Abstract

When an optical fiber is dipped in an etching solution, the internal stress profile in the fiber varies with the fiber diameter. We observed a physical contraction as much as 0.2% in the fiber axial dimension when the fiber was reduced from its original diameter to ~6 µm through analysis using high resolution microscope images of the grating period of an etched FBG at different fiber diameters. This axial contraction is related to the varying axial stress profile in the fiber when the fiber diameter is reduced. On top of that, the refractive index of fiber core increases with reducing fiber diameter due to stress-optic effect. The calculated index increment is as much as 1.8 × 10−3 at the center of fiber core after the diameter is reduced down to ~6 µm. In comparison with the conventional model that assumes constant grating period and neglects the variation in stress-induced index change in fiber core, our proposed model indicates a discrepancy as much as 3nm in Bragg wavelength at a fiber diameter of ~6 µm.

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References

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  1. A. Gillooly, “Photosensitive fibers: Growing gratings,” Nat. Photonics5(8), 468–469 (2011).
    [CrossRef]
  2. P. Y. Fonjallaz, H. G. Limberger, R. P. Salathé, F. Cochet, and B. Leuenberger, “Tension increase correlated to refractive-index change in fibers containing UV-written Bragg gratings,” Opt. Lett.20(11), 1346–1348 (1995).
    [CrossRef] [PubMed]
  3. N. H. Ky, H. G. Limberger, R. P. Salathé, and F. Cochet, “Effects of drawing tension on the photosensitivity of Sn-Ge- and B-Ge-codoped core fibers,” Opt. Lett.23(17), 1402–1404 (1998).
    [CrossRef] [PubMed]
  4. K. Tsujikawa, M. Ohashi, K. Shiraki, and M. Tateda, “Scattering property of F and GeO2 codoped silica glasses,” Electron. Lett.30(4), 351–352 (1994).
    [CrossRef]
  5. M. G. Sceats, G. R. Atkins, and S. B. Poole, “Photolytic index changes in optical fibers,” Annu. Rev. Mater. Sci.23(1), 381–410 (1993).
    [CrossRef]
  6. P. K. Bachmann, W. Hermann, H. Wehr, and D. U. Wiechert, “Stress in optical waveguides. 2: Fibers,” Appl. Opt.26(7), 1175–1182 (1987).
    [CrossRef] [PubMed]
  7. A. Barlow and D. Payne, “The stress-optic effect in optical fibers,” IEEE J. Quantum Electron.19(5), 834–839 (1983).
    [CrossRef]
  8. J. Sakai and T. Kimura, “Birefringence caused by thermal stress in elliptically deformed core optical fibers,” IEEE J. Quantum Electron.18(11), 1899–1909 (1982).
    [CrossRef]
  9. Y. Park, U.-C. Paek, and D. Y. Kim, “Complete determination of the stress tensor of a polarization-maintaining fiber by photoelastic tomography,” Opt. Lett.27(14), 1217–1219 (2002).
    [CrossRef] [PubMed]
  10. P. Chu and R. Sammut, “Analytical method for calculation of stresses and material birefringence in polarization-maintaining optical fiber,” J. Lightwave Technol.2(5), 650–662 (1984).
    [CrossRef]
  11. I. H. Shin, B. H. Kim, S. P. Veetil, W.-T. Han, and D. Y. Kim, “Residual stress relaxation in cleaved fibers,” Opt. Commun.281(1), 75–79 (2008).
    [CrossRef]
  12. Y. Mohanna, J. M. Saugrain, J. C. Rousseau, and P. Ledoux, “Relaxation of internal stresses in optical fibers,” J. Lightwave Technol.8(12), 1799–1802 (1990).
    [CrossRef]
  13. B. H. Kim, Y. Park, T. J. Ahn, D. Y. Kim, B. H. Lee, Y. Chung, U. C. Paek, and W. T. Han, “Residual stress relaxation in the core of optical fiber by CO2 laser irradiation,” Opt. Lett.26(21), 1657–1659 (2001).
    [CrossRef] [PubMed]
  14. N. H. Ky, H. G. Limberger, R. P. Salathe, F. Cochet, and L. Dong, “Hydrogen-induced reduction of axial stress in optical fiber cores,” Appl. Phys. Lett.74(4), 516–518 (1999).
    [CrossRef]
  15. N. Shibata, M. Tateda, and S. Seikai, “Polarization mode dispersion measurement in elliptical core single-mode fibers by a spatial technique,” IEEE J. Quantum Electron.18(1), 53–58 (1982).
    [CrossRef]
  16. I. Abe, O. Frazão, M. W. Schiller, R. N. Nogueira, H. J. Kalinowski, and J. L. Pinto, “Bragg gratings in normal and reduced diameter high birefringence fiber optics,” Meas. Sci. Technol.17(6), 1477–1484 (2006).
    [CrossRef]
  17. H. K. Bal, Z. Brodzeli, N. M. Dragomir, S. F. Collins, and F. Sidiroglou, “Uniformly thinned optical fibers produced via HF etching with spectral and microscopic verification,” Appl. Opt.51(13), 2282–2287 (2012).
    [CrossRef] [PubMed]
  18. N. M. Dragomir, C. Rollinson, S. A. Wade, A. J. Stevenson, S. F. Collins, G. W. Baxter, P. M. Farrell, and A. Roberts, “Nondestructive imaging of a type I optical fiber Bragg grating,” Opt. Lett.28(10), 789–791 (2003).
    [CrossRef] [PubMed]
  19. B. P. Kouskousis, C. M. Rollinson, D. J. Kitcher, S. F. Collins, G. W. Baxter, S. A. Wade, N. M. Dragomir, and A. Roberts, “Quantitative investigation of the refractive-index modulation within the core of a fiber Bragg grating,” Opt. Express14(22), 10332–10338 (2006).
    [CrossRef] [PubMed]
  20. F. Kherbouche and B. Poumellec, “UV-induced stress field during Bragg grating inscription in optical fibers,” J. Opt. A, Pure Appl. Opt.3(6), 429–439 (2001).
    [CrossRef]
  21. A. C. Ugural and S. K. Fenster, “Two dimensional problems in elasticity,” in Advanced Strength and Applied Elasticity (Pearson Education, 2003).
  22. B. Malo, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, “Single-excimer-pulse writing of fiber gratings by use of a zero-order nulled phase mask: grating spectral response and visualization of index perturbations,” Opt. Lett.18(15), 1277–1279 (1993).
    [CrossRef] [PubMed]
  23. A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

