Abstract

Fiber Bragg gratings (FBG) in multimode optical fibers provide a means for cost-effictive devices resulting in simplified and robust optic sensor systems. Parasitic mode effects in optical components of the entire measurement system strongly influence the measured multi-resonance reflection spectrum. Using a mode transfer matrix formalism we can describe these complex mode coupling effects in multimode optical systems in more detail. We demonstrate the accordance of the theory by two experiments. With this formalism it is possible to understand and optimize mode effects in multimode fiber optic systems.

© 2015 Optical Society of America

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References

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  1. T. Mizunami, T. V. Djambova, T. Niiho, and S. Gupta, “Bragg gratings in multimode and few-mode optical fibers,” J. Lightwave Technol. 18(2), 230–235 (2000).
    [Crossref]
  2. R. M. Cazo, O. Lisba, H. T. Hattori, V. M. Schneider, C. L. Barbosa, R. C. Rabelo, and J. L. S. Ferreira, “Experimental analysis of reflected modes in a multimode strained grating,” Microw. Opt. Technol. Lett. 28(1), 4–8 (2001).
    [Crossref]
  3. T. Liu, D. Wang, R. Raenaei, X. Chen, L. Zhang, and I. Bennion, “A low-cost multimode fiber Bragg grating sensor system,” Proc. SPIE 5634, 54–61 (2005).
    [Crossref]
  4. J. Lim, Q. Yang, B. E. Jones, and P. R. Jackson, “Strain and temperature sensors using multimode optical fiber Bragg gratings and correlation signal processing,” IEEE Trans. Instrum. Meas. 51(4), 622–627 (2002).
    [Crossref]
  5. X. Sang, C. Yu, and B. Yan, “Bragg gratings in multimode optical fibers and their applications,” J. Optoelectron. Adv. M. 8(4), 1616–1621 (2006).
  6. Y. Liu, J. Lit, X. Gu, and L. Wei, “Fiber comb filters based on UV-writing Bragg gratings in graded-index multimode fibers,” Opt. Express 13(21), 8508–8513 (2005).
    [Crossref] [PubMed]
  7. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
    [Crossref]
  8. J.-M. Liu, Photonic Devices (Cambridge University, 2005).
    [Crossref]
  9. C. Lu and Y. Cui, “Fiber Bragg grating spectra in multimode optical fibers,” J. Lightwave Technol. 24(1), 598–604 (2006).
    [Crossref]
  10. E. Hecht, Optics (Addison-Wesley, 2002).
  11. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quant. Electron. QE-9(9), 919–933 (1973).
    [Crossref]
  12. 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. Quant. Electron. 45(9), 1142–1148 (2009).
    [Crossref]
  13. T. Komuro and M. Akiyama, “Analysis and realization of cascaded transmission-line networks by the transfer scattering matrix,” IEEE T. Circuits Syst. CAS-32(11), 1166–1169 (1985).
    [Crossref]
  14. W. Zhao and R. O. Claus, “Optical fiber grating sensors in multimode fibers,” Smart Mater. Struct. 9, 212–214 (2000).
    [Crossref]
  15. X. Yang, C. Zhao, J. Zhou, X. Guo, J. Ng, X. Zhou, and C. Lu, “The characteristics of fiber slanted gratings in multimode fiber,” Opt. Commun. 229, 161–165 (2004).
    [Crossref]

2009 (1)

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. Quant. Electron. 45(9), 1142–1148 (2009).
[Crossref]

2006 (2)

X. Sang, C. Yu, and B. Yan, “Bragg gratings in multimode optical fibers and their applications,” J. Optoelectron. Adv. M. 8(4), 1616–1621 (2006).

