ε)_{D}*ε*of the HC-PBF as the function of

_{r}^{1}*r*, when the fiber is subjected to axial straining of 1με. The material index change is mainly due to the relatively large strain along the axial direction. Substituting

*dn*into Eq. (3),

_{eff}/ε*S*,

_{L}*S*, and

_{n}*S*of HC-1550-02 may be estimated to be 1 ε

^{−1}, −0.0203 ε

^{−1}, 0.9797 ε

^{−1}, respectively.

*(dn*,

_{eff}/dP)_{D}*(dn*and

_{eff}/dP)_{N}*dn*, are estimated to be 3.97*10

_{eff}/dP^{−12}Pa

^{−1}, 6.66*10

^{−13}Pa

^{−1}and 4.64*10

^{−12}Pa

^{−1}, respectively. The geometry deformation contribution

*(dn*is dominant and is ~6 times larger than material index contribution

_{eff}/dP)_{D}*(dn*. This is because that the acoustic pressure is applied to both axial and radial directions [4

_{eff}/dP)_{N}**17**(13), 11088–11097 (2009). [CrossRef] [PubMed]

*ε*) increases as

_{r}^{1}*r*decreases and reaches its maximum when approaching to the core-cladding interface. Figure 7(2) shows

*ε*as function of

_{r}^{1}*r*acoustic pressure of P = 10

^{3}Pa. The large deformation near the core region of air-silica cladding makes the fiber have a high

*(dn*. And as discussed in Section 2.2, this large deformation in air-silica cladding mainly comes from the bending of the honeycomb walls, but not the silica material strain.

_{eff}/dP)_{D}*S*,

*S*and

_{L}*S*can be calculated from Eq. (2) to be −2.216*10

_{n}^{−11}Pa

^{−1}, −2.68*10

^{−11}Pa

^{−1}and 4.64*10

^{−12}Pa

^{−1}, respectively. The predicted

*NR*(20log|

*S*|) of HC-1550-02 PBF to acoustic pressure, is then ~-333.1dB re 1μPa

^{−1}.

*S*and

_{L}*S*for a HC-PBF with the same outer diameter (including jacket), core size and air-filling ratio of 70 - 94% but with the thickness of the outer silica cladding

_{n}*(c-b)*increased from 1 to 30 μm. The results are shown in Fig. 8 (1) and 8 (2) respectively. The results show that the NR of HC-PBF to acoustic pressure can be improved by optimizing the thickness of the fiber’s silica cladding and the air-filling ratio; however the relative contribution of the refractive index, as compared to physical change of the fiber length,

*S*/

_{n}*S*remains being around −15%. Figure 9 shows the NRs as functions of the outer silica cladding thickness

_{L}*(c-b)*with/without considering the effective index contribution (

*S*), when the air-filling ratio is fixed at 94%. The calculated NR with

_{n}*S*considered is ~1.1 – 1.8 dB lower than without considering

_{n}*S*when the thickness of the outer silica-cladding

_{n}*(c-b)*is varied from 1 to 30 μm.

*E*,

_{3}*ν*) of the HC-PBF’s polymer jacket material is uncertain because of the polymer material’s flexible ingredient and fabrication process, and those properties may change with the environment. However, as mentioned in Ref. 4

_{3}**17**(13), 11088–11097 (2009). [CrossRef] [PubMed]

*E*is varied from 0.5 to 1 MPa and

_{3}*ν*from 0.35 to 0.37, calculation shows that the NR of HC-1550-02 is changed by only ~0.4 dB.

_{3}14. Crystal Fiber website, http://www.nktphotonics.com/

*dn*) is ~-0.314 ε

_{eff}/ε^{−1}. By using Eq. (3), the stain-sensitivity (

*S*,

*S*and

_{L}*S*) are calculated to be 0.7773 ε

_{n}^{−1}, 1 ε

^{−1}and −0.2227 ε

^{−1}, respectively. This is in good agreement with experimental result as will be presented in section 3.

## 3. Experiment and results

*L*. To measure phase sensitivity (

*S*), one end of the rail was fixed, while the other end was computer controlled to move a known distance (

*ΔL*). The output power variation of the interferometer and the moving distance of the test-rail were simultaneously recorded by the computer. The axial strain acting on the sensing fiber can be calculated by using

*ε = ΔL/L*. The phase sensitivity was then calculated from the measured number of fringes (

*N*) for a given

_{f}*ΔL*.where the term (

*1/2*) in Eq. (20) is because the measurement was taken with a Michelson interferometer, in which the light goes through the sensing fiber twice. The experimental results agree well with the predicted values as summarized in Table 3 . For sensitivity measurements of HC-1550-02 and NL-3.3 fibers, 258 and 129 fringes were respectively recorded with experimental inaccuracy of smaller than ± 1 fringes, corresponding to inaccuracy in

*S*of ~ ± 0.004 for HC-1550-02 and ~ ± 0.006 for NL-3.3.