2012 (1)

2011 (1)

A. Gillooly, “Photosensitive fibers: Growing gratings,” Nat. Photonics5(8), 468–469 (2011).
[CrossRef]

2008 (1)

I. H. Shin, B. H. Kim, S. P. Veetil, W.-T. Han, and D. Y. Kim, “Residual stress relaxation in cleaved fibers,” Opt. Commun.281(1), 75–79 (2008).
[CrossRef]

2006 (2)

B. P. Kouskousis, C. M. Rollinson, D. J. Kitcher, S. F. Collins, G. W. Baxter, S. A. Wade, N. M. Dragomir, and A. Roberts, “Quantitative investigation of the refractive-index modulation within the core of a fiber Bragg grating,” Opt. Express14(22), 10332–10338 (2006).
[CrossRef] [PubMed]

I. Abe, O. Frazão, M. W. Schiller, R. N. Nogueira, H. J. Kalinowski, and J. L. Pinto, “Bragg gratings in normal and reduced diameter high birefringence fiber optics,” Meas. Sci. Technol.17(6), 1477–1484 (2006).
[CrossRef]

2003 (1)

2002 (1)

2001 (2)

F. Kherbouche and B. Poumellec, “UV-induced stress field during Bragg grating inscription in optical fibers,” J. Opt. A, Pure Appl. Opt.3(6), 429–439 (2001).
[CrossRef]

B. H. Kim, Y. Park, T. J. Ahn, D. Y. Kim, B. H. Lee, Y. Chung, U. C. Paek, and W. T. Han, “Residual stress relaxation in the core of optical fiber by CO2 laser irradiation,” Opt. Lett.26(21), 1657–1659 (2001).
[CrossRef] [PubMed]