C. Lu and Y. Cui, “Fiber Bragg grating spectra in multimode optical fibers,” J. Lightwave Technol. 24(1), 598–604 (2006).
[Crossref]

2005 (2)

Y. Liu, J. Lit, X. Gu, and L. Wei, “Fiber comb filters based on UV-writing Bragg gratings in graded-index multimode fibers,” Opt. Express 13(21), 8508–8513 (2005).
[Crossref] [PubMed]

T. Liu, D. Wang, R. Raenaei, X. Chen, L. Zhang, and I. Bennion, “A low-cost multimode fiber Bragg grating sensor system,” Proc. SPIE 5634, 54–61 (2005).
[Crossref]

2004 (1)

X. Yang, C. Zhao, J. Zhou, X. Guo, J. Ng, X. Zhou, and C. Lu, “The characteristics of fiber slanted gratings in multimode fiber,” Opt. Commun. 229, 161–165 (2004).
[Crossref]

2002 (1)

J. Lim, Q. Yang, B. E. Jones, and P. R. Jackson, “Strain and temperature sensors using multimode optical fiber Bragg gratings and correlation signal processing,” IEEE Trans. Instrum. Meas. 51(4), 622–627 (2002).
[Crossref]

2001 (1)

R. M. Cazo, O. Lisba, H. T. Hattori, V. M. Schneider, C. L. Barbosa, R. C. Rabelo, and J. L. S. Ferreira, “Experimental analysis of reflected modes in a multimode strained grating,” Microw. Opt. Technol. Lett. 28(1), 4–8 (2001).
[Crossref]

2000 (2)

W. Zhao and R. O. Claus, “Optical fiber grating sensors in multimode fibers,” Smart Mater. Struct. 9, 212–214 (2000).
[Crossref]

T. Mizunami, T. V. Djambova, T. Niiho, and S. Gupta, “Bragg gratings in multimode and few-mode optical fibers,” J. Lightwave Technol. 18(2), 230–235 (2000).
[Crossref]

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

1985 (1)

T. Komuro and M. Akiyama, “Analysis and realization of cascaded transmission-line networks by the transfer scattering matrix,” IEEE T. Circuits Syst. CAS-32(11), 1166–1169 (1985).
[Crossref]

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quant. Electron. QE-9(9), 919–933 (1973).
[Crossref]

Akiyama, M.

T. Komuro and M. Akiyama, “Analysis and realization of cascaded transmission-line networks by the transfer scattering matrix,” IEEE T. Circuits Syst. CAS-32(11), 1166–1169 (1985).
[Crossref]

Barbosa, C. L.

R. M. Cazo, O. Lisba, H. T. Hattori, V. M. Schneider, C. L. Barbosa, R. C. Rabelo, and J. L. S. Ferreira, “Experimental analysis of reflected modes in a multimode strained grating,” Microw. Opt. Technol. Lett. 28(1), 4–8 (2001).
[Crossref]

Bennion, I.

T. Liu, D. Wang, R. Raenaei, X. Chen, L. Zhang, and I. Bennion, “A low-cost multimode fiber Bragg grating sensor system,” Proc. SPIE 5634, 54–61 (2005).
[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. Quant. Electron. 45(9), 1142–1148 (2009).
[Crossref]

Cazo, R. M.

R. M. Cazo, O. Lisba, H. T. Hattori, V. M. Schneider, C. L. Barbosa, R. C. Rabelo, and J. L. S. Ferreira, “Experimental analysis of reflected modes in a multimode strained grating,” Microw. Opt. Technol. Lett. 28(1), 4–8 (2001).
[Crossref]

Chen, X.

T. Liu, D. Wang, R. Raenaei, X. Chen, L. Zhang, and I. Bennion, “A low-cost multimode fiber Bragg grating sensor system,” Proc. SPIE 5634, 54–61 (2005).
[Crossref]

Claus, R. O.

W. Zhao and R. O. Claus, “Optical fiber grating sensors in multimode fibers,” Smart Mater. Struct. 9, 212–214 (2000).
[Crossref]

Cui, Y.

Djambova, T. V.

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. Quant. Electron. 45(9), 1142–1148 (2009).
[Crossref]

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

Ferreira, J. L. S.

R. M. Cazo, O. Lisba, H. T. Hattori, V. M. Schneider, C. L. Barbosa, R. C. Rabelo, and J. L. S. Ferreira, “Experimental analysis of reflected modes in a multimode strained grating,” Microw. Opt. Technol. Lett. 28(1), 4–8 (2001).
[Crossref]

Gu, X.

Guo, X.

X. Yang, C. Zhao, J. Zhou, X. Guo, J. Ng, X. Zhou, and C. Lu, “The characteristics of fiber slanted gratings in multimode fiber,” Opt. Commun. 229, 161–165 (2004).
[Crossref]

Gupta, S.

Hattori, H. T.