**17**(13), 11088–11097 (2009). [CrossRef] [PubMed]

*NR =*−334.4 dB re 1μPa

^{−1}is in good agreement with the theoretically result of −333.1dB re 1μPa

^{−1}. The prediction from the model with effective index contribution is closer to the experimental value than the result without considering the index term (−331.6dB re 1μPa

^{−1}) [4

**17**(13), 11088–11097 (2009). [CrossRef] [PubMed]

## 4. Conclusion

^{−1}and −2.216*10

^{−11}Pa

^{−1}(−334.4dB re 1μPa

^{−1}). These values agree well with our theoretical predictions. The theoretical model is also applicable to other hollow-core or solid-core photonic crystal fibers with high air-filling ratios.

## Acknowledgement

## References and links

1. | C. K. Kirkendall and A. Dandridge, “Overview of high performance fiber-optic sensing,” J. Phys. D |

2. | V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Phase sensitivity to temperature of the fundamental mode in air-guiding photonic-bandgap fibers,” Opt. Express |

3. | H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Air-core photonic-bandgap fiber-optic Gyroscope,” J. Lightwave Technol. |

4. | M. Pang and W. Jin, “Detection of acoustic pressure with hollow-core photonic bandgap fiber,” Opt. Express |

5. | L. J. Gibson, and M. F. Ashby, |

6. | R. M. Christensen, “Mechanics of cellular and other low-density materials,” Int. J. Solids Struct. |

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

8. | A. Cucnotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. |

9. | J. Ju, W. Jin, and M. S. Demokan, “Properties of a Highly Birefringent Photonic Crystal Fiber,” IEEE Photon. Technol. Lett. |

10. | J. D. Shephard, P. J. Roberts, J. D. C. Jones, J. C. Knight, and D. P. Hand, “Measuring Beam Quality of Hollow Core Photonic Crystal Fibers,” J. Lightwave Technol. |

11. | Y. L. Su, Y. Q. Wang, Z. G. Zhao, and Y. L. Kang, |

12. | K. Saitoh, N. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express |

13. | C. D. Butter and G. B. Hocker, “Fiber optics strain gauge,” Appl. Opt. |

14. | Crystal Fiber website, http://www.nktphotonics.com/ |

**OCIS Codes**

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

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(060.5295) Fiber optics and optical communications : Photonic crystal fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: March 12, 2010

Revised Manuscript: June 4, 2010

Manuscript Accepted: June 4, 2010

Published: June 15, 2010

**Citation**

M. Pang, H. F. Xuan, J. Ju, and W. Jin, "Influence of strain and pressure to the effective refractive index of the fundamental mode of hollow-core photonic bandgap fibers," Opt. Express **18**, 14041-14055 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-14041

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### References

- C. K. Kirkendall and A. Dandridge, “Overview of high performance fiber-optic sensing,” J. Phys. D 37(18), R197–R216 (2004). [CrossRef]
- V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Phase sensitivity to temperature of the fundamental mode in air-guiding photonic-bandgap fibers,” Opt. Express 13(18), 6669–6684 (2005). [CrossRef] [PubMed]
- H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Air-core photonic-bandgap fiber-optic Gyroscope,” J. Lightwave Technol. 24(8), 3169–3174 (2006). [CrossRef]
- M. Pang and W. Jin, “Detection of acoustic pressure with hollow-core photonic bandgap fiber,” Opt. Express 17(13), 11088–11097 (2009). [CrossRef] [PubMed]
- L. J. Gibson, and M. F. Ashby, Cellular solids: structure and properties, second edition, (Cambridge University Press, New York 1997).
- R. M. Christensen, “Mechanics of cellular and other low-density materials,” Int. J. Solids Struct. 37(1-2), 93–104 (2000). [CrossRef]
- S. P. Timoshenko and J. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970).
- A. Cucnotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. 14(11), 1530–1532 (2002). [CrossRef]
- J. Ju, W. Jin, and M. S. Demokan, “Properties of a Highly Birefringent Photonic Crystal Fiber,” IEEE Photon. Technol. Lett. 15(10), 1375–1377 (2003). [CrossRef]
- J. D. Shephard, P. J. Roberts, J. D. C. Jones, J. C. Knight, and D. P. Hand, “Measuring Beam Quality of Hollow Core Photonic Crystal Fibers,” J. Lightwave Technol. 24(10), 3761–3769 (2006). [CrossRef]
- Y. L. Su, Y. Q. Wang, Z. G. Zhao, and Y. L. Kang, Mechanics of Materials (Tianjin Univ. Press, China 2001). [PubMed]
- K. Saitoh, N. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express 12(3), 394–400 (2004). [CrossRef] [PubMed]
- C. D. Butter and G. B. Hocker, “Fiber optics strain gauge,” Appl. Opt. 17(18), 2867–2869 (1978). [CrossRef] [PubMed]
- Crystal Fiber website, http://www.nktphotonics.com/

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