1999 (1)

N. H. Ky, H. G. Limberger, R. P. Salathe, F. Cochet, and L. Dong, “Hydrogen-induced reduction of axial stress in optical fiber cores,” Appl. Phys. Lett.74(4), 516–518 (1999).
[CrossRef]

1998 (1)

1995 (1)

1994 (1)

K. Tsujikawa, M. Ohashi, K. Shiraki, and M. Tateda, “Scattering property of F and GeO2 codoped silica glasses,” Electron. Lett.30(4), 351–352 (1994).
[CrossRef]

1993 (2)

1990 (1)

Y. Mohanna, J. M. Saugrain, J. C. Rousseau, and P. Ledoux, “Relaxation of internal stresses in optical fibers,” J. Lightwave Technol.8(12), 1799–1802 (1990).
[CrossRef]

1987 (1)

1984 (1)

P. Chu and R. Sammut, “Analytical method for calculation of stresses and material birefringence in polarization-maintaining optical fiber,” J. Lightwave Technol.2(5), 650–662 (1984).
[CrossRef]

1983 (1)

A. Barlow and D. Payne, “The stress-optic effect in optical fibers,” IEEE J. Quantum Electron.19(5), 834–839 (1983).
[CrossRef]

1982 (2)

J. Sakai and T. Kimura, “Birefringence caused by thermal stress in elliptically deformed core optical fibers,” IEEE J. Quantum Electron.18(11), 1899–1909 (1982).
[CrossRef]

N. Shibata, M. Tateda, and S. Seikai, “Polarization mode dispersion measurement in elliptical core single-mode fibers by a spatial technique,” IEEE J. Quantum Electron.18(1), 53–58 (1982).
[CrossRef]

Abe, I.

I. Abe, O. Frazão, M. W. Schiller, R. N. Nogueira, H. J. Kalinowski, and J. L. Pinto, “Bragg gratings in normal and reduced diameter high birefringence fiber optics,” Meas. Sci. Technol.17(6), 1477–1484 (2006).
[CrossRef]

Ahn, T. J.

Albert, J.

Atkins, G. R.

M. G. Sceats, G. R. Atkins, and S. B. Poole, “Photolytic index changes in optical fibers,” Annu. Rev. Mater. Sci.23(1), 381–410 (1993).
[CrossRef]

Bachmann, P. K.

Bal, H. K.

Barlow, A.

A. Barlow and D. Payne, “The stress-optic effect in optical fibers,” IEEE J. Quantum Electron.19(5), 834–839 (1983).
[CrossRef]

Baxter, G. W.

Bilodeau, F.

Brodzeli, Z.

Chu, P.

P. Chu and R. Sammut, “Analytical method for calculation of stresses and material birefringence in polarization-maintaining optical fiber,” J. Lightwave Technol.2(5), 650–662 (1984).
[CrossRef]

Chung, Y.

Cochet, F.

Collins, S. F.

Dong, L.

N. H. Ky, H. G. Limberger, R. P. Salathe, F. Cochet, and L. Dong, “Hydrogen-induced reduction of axial stress in optical fiber cores,” Appl. Phys. Lett.74(4), 516–518 (1999).
[CrossRef]

Dragomir, N. M.

Farrell, P. M.

Fonjallaz, P. Y.

Frazão, O.

I. Abe, O. Frazão, M. W. Schiller, R. N. Nogueira, H. J. Kalinowski, and J. L. Pinto, “Bragg gratings in normal and reduced diameter high birefringence fiber optics,” Meas. Sci. Technol.17(6), 1477–1484 (2006).
[CrossRef]

Gillooly, A.

A. Gillooly, “Photosensitive fibers: Growing gratings,” Nat. Photonics5(8), 468–469 (2011).
[CrossRef]

Han, W. T.

Han, W.-T.

I. H. Shin, B. H. Kim, S. P. Veetil, W.-T. Han, and D. Y. Kim, “Residual stress relaxation in cleaved fibers,” Opt. Commun.281(1), 75–79 (2008).
[CrossRef]

Hermann, W.