R. M. Cazo, O. Lisba, H. T. Hattori, V. M. Schneider, C. L. Barbosa, R. C. Rabelo, and J. L. S. Ferreira, “Experimental analysis of reflected modes in a multimode strained grating,” Microw. Opt. Technol. Lett. 28(1), 4–8 (2001).
[Crossref]

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2002).

Jackson, P. R.

J. Lim, Q. Yang, B. E. Jones, and P. R. Jackson, “Strain and temperature sensors using multimode optical fiber Bragg gratings and correlation signal processing,” IEEE Trans. Instrum. Meas. 51(4), 622–627 (2002).
[Crossref]

Jones, B. E.

J. Lim, Q. Yang, B. E. Jones, and P. R. Jackson, “Strain and temperature sensors using multimode optical fiber Bragg gratings and correlation signal processing,” IEEE Trans. Instrum. Meas. 51(4), 622–627 (2002).
[Crossref]

Koch, A. W.

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. Quant. Electron. 45(9), 1142–1148 (2009).
[Crossref]

Komuro, T.

T. Komuro and M. Akiyama, “Analysis and realization of cascaded transmission-line networks by the transfer scattering matrix,” IEEE T. Circuits Syst. CAS-32(11), 1166–1169 (1985).
[Crossref]

Lim, J.

J. Lim, Q. Yang, B. E. Jones, and P. R. Jackson, “Strain and temperature sensors using multimode optical fiber Bragg gratings and correlation signal processing,” IEEE Trans. Instrum. Meas. 51(4), 622–627 (2002).
[Crossref]

Lisba, O.

R. M. Cazo, O. Lisba, H. T. Hattori, V. M. Schneider, C. L. Barbosa, R. C. Rabelo, and J. L. S. Ferreira, “Experimental analysis of reflected modes in a multimode strained grating,” Microw. Opt. Technol. Lett. 28(1), 4–8 (2001).
[Crossref]

Lit, J.

Liu, J.-M.

J.-M. Liu, Photonic Devices (Cambridge University, 2005).
[Crossref]

Liu, T.

T. Liu, D. Wang, R. Raenaei, X. Chen, L. Zhang, and I. Bennion, “A low-cost multimode fiber Bragg grating sensor system,” Proc. SPIE 5634, 54–61 (2005).
[Crossref]

Liu, Y.

Lu, C.

C. Lu and Y. Cui, “Fiber Bragg grating spectra in multimode optical fibers,” J. Lightwave Technol. 24(1), 598–604 (2006).
[Crossref]

X. Yang, C. Zhao, J. Zhou, X. Guo, J. Ng, X. Zhou, and C. Lu, “The characteristics of fiber slanted gratings in multimode fiber,” Opt. Commun. 229, 161–165 (2004).
[Crossref]

Mizunami, T.

Müller, M. S.

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. Quant. Electron. 45(9), 1142–1148 (2009).
[Crossref]

Ng, J.

X. Yang, C. Zhao, J. Zhou, X. Guo, J. Ng, X. Zhou, and C. Lu, “The characteristics of fiber slanted gratings in multimode fiber,” Opt. Commun. 229, 161–165 (2004).
[Crossref]

Niiho, T.

Rabelo, R. C.

R. M. Cazo, O. Lisba, H. T. Hattori, V. M. Schneider, C. L. Barbosa, R. C. Rabelo, and J. L. S. Ferreira, “Experimental analysis of reflected modes in a multimode strained grating,” Microw. Opt. Technol. Lett. 28(1), 4–8 (2001).
[Crossref]

Raenaei, R.

T. Liu, D. Wang, R. Raenaei, X. Chen, L. Zhang, and I. Bennion, “A low-cost multimode fiber Bragg grating sensor system,” Proc. SPIE 5634, 54–61 (2005).
[Crossref]

Sang, X.

X. Sang, C. Yu, and B. Yan, “Bragg gratings in multimode optical fibers and their applications,” J. Optoelectron. Adv. M. 8(4), 1616–1621 (2006).

Schneider, V. M.

R. M. Cazo, O. Lisba, H. T. Hattori, V. M. Schneider, C. L. Barbosa, R. C. Rabelo, and J. L. S. Ferreira, “Experimental analysis of reflected modes in a multimode strained grating,” Microw. Opt. Technol. Lett. 28(1), 4–8 (2001).
[Crossref]

Wang, D.

