Abstract

Coupling coefficients of various grating types and strengths are calculated from measurements of the complex reflectivity using an applied thermal chirp and optical frequency domain reflectometry (OFDR). The complex reflectivity is then utilized by a layer peeling algorithm to determine the coupling coefficient of the thermally chirped grating. A guess of the temperature profile enables the coupling coefficient of the unchirped grating to be estimated. An iterative algorithm is then used to converge on the exact coupling coefficient, employing an error minimization method applied to the reflectivity spectra. This technique removes the need for a reference grating while preserving the spatial resolution obtained with the initial OFDR measurement. Successful reconstruction of gratings with integrated |κ|L9.0 are demonstrated with a spatial resolution of less than 100μm.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2009 (1)

2008 (1)

G. A. Miller, G. M. H. Flockhart, and G. A. Cranch, “Technique for correcting systematic phase errors during fiber Bragg grating inscription,” Electron. Lett. 44, 1399–1401 (2008).
[CrossRef]

2007 (1)

2005 (1)

2004 (1)

2003 (1)

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
[CrossRef]

2002 (2)

2001 (1)

J. Skaar, L. G. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

2000 (1)

1998 (2)

J. Skaar and K. M. Risvik, “A genetic algorithm for the inverse problem in synthesis of fiber gratings,” J. Lightwave Technol. 16, 1928–1932 (1998).
[CrossRef]

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

1997 (1)

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

Aksnes, K.

Bruvik, E. M.

O. H. Waagaard, J. T. Kringlebotn, and E. M. Bruvik, “Spatial characterization of strong FBGs using thermal linear chirp and optical frequency domain reflectometry,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, Technical Digest (Optical Society of America, 2003), paper WC2.

Casagrande, F.

Cranch, G. A.

G. A. Cranch and G. A. Miller, “Improved implementation of optical space domain reflectometry for characterizing the complex coupling coefficient of strong fiber Bragg gratings,” Appl. Opt. 48, 4506–4513 (2009).
[CrossRef] [PubMed]

G. A. Miller, G. M. H. Flockhart, and G. A. Cranch, “Technique for correcting systematic phase errors during fiber Bragg grating inscription,” Electron. Lett. 44, 1399–1401 (2008).
[CrossRef]

Crespi, P.

Erdogan, T.

J. Skaar, L. G. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

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

Feced, R.

Flockhart, G. M. H.

G. A. Miller, G. M. H. Flockhart, and G. A. Cranch, “Technique for correcting systematic phase errors during fiber Bragg grating inscription,” Electron. Lett. 44, 1399–1401 (2008).
[CrossRef]

Grassi, A. M.

Horowitz, M.

A. Sherman, A. Rosenthal, and M. Horowitz, “Extracting the structure of highly reflecting fiber Bragg gratings by measuring both the transmission and the reflection spectra,” Opt. Lett. 32, 457–459 (2007).
[CrossRef] [PubMed]

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
[CrossRef]

Kenny, R. P.

Kringlebotn, J. T.

O. H. Waagaard, J. T. Kringlebotn, and E. M. Bruvik, “Spatial characterization of strong FBGs using thermal linear chirp and optical frequency domain reflectometry,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, Technical Digest (Optical Society of America, 2003), paper WC2.

O. H. Waagaard, E. Rønnekleiv, and J. T. Kringlebotn, “Spatial characterization of FBGs using layer peeling,” in Proceedings of European Conference of Optical Communication (ECOC) (AEI-Ufficio Centrale, 2002), paper P1.32.

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Lulli, A.

Miller, G. A.

G. A. Cranch and G. A. Miller, “Improved implementation of optical space domain reflectometry for characterizing the complex coupling coefficient of strong fiber Bragg gratings,” Appl. Opt. 48, 4506–4513 (2009).
[CrossRef] [PubMed]

G. A. Miller, G. M. H. Flockhart, and G. A. Cranch, “Technique for correcting systematic phase errors during fiber Bragg grating inscription,” Electron. Lett. 44, 1399–1401 (2008).
[CrossRef]

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Risvik, K. M.