Hill, K. O.

Johnson, D. C.

Kalinowski, H. J.

I. Abe, O. Frazão, M. W. Schiller, R. N. Nogueira, H. J. Kalinowski, and J. L. Pinto, “Bragg gratings in normal and reduced diameter high birefringence fiber optics,” Meas. Sci. Technol.17(6), 1477–1484 (2006).
[CrossRef]

Kherbouche, F.

F. Kherbouche and B. Poumellec, “UV-induced stress field during Bragg grating inscription in optical fibers,” J. Opt. A, Pure Appl. Opt.3(6), 429–439 (2001).
[CrossRef]

Kim, B. H.

Kim, D. Y.

Kimura, T.

J. Sakai and T. Kimura, “Birefringence caused by thermal stress in elliptically deformed core optical fibers,” IEEE J. Quantum Electron.18(11), 1899–1909 (1982).
[CrossRef]

Kitcher, D. J.

Kouskousis, B. P.

Ky, N. H.

N. H. Ky, H. G. Limberger, R. P. Salathe, F. Cochet, and L. Dong, “Hydrogen-induced reduction of axial stress in optical fiber cores,” Appl. Phys. Lett.74(4), 516–518 (1999).
[CrossRef]

N. H. Ky, H. G. Limberger, R. P. Salathé, and F. Cochet, “Effects of drawing tension on the photosensitivity of Sn-Ge- and B-Ge-codoped core fibers,” Opt. Lett.23(17), 1402–1404 (1998).
[CrossRef] [PubMed]

Ledoux, P.

Y. Mohanna, J. M. Saugrain, J. C. Rousseau, and P. Ledoux, “Relaxation of internal stresses in optical fibers,” J. Lightwave Technol.8(12), 1799–1802 (1990).
[CrossRef]

Lee, B. H.

Leuenberger, B.

Limberger, H. G.

Malo, B.

Mohanna, Y.

Y. Mohanna, J. M. Saugrain, J. C. Rousseau, and P. Ledoux, “Relaxation of internal stresses in optical fibers,” J. Lightwave Technol.8(12), 1799–1802 (1990).
[CrossRef]

Nogueira, R. N.

I. Abe, O. Frazão, M. W. Schiller, R. N. Nogueira, H. J. Kalinowski, and J. L. Pinto, “Bragg gratings in normal and reduced diameter high birefringence fiber optics,” Meas. Sci. Technol.17(6), 1477–1484 (2006).
[CrossRef]

Ohashi, M.

K. Tsujikawa, M. Ohashi, K. Shiraki, and M. Tateda, “Scattering property of F and GeO2 codoped silica glasses,” Electron. Lett.30(4), 351–352 (1994).
[CrossRef]

Paek, U. C.

Paek, U.-C.

Park, Y.

Payne, D.

A. Barlow and D. Payne, “The stress-optic effect in optical fibers,” IEEE J. Quantum Electron.19(5), 834–839 (1983).
[CrossRef]

Pinto, J. L.

I. Abe, O. Frazão, M. W. Schiller, R. N. Nogueira, H. J. Kalinowski, and J. L. Pinto, “Bragg gratings in normal and reduced diameter high birefringence fiber optics,” Meas. Sci. Technol.17(6), 1477–1484 (2006).
[CrossRef]

Poole, S. B.

M. G. Sceats, G. R. Atkins, and S. B. Poole, “Photolytic index changes in optical fibers,” Annu. Rev. Mater. Sci.23(1), 381–410 (1993).
[CrossRef]

Poumellec, B.

F. Kherbouche and B. Poumellec, “UV-induced stress field during Bragg grating inscription in optical fibers,” J. Opt. A, Pure Appl. Opt.3(6), 429–439 (2001).
[CrossRef]

Roberts, A.

Rollinson, C.

Rollinson, C. M.

Rousseau, J. C.

Y. Mohanna, J. M. Saugrain, J. C. Rousseau, and P. Ledoux, “Relaxation of internal stresses in optical fibers,” J. Lightwave Technol.8(12), 1799–1802 (1990).
[CrossRef]

Sakai, J.