T. Liu, D. Wang, R. Raenaei, X. Chen, L. Zhang, and I. Bennion, “A low-cost multimode fiber Bragg grating sensor system,” Proc. SPIE 5634, 54–61 (2005).
[Crossref]

Wei, L.

Yan, B.

X. Sang, C. Yu, and B. Yan, “Bragg gratings in multimode optical fibers and their applications,” J. Optoelectron. Adv. M. 8(4), 1616–1621 (2006).

Yang, Q.

J. Lim, Q. Yang, B. E. Jones, and P. R. Jackson, “Strain and temperature sensors using multimode optical fiber Bragg gratings and correlation signal processing,” IEEE Trans. Instrum. Meas. 51(4), 622–627 (2002).
[Crossref]

Yang, X.

X. Yang, C. Zhao, J. Zhou, X. Guo, J. Ng, X. Zhou, and C. Lu, “The characteristics of fiber slanted gratings in multimode fiber,” Opt. Commun. 229, 161–165 (2004).
[Crossref]

Yariv, A.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quant. Electron. QE-9(9), 919–933 (1973).
[Crossref]

Yu, C.

X. Sang, C. Yu, and B. Yan, “Bragg gratings in multimode optical fibers and their applications,” J. Optoelectron. Adv. M. 8(4), 1616–1621 (2006).

Zhang, L.

T. Liu, D. Wang, R. Raenaei, X. Chen, L. Zhang, and I. Bennion, “A low-cost multimode fiber Bragg grating sensor system,” Proc. SPIE 5634, 54–61 (2005).
[Crossref]

Zhao, C.

X. Yang, C. Zhao, J. Zhou, X. Guo, J. Ng, X. Zhou, and C. Lu, “The characteristics of fiber slanted gratings in multimode fiber,” Opt. Commun. 229, 161–165 (2004).
[Crossref]

Zhao, W.

W. Zhao and R. O. Claus, “Optical fiber grating sensors in multimode fibers,” Smart Mater. Struct. 9, 212–214 (2000).
[Crossref]

Zhou, J.

X. Yang, C. Zhao, J. Zhou, X. Guo, J. Ng, X. Zhou, and C. Lu, “The characteristics of fiber slanted gratings in multimode fiber,” Opt. Commun. 229, 161–165 (2004).
[Crossref]

Zhou, X.

X. Yang, C. Zhao, J. Zhou, X. Guo, J. Ng, X. Zhou, and C. Lu, “The characteristics of fiber slanted gratings in multimode fiber,” Opt. Commun. 229, 161–165 (2004).
[Crossref]

IEEE J. Quant. Electron. (2)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quant. Electron. QE-9(9), 919–933 (1973).
[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. Quant. Electron. 45(9), 1142–1148 (2009).
[Crossref]

IEEE T. Circuits Syst. (1)

T. Komuro and M. Akiyama, “Analysis and realization of cascaded transmission-line networks by the transfer scattering matrix,” IEEE T. Circuits Syst. CAS-32(11), 1166–1169 (1985).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

J. Lim, Q. Yang, B. E. Jones, and P. R. Jackson, “Strain and temperature sensors using multimode optical fiber Bragg gratings and correlation signal processing,” IEEE Trans. Instrum. Meas. 51(4), 622–627 (2002).
[Crossref]

J. Lightwave Technol. (3)

J. Optoelectron. Adv. M. (1)

X. Sang, C. Yu, and B. Yan, “Bragg gratings in multimode optical fibers and their applications,” J. Optoelectron. Adv. M. 8(4), 1616–1621 (2006).

Microw. Opt. Technol. Lett. (1)

R. M. Cazo, O. Lisba, H. T. Hattori, V. M. Schneider, C. L. Barbosa, R. C. Rabelo, and J. L. S. Ferreira, “Experimental analysis of reflected modes in a multimode strained grating,” Microw. Opt. Technol. Lett. 28(1), 4–8 (2001).
[Crossref]

Opt. Commun. (1)

X. Yang, C. Zhao, J. Zhou, X. Guo, J. Ng, X. Zhou, and C. Lu, “The characteristics of fiber slanted gratings in multimode fiber,” Opt. Commun. 229, 161–165 (2004).
[Crossref]

