Abstract

The polarization-dependant reflection spectra of fiber Bragg grating (FBG) sensors in polarization-maintaining fibers are influenced by shear strain. This influence can be evaluated from a tensorial coupled-mode theory approach. Yet, this approach requires the numerical integration of the four coupled-mode equations. We present an easy to handle, completely analytical treatment of the polarization-dependent reflection spectra of FBGs. We derive the required equations and compare the results to the numerical integration of the four tensorial coupled mode equations.

© 2009 Optical Society of America

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References

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  1. C. Lawrence, D. Nelson, E. Udd, and T. Bennett, Exp. Mech. 39, 202 (1999).
    [CrossRef]
  2. T. Mawatari and D. Nelson, Smart Mater. Struct. 17, 19 (2008).
    [CrossRef]
  3. F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. Limberger, and R. Salathe, Smart Mater. Struct. 12, 925 (2003).
    [CrossRef]
  4. E. Chehura, C. Ye, S. Staines, S. James, and R. Tatam, Smart Mater. Struct. 13, 888 (2004).
    [CrossRef]
  5. M. Prabhugoud and K. Peters, Smart Mater. Struct. 15, 550 (2006).
    [CrossRef]
  6. E. Udd, W. Schulz, J. Seim, and E. Haugse, in Proc. SPIE 3986, 254 (2000).
    [CrossRef]
  7. M. S. Müller, L. Hoffmann, A. Sandmair, and A. W. Koch, IEEE J. Quantum Electron. 45, 547 (2009).
    [CrossRef]
  8. M. S. Müller, H. El-Khozondar, A. Bernardini, and A. W. Koch, “Transfer matrix approach to four mode coupling in fiber Bragg gratings,” IEEE J. Quantum Electron. (to be published).
  9. M. S. Müller, T. C. Buck, H. J. El-Khozondar, and A. W. Koch., “Shear-strain influence on fiber Bragg grating measurement systems,” J. Lightwave Technol. (to be published).
  10. R. J. Van Steenkiste and G. S. Springer, Strain and Temperature Measurement with Fiber Optic Sensors (Technomic, 1997).
  11. T. Narasimhamutry, Photoelastic and Electro-Optic Properties of Crystals (Plenum, 1981).
  12. A. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, 1983).
  13. P. Yeh, J. Opt. Soc. Am. 69, 742 (1979).
    [CrossRef]
  14. T. Erdogan, J. Lightwave Technol. 15, 1277 (1997).
    [CrossRef]
  15. M. McCall, J. Lightwave Technol. 18, 236 (2000).
    [CrossRef]
  16. J. A. Guemes and J. M. Menendez, Compos. Sci. Technol. 62, 959 (2002).
    [CrossRef]

2009 (1)

M. S. Müller, L. Hoffmann, A. Sandmair, and A. W. Koch, IEEE J. Quantum Electron. 45, 547 (2009).
[CrossRef]

2008 (1)

T. Mawatari and D. Nelson, Smart Mater. Struct. 17, 19 (2008).
[CrossRef]

2006 (1)

M. Prabhugoud and K. Peters, Smart Mater. Struct. 15, 550 (2006).
[CrossRef]

2004 (1)

E. Chehura, C. Ye, S. Staines, S. James, and R. Tatam, Smart Mater. Struct. 13, 888 (2004).
[CrossRef]

2003 (1)

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. Limberger, and R. Salathe, Smart Mater. Struct. 12, 925 (2003).
[CrossRef]

2002 (1)

J. A. Guemes and J. M. Menendez, Compos. Sci. Technol. 62, 959 (2002).
[CrossRef]

2000 (2)

M. McCall, J. Lightwave Technol. 18, 236 (2000).
[CrossRef]

E. Udd, W. Schulz, J. Seim, and E. Haugse, in Proc. SPIE 3986, 254 (2000).
[CrossRef]

1999 (1)

C. Lawrence, D. Nelson, E. Udd, and T. Bennett, Exp. Mech. 39, 202 (1999).
[CrossRef]

1997 (1)

T. Erdogan, J. Lightwave Technol. 15, 1277 (1997).
[CrossRef]

1979 (1)

Bennett, T.

C. Lawrence, D. Nelson, E. Udd, and T. Bennett, Exp. Mech. 39, 202 (1999).
[CrossRef]

Bernardini, A.

M. S. Müller, H. El-Khozondar, A. Bernardini, and A. W. Koch, “Transfer matrix approach to four mode coupling in fiber Bragg gratings,” IEEE J. Quantum Electron. (to be published).

Bosia, F.

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. Limberger, and R. Salathe, Smart Mater. Struct. 12, 925 (2003).
[CrossRef]

Botsis, J.

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. Limberger, and R. Salathe, Smart Mater. Struct. 12, 925 (2003).
[CrossRef]

Buck, T. C.

M. S. Müller, T. C. Buck, H. J. El-Khozondar, and A. W. Koch., “Shear-strain influence on fiber Bragg grating measurement systems,” J. Lightwave Technol. (to be published).