J. Sakai and T. Kimura, “Birefringence caused by thermal stress in elliptically deformed core optical fibers,” IEEE J. Quantum Electron.18(11), 1899–1909 (1982).
[CrossRef]

Salathe, R. P.

N. H. Ky, H. G. Limberger, R. P. Salathe, F. Cochet, and L. Dong, “Hydrogen-induced reduction of axial stress in optical fiber cores,” Appl. Phys. Lett.74(4), 516–518 (1999).
[CrossRef]

Salathé, R. P.

Sammut, R.

P. Chu and R. Sammut, “Analytical method for calculation of stresses and material birefringence in polarization-maintaining optical fiber,” J. Lightwave Technol.2(5), 650–662 (1984).
[CrossRef]

Saugrain, J. M.

Y. Mohanna, J. M. Saugrain, J. C. Rousseau, and P. Ledoux, “Relaxation of internal stresses in optical fibers,” J. Lightwave Technol.8(12), 1799–1802 (1990).
[CrossRef]

Sceats, M. G.

M. G. Sceats, G. R. Atkins, and S. B. Poole, “Photolytic index changes in optical fibers,” Annu. Rev. Mater. Sci.23(1), 381–410 (1993).
[CrossRef]

Schiller, M. W.

I. Abe, O. Frazão, M. W. Schiller, R. N. Nogueira, H. J. Kalinowski, and J. L. Pinto, “Bragg gratings in normal and reduced diameter high birefringence fiber optics,” Meas. Sci. Technol.17(6), 1477–1484 (2006).
[CrossRef]

Seikai, S.

N. Shibata, M. Tateda, and S. Seikai, “Polarization mode dispersion measurement in elliptical core single-mode fibers by a spatial technique,” IEEE J. Quantum Electron.18(1), 53–58 (1982).
[CrossRef]

Shibata, N.

N. Shibata, M. Tateda, and S. Seikai, “Polarization mode dispersion measurement in elliptical core single-mode fibers by a spatial technique,” IEEE J. Quantum Electron.18(1), 53–58 (1982).
[CrossRef]

Shin, I. H.

I. H. Shin, B. H. Kim, S. P. Veetil, W.-T. Han, and D. Y. Kim, “Residual stress relaxation in cleaved fibers,” Opt. Commun.281(1), 75–79 (2008).
[CrossRef]

Shiraki, K.

K. Tsujikawa, M. Ohashi, K. Shiraki, and M. Tateda, “Scattering property of F and GeO2 codoped silica glasses,” Electron. Lett.30(4), 351–352 (1994).
[CrossRef]

Sidiroglou, F.

Stevenson, A. J.

Tateda, M.

K. Tsujikawa, M. Ohashi, K. Shiraki, and M. Tateda, “Scattering property of F and GeO2 codoped silica glasses,” Electron. Lett.30(4), 351–352 (1994).
[CrossRef]

N. Shibata, M. Tateda, and S. Seikai, “Polarization mode dispersion measurement in elliptical core single-mode fibers by a spatial technique,” IEEE J. Quantum Electron.18(1), 53–58 (1982).
[CrossRef]

Tsujikawa, K.

K. Tsujikawa, M. Ohashi, K. Shiraki, and M. Tateda, “Scattering property of F and GeO2 codoped silica glasses,” Electron. Lett.30(4), 351–352 (1994).
[CrossRef]

Veetil, S. P.

I. H. Shin, B. H. Kim, S. P. Veetil, W.-T. Han, and D. Y. Kim, “Residual stress relaxation in cleaved fibers,” Opt. Commun.281(1), 75–79 (2008).
[CrossRef]

Wade, S. A.

Wehr, H.

Wiechert, D. U.