Opt. Express (1)

Proc. SPIE (1)

T. Liu, D. Wang, R. Raenaei, X. Chen, L. Zhang, and I. Bennion, “A low-cost multimode fiber Bragg grating sensor system,” Proc. SPIE 5634, 54–61 (2005).
[Crossref]

Smart Mater. Struct. (1)

W. Zhao and R. O. Claus, “Optical fiber grating sensors in multimode fibers,” Smart Mater. Struct. 9, 212–214 (2000).
[Crossref]

Other (2)

E. Hecht, Optics (Addison-Wesley, 2002).

J.-M. Liu, Photonic Devices (Cambridge University, 2005).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 The figure shows the transfer matrix Θ with the incident mode fields A⃗(z1)+ and A⃗(z2), and the emergent mode fields A⃗(z1) and A⃗(z2)+.
Fig. 2
Fig. 2 The multimode fiber Bragg grating measurement system. The singlemode Erbium-doped fiber amplifier tunable filter combination (EDFA/TF) couples light over a single-to-multimode splice into the mode scrambler. With the mode scrambler we excite higher order modes in the multimode fiber. The multimode fiber coupler guides this light onto the multimode fiber Bragg grating (MMFBG) and the reflection spectrum back to a photodiode (multimode detector).
Fig. 3
Fig. 3 Simulated and measured reflection spectrum of a multimode fiber Bragg grating. The spectrum is recorded with measurement system A including a multimode photo detector and a scanning tunable filter.
Fig. 4
Fig. 4 The FBG interrogator sm125 by Micron Optics connected to a multimode fiber Bragg grating. We couple the singlemode scanning laser over a single-to-multimode splice into our mode scrambler, which excites higher order modes of the incident light. The multimode FBG (MMFBG) reflects the characteristic Bragg spectrum. The backscattered light travels back over the mode scrambler and the single-to-multimode splice to the sm125. The measurement system thereby records just the intensity of the fundamental mode, because of singlemode characteristic.
Fig. 5
Fig. 5 Simulated and measured reflection spectrum of the multimode fiber Bragg grating. The spectrum is recorded using the singlemode measurement system sm125.
Fig. 6
Fig. 6 Detailed view of the simulated and measured reflection spectrum of the multimode fiber Bragg grating. The spectrum is recorded by a singlemode measurement system sm125.

Equations (19)

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β u ( λ ) = β v ( λ ) + m 2 π Λ
β u 2 ω μ 0 ^ u ^ v * d x d y = δ u v ,
E = u = 1 N α u ( z ) ^ u ( x , y ) e j β u z e j ω t
d d z α u = j v κ u v ( z ) α v e j ( β v β u ) z .
κ u v ( z ) = ω 4 ( ^ u * ( x , y ) ) T Δ ε ¯ ( x , y , z ) ^ v ( x , y ) d x d y
d d z α u Δ α u Δ z = α u ( z i + Δ z ) α u ( z i ) Δ z .
α u ( z i + Δ z ) α u ( z i ) = v κ u v e j ( β u β v ) z i α v ( z i ) Δ z
α u ( z i + Δ z ) = v ( κ u v e j ( β u β v ) z i Δ z + δ u v ) α v ( z i ) .
Θ : ϑ u v ( z i ) = κ u v ( z i ) e j ( β u β v ) z i Δ z + δ u v ,
A = ( α 1 + , α 2 + , α N + , α 1 , α 2 , α N ) T
A ( z 1 + M Δ z ) = ( i = 1 M Θ ( z i ) ) A ( z 1 ) = Θ A ( z 1 ) .
( A ( z 2 ) + A ( z 2 ) ) = Θ = ( T + + T + T + T ) ( A ( z 1 ) + A ( z 1 ) )
( A ( z 2 ) A ( z 2 ) + ) = Σ = ( S + S S + + S + ) ( A ( z 1 ) + A ( z 1 ) ) .
A ( z 2 ) + = T + + A ( z 1 ) +
A ( z 2 ) = T A ( z 1 ) .
P = ( A * ) T A = | A | 2 .
A out = T MFC 23 S FBG T MFC 12 T MS T BT A in .
f = f os + f max 1 + [ FWHM ( λ λ 0 ) ] 2
A out = T BT T MS S FBG T MS T BT A in

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