Chehura, E.

E. Chehura, C. Ye, S. Staines, S. James, and R. Tatam, Smart Mater. Struct. 13, 888 (2004).
[CrossRef]

El-Khozondar, H.

M. S. Müller, H. El-Khozondar, A. Bernardini, and A. W. Koch, “Transfer matrix approach to four mode coupling in fiber Bragg gratings,” IEEE J. Quantum Electron. (to be published).

El-Khozondar, H. J.

M. S. Müller, T. C. Buck, H. J. El-Khozondar, and A. W. Koch., “Shear-strain influence on fiber Bragg grating measurement systems,” J. Lightwave Technol. (to be published).

Erdogan, T.

T. Erdogan, J. Lightwave Technol. 15, 1277 (1997).
[CrossRef]

Facchini, M.

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. Limberger, and R. Salathe, Smart Mater. Struct. 12, 925 (2003).
[CrossRef]

Giaccari, P.

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. Limberger, and R. Salathe, Smart Mater. Struct. 12, 925 (2003).
[CrossRef]

Guemes, J. A.

J. A. Guemes and J. M. Menendez, Compos. Sci. Technol. 62, 959 (2002).
[CrossRef]

Haugse, E.

E. Udd, W. Schulz, J. Seim, and E. Haugse, in Proc. SPIE 3986, 254 (2000).
[CrossRef]

Hoffmann, L.

M. S. Müller, L. Hoffmann, A. Sandmair, and A. W. Koch, IEEE J. Quantum Electron. 45, 547 (2009).
[CrossRef]

James, S.

E. Chehura, C. Ye, S. Staines, S. James, and R. Tatam, Smart Mater. Struct. 13, 888 (2004).
[CrossRef]

Koch, A. W.

M. S. Müller, L. Hoffmann, A. Sandmair, and A. W. Koch, IEEE J. Quantum Electron. 45, 547 (2009).
[CrossRef]

M. S. Müller, H. El-Khozondar, A. Bernardini, and A. W. Koch, “Transfer matrix approach to four mode coupling in fiber Bragg gratings,” IEEE J. Quantum Electron. (to be published).

Koch., A. W.

M. S. Müller, T. C. Buck, H. J. El-Khozondar, and A. W. Koch., “Shear-strain influence on fiber Bragg grating measurement systems,” J. Lightwave Technol. (to be published).

Lawrence, C.

C. Lawrence, D. Nelson, E. Udd, and T. Bennett, Exp. Mech. 39, 202 (1999).
[CrossRef]

Limberger, H.

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. Limberger, and R. Salathe, Smart Mater. Struct. 12, 925 (2003).
[CrossRef]

Love, J.

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

Mawatari, T.

T. Mawatari and D. Nelson, Smart Mater. Struct. 17, 19 (2008).
[CrossRef]

McCall, M.

Menendez, J. M.

J. A. Guemes and J. M. Menendez, Compos. Sci. Technol. 62, 959 (2002).
[CrossRef]

Müller, M. S.

M. S. Müller, L. Hoffmann, A. Sandmair, and A. W. Koch, IEEE J. Quantum Electron. 45, 547 (2009).
[CrossRef]

M. S. Müller, T. C. Buck, H. J. El-Khozondar, and A. W. Koch., “Shear-strain influence on fiber Bragg grating measurement systems,” J. Lightwave Technol. (to be published).

M. S. Müller, H. El-Khozondar, A. Bernardini, and A. W. Koch, “Transfer matrix approach to four mode coupling in fiber Bragg gratings,” IEEE J. Quantum Electron. (to be published).

Narasimhamutry, T.

T. Narasimhamutry, Photoelastic and Electro-Optic Properties of Crystals (Plenum, 1981).

Nelson, D.

T. Mawatari and D. Nelson, Smart Mater. Struct. 17, 19 (2008).
[CrossRef]

C. Lawrence, D. Nelson, E. Udd, and T. Bennett, Exp. Mech. 39, 202 (1999).
[CrossRef]

Peters, K.

M. Prabhugoud and K. Peters, Smart Mater. Struct. 15, 550 (2006).
[CrossRef]

Prabhugoud, M.

M. Prabhugoud and K. Peters, Smart Mater. Struct. 15, 550 (2006).
[CrossRef]

Salathe, R.

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. Limberger, and R. Salathe, Smart Mater. Struct. 12, 925 (2003).
[CrossRef]

Sandmair, A.

M. S. Müller, L. Hoffmann, A. Sandmair, and A. W. Koch, IEEE J. Quantum Electron. 45, 547 (2009).
[CrossRef]

Schulz, W.

E. Udd, W. Schulz, J. Seim, and E. Haugse, in Proc. SPIE 3986, 254 (2000).
[CrossRef]

Seim, J.

E. Udd, W. Schulz, J. Seim, and E. Haugse, in Proc. SPIE 3986, 254 (2000).
[CrossRef]

Snyder, A.

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

Springer, G. S.