Annu. Rev. Mater. Sci. (1)

M. G. Sceats, G. R. Atkins, and S. B. Poole, “Photolytic index changes in optical fibers,” Annu. Rev. Mater. Sci.23(1), 381–410 (1993).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

N. H. Ky, H. G. Limberger, R. P. Salathe, F. Cochet, and L. Dong, “Hydrogen-induced reduction of axial stress in optical fiber cores,” Appl. Phys. Lett.74(4), 516–518 (1999).
[CrossRef]

Electron. Lett. (1)

K. Tsujikawa, M. Ohashi, K. Shiraki, and M. Tateda, “Scattering property of F and GeO2 codoped silica glasses,” Electron. Lett.30(4), 351–352 (1994).
[CrossRef]

IEEE J. Quantum Electron. (3)

N. Shibata, M. Tateda, and S. Seikai, “Polarization mode dispersion measurement in elliptical core single-mode fibers by a spatial technique,” IEEE J. Quantum Electron.18(1), 53–58 (1982).
[CrossRef]

A. Barlow and D. Payne, “The stress-optic effect in optical fibers,” IEEE J. Quantum Electron.19(5), 834–839 (1983).
[CrossRef]

J. Sakai and T. Kimura, “Birefringence caused by thermal stress in elliptically deformed core optical fibers,” IEEE J. Quantum Electron.18(11), 1899–1909 (1982).
[CrossRef]

J. Lightwave Technol. (2)

P. Chu and R. Sammut, “Analytical method for calculation of stresses and material birefringence in polarization-maintaining optical fiber,” J. Lightwave Technol.2(5), 650–662 (1984).
[CrossRef]

Y. Mohanna, J. M. Saugrain, J. C. Rousseau, and P. Ledoux, “Relaxation of internal stresses in optical fibers,” J. Lightwave Technol.8(12), 1799–1802 (1990).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

F. Kherbouche and B. Poumellec, “UV-induced stress field during Bragg grating inscription in optical fibers,” J. Opt. A, Pure Appl. Opt.3(6), 429–439 (2001).
[CrossRef]

Meas. Sci. Technol. (1)

I. Abe, O. Frazão, M. W. Schiller, R. N. Nogueira, H. J. Kalinowski, and J. L. Pinto, “Bragg gratings in normal and reduced diameter high birefringence fiber optics,” Meas. Sci. Technol.17(6), 1477–1484 (2006).
[CrossRef]

Nat. Photonics (1)

A. Gillooly, “Photosensitive fibers: Growing gratings,” Nat. Photonics5(8), 468–469 (2011).
[CrossRef]

Opt. Commun. (1)

I. H. Shin, B. H. Kim, S. P. Veetil, W.-T. Han, and D. Y. Kim, “Residual stress relaxation in cleaved fibers,” Opt. Commun.281(1), 75–79 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Other (2)

A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

A. C. Ugural and S. K. Fenster, “Two dimensional problems in elasticity,” in Advanced Strength and Applied Elasticity (Pearson Education, 2003).

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Figures (7)

Fig. 1
Fig. 1

Stress profiles of (a) radial (b) circumferential and (c) axial stress for a step index fiber (γ = ∞) of different fiber diameter (FD). Parameters used for the modelingare: a = 3µm, b = 62.5 µm, E = 7830kg/mm2, γ = ∞, ν = 0.186, T = −850°C, α1 = 4.0 × 10−6/þC and α2 = 5.4 × 10−7/þC

Fig. 2
Fig. 2

(a) The relationship between axial stress at the center of core and the varying fiber diameter for different profile parameters γ. (b) Stress-induced index change at different fiber diameter for different parameter γ. Stress-optic coefficients are C1 = 7.42 × 10−6 mm2/kg and C2 = 4.102 × 10−5 mm2/kg

Fig. 3
Fig. 3

shows (a) a DIC microscope image of a grating structure with clear dark and bright regions perpendicular to the fiber axial direction. The spatial resolution of image is 31.98nm/pixel. (b) The intensity profile (solid) and its smoothened profile (dotted) of a small sample image of grating structure taken from the microscope image. The red circles mark the positions of the peaks in the profile. (c) The graph of standard deviation of peak-to-peak spacing calculated from the corresponding intensity profile. The position of the standard deviation minima in the graph coincides with the position grating structure in the microscope image.

Fig. 4
Fig. 4

Optical microscope images of an etched FBG at different fiber diameter (a) 125µm (b) 18µm and (c) 6 µm. The calculated grating periods for the three different fiber diameters are 1.0764µm, 1.0750µm and 1.0743µm respectively.

Fig. 5
Fig. 5

Sample image of the grating structure taken from the microscope images in Fig. 4. The sample images (i) and (ii) are axially spaced at a distance of 103 µm in the original images in Fig. 4(a), 4(b) and 4(c).

Fig. 6
Fig. 6

(a) Schematic illustration of axial contraction in etched FBG (b) The relationship between grating period difference and fiber diameter. The contraction is large when the fiber diameter is smaller than 20µm.

Fig. 7
Fig. 7

compares fiber diameter dependence of Bragg wavelength of original model (dotted) and proposed model (solid). The experimental data (circles) were taken from an FBG etched in BOE solution for 8 hours.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

σ r = E 1+ν 1 r [ χ r + 2 χ r θ 2 ]
σ θ = E 1+ν 2 χ r 2
τ rθ = E 1+ν r ( 1 r χ θ )
χ=ϕ+F
2 ϕ={ β[ 1 (r/a) γ ]T 0ra 0 a<rb
β= 1+ν 1ν ( α 1 α 2 )
σ r (r)={ EβT 1+ν [ 1 2 ( r/a ) γ γ+2 a 2 b 2 ( 1 2 1 γ+2 ) ] 0ra EβT 1+ν ( 1 2 1 γ+2 )( a 2 r 2 a 2 b 2 ) a<rb
σ θ (r)={ EβT 1+ν [ 1 2 γ+1 γ+2 ( r a ) γ a 2 b 2 ( 1 2 1 γ+2 ) ] 0ra EβT 1+ν ( 1 2 1 γ+2 )( a 2 r 2 + a 2 b 2 ) a<rb
τ rθ =0
σ z (r)= σ r (r)+ σ θ (r)
σ z (r)={ EβT 1+ν [ 1 a 2 b 2 ( 1 2 γ+2 ) ( r a ) γ ] 0ra EβT 1+ν a 2 b 2 ( 1 2 γ+2 ) a<rb
A σ z,etched dA =0
σ z,etched (r,ξ)= σ z (r)Eε
0 ξ σ z (r)2πrdr 0 ξ Eε2πrdr =0
ε(ξ)= βT 1+ν { ( 1 2 γ+2 )( 1 a 2 b 2 )+ 2 γ+2 [ 1 ( ξ a ) γ ] 0ξa a 2 b 2 ( b 2 ξ 2 1 )( 1 2 γ+2 ) a<ξb
σ z,etched (r,ξ)={ EβT 1+ν [ 1 a 2 ξ 2 ( 1 2 γ+2 ) ( r a ) γ ] 0ra EβT 1+ν ( 1 2 γ+2 ) ( a ξ ) 2 a<rξb
σ z,etched (r,ξ)= EβT 1+ν [ ( r a ) γ 2 γ+2 ( ξ a ) γ ] 0rξa
σ x (r,θ)= σ r (r) cos 2 θ+ σ θ (r) sin 2 θ
σ y (r,θ)= σ r (r) sin 2 θ+ σ θ (r) cos 2 θ
Δ n x (r,θ)= C 1 σ x (r,θ) C 2 [ σ y (r,θ)+ σ z (r) ]
Δ n x (r,θ)= C 1 [ σ r (r) cos 2 θ+ σ θ (r) sin 2 θ ] C 2 [ σ r (r) sin 2 θ+ σ θ (r) cos 2 θ+ σ z (r) ]
Δ n ¯ x (r)= 1 2π 0 2π Δ n x (r,θ) dθ = C 1 [ π σ r (r)+π σ θ (r) ] C 2 [ π σ r (r)+π σ θ (r)+2π σ z (r) ] 2π
Δ n ¯ x (ξ)=Δ n ¯ x,0 ( C 1 +3 C 2 ) σ z,etched (r=0,ξ)/2
Λ Etched Λ 0 = Λ 0 ε(ξ)
λ B = 2( n eff + ΓΔ n ¯ x )( Λ Etched /2)

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