R. J. Van Steenkiste and G. S. Springer, Strain and Temperature Measurement with Fiber Optic Sensors (Technomic, 1997).

Staines, S.

E. Chehura, C. Ye, S. Staines, S. James, and R. Tatam, Smart Mater. Struct. 13, 888 (2004).
[CrossRef]

Tatam, R.

E. Chehura, C. Ye, S. Staines, S. James, and R. Tatam, Smart Mater. Struct. 13, 888 (2004).
[CrossRef]

Udd, E.

E. Udd, W. Schulz, J. Seim, and E. Haugse, in Proc. SPIE 3986, 254 (2000).
[CrossRef]

C. Lawrence, D. Nelson, E. Udd, and T. Bennett, Exp. Mech. 39, 202 (1999).
[CrossRef]

Van Steenkiste, R. J.

R. J. Van Steenkiste and G. S. Springer, Strain and Temperature Measurement with Fiber Optic Sensors (Technomic, 1997).

Ye, C.

E. Chehura, C. Ye, S. Staines, S. James, and R. Tatam, Smart Mater. Struct. 13, 888 (2004).
[CrossRef]

Yeh, P.

Compos. Sci. Technol. (1)

J. A. Guemes and J. M. Menendez, Compos. Sci. Technol. 62, 959 (2002).
[CrossRef]

Exp. Mech. (1)

C. Lawrence, D. Nelson, E. Udd, and T. Bennett, Exp. Mech. 39, 202 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. S. Müller, L. Hoffmann, A. Sandmair, and A. W. Koch, IEEE J. Quantum Electron. 45, 547 (2009).
[CrossRef]

J. Lightwave Technol. (2)

M. McCall, J. Lightwave Technol. 18, 236 (2000).
[CrossRef]

T. Erdogan, J. Lightwave Technol. 15, 1277 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. SPIE (1)

E. Udd, W. Schulz, J. Seim, and E. Haugse, in Proc. SPIE 3986, 254 (2000).
[CrossRef]

Smart Mater. Struct. (4)

T. Mawatari and D. Nelson, Smart Mater. Struct. 17, 19 (2008).
[CrossRef]

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. Limberger, and R. Salathe, Smart Mater. Struct. 12, 925 (2003).
[CrossRef]

E. Chehura, C. Ye, S. Staines, S. James, and R. Tatam, Smart Mater. Struct. 13, 888 (2004).
[CrossRef]

M. Prabhugoud and K. Peters, Smart Mater. Struct. 15, 550 (2006).
[CrossRef]

Other (5)

M. S. Müller, H. El-Khozondar, A. Bernardini, and A. W. Koch, “Transfer matrix approach to four mode coupling in fiber Bragg gratings,” IEEE J. Quantum Electron. (to be published).

M. S. Müller, T. C. Buck, H. J. El-Khozondar, and A. W. Koch., “Shear-strain influence on fiber Bragg grating measurement systems,” J. Lightwave Technol. (to be published).

R. J. Van Steenkiste and G. S. Springer, Strain and Temperature Measurement with Fiber Optic Sensors (Technomic, 1997).

T. Narasimhamutry, Photoelastic and Electro-Optic Properties of Crystals (Plenum, 1981).

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

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Figures (4)

Fig. 1
Fig. 1

FBG subject to a mechanical load ( e ¯ ) . The fundamental modes of the FBG change their polarization due to the mechanical perturbation B ¯ .

Fig. 2
Fig. 2

Results for a grating with L B = 3.5 mm , e x y = 1 × 10 3 , L = 3 mm , and Δ n = 1 × 10 4 .

Fig. 3
Fig. 3

Results for a grating with L B = 4.2 mm , e x y = 1 × 10 3 , L = 3 mm , and Δ n = 0.5 × 10 4 .

Fig. 4
Fig. 4

Results for a grating with L B = 10.6 mm , e x y = 0.5 × 10 3 , L = 7 mm , and Δ n = 0.3 × 10 4 .

Equations (8)

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

( B x x n e , i 2 B x y B x y B y y n e , i 2 ) D t = M ̱ i D t = 0 .
n e , p s 2 = ( B x x + B y y ) ± ( B x x B y y ) 2 + 4 B x y 2 2 .
D ̃ p = { M 1 , 12 M 1 , 11 , 1 } T , D ̃ s = { M 2 , 12 M 2 , 11 , 1 } T ,
E ̃ j = B ¯ D ̃ j | B ¯ D ̃ j | .
A + = ( E ̃ p , x E ̃ s , x E ̃ p , y E ̃ s , y ) A ̃ + = Q ̱ A ̃ + .
ρ ̱ ̃ = diag { ρ ( λ , Λ , n e , p ) , ρ ( λ , Λ , n e , s ) } T .
A ̃ = ρ ̱ ̃ A ̃ + .
A = Q ̱ A ̃ = Q ̱ ρ ̱ ̃ A ̃ + = Q ̱ ρ ̱ ̃ Q ̱ 1 A +

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