Rønnekleiv, E.

O. H. Waagaard, E. Rønnekleiv, and J. T. Kringlebotn, “Spatial characterization of FBGs using layer peeling,” in Proceedings of European Conference of Optical Communication (ECOC) (AEI-Ufficio Centrale, 2002), paper P1.32.

Rosenthal, A.

A. Sherman, A. Rosenthal, and M. Horowitz, “Extracting the structure of highly reflecting fiber Bragg gratings by measuring both the transmission and the reflection spectra,” Opt. Lett. 32, 457–459 (2007).
[CrossRef] [PubMed]

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
[CrossRef]

Sherman, A.

Skaar, J.

Waagaard, O. H.

O. H. Waagaard, “Spatial characterization of strong fiber Bragg gratings using thermal chirp and optical-frequency domain reflectometry,” J. Lightwave Technol. 23, 909–914(2005).
[CrossRef]

O. H. Waagaard, E. Rønnekleiv, and J. T. Kringlebotn, “Spatial characterization of FBGs using layer peeling,” in Proceedings of European Conference of Optical Communication (ECOC) (AEI-Ufficio Centrale, 2002), paper P1.32.

O. H. Waagaard, J. T. Kringlebotn, and E. M. Bruvik, “Spatial characterization of strong FBGs using thermal linear chirp and optical frequency domain reflectometry,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, Technical Digest (Optical Society of America, 2003), paper WC2.

Wang, L. G.

J. Skaar, L. G. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

Whelan, M. P.

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Zervas, M. N.

Appl. Opt. (3)

Electron. Lett. (1)

G. A. Miller, G. M. H. Flockhart, and G. A. Cranch, “Technique for correcting systematic phase errors during fiber Bragg grating inscription,” Electron. Lett. 44, 1399–1401 (2008).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
[CrossRef]

J. Skaar, L. G. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

J. Lightwave Technol. (4)

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

SIAM J. Optim. (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Other (3)

O. H. Waagaard, E. Rønnekleiv, and J. T. Kringlebotn, “Spatial characterization of FBGs using layer peeling,” in Proceedings of European Conference of Optical Communication (ECOC) (AEI-Ufficio Centrale, 2002), paper P1.32.

O. H. Waagaard, J. T. Kringlebotn, and E. M. Bruvik, “Spatial characterization of strong FBGs using thermal linear chirp and optical frequency domain reflectometry,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, Technical Digest (Optical Society of America, 2003), paper WC2.

J. Skaar, “Synthesis and characterization of fiber Bragg gratings,” Ph.D. dissertation (Norwegian University of Science and Technology, 2000).

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Figures (9)

Fig. 1
Fig. 1

Top: operating configurations for the thermal chirp setup. Bottom: schematic of the experimental apparatus.

Fig. 2
Fig. 2

Annotated photograph of the experimental setup.

Fig. 3
Fig. 3

Diagram of the IA. The process flows clockwise from the top-left corner.

Fig. 4
Fig. 4

Top: temperature profile along the stainless steel beam as measured by reference grating (gray) and third-order polynomial fit (black). Bottom: transmission spectrum of triangular apodized grating before (black) and after (gray) thermal chirp.

Fig. 5
Fig. 5

Top and middle: amplitude and phase of coupling coefficient, before (right axis) and after (left axis) subtraction of the excess phase due to temperature, for the weak grating using direct DLP (gray) and using the IA (black). Bottom: phase error attributed to the IA.

Fig. 6
Fig. 6

Top and middle: amplitude and phase of coupling coefficient for a triangular apodized grating calculated using the unchirped spectrum (gray), the reference grating (black), and the IA (dotted). Bottom: reflection spectra of the unchirped grating (gray) and that calculated from the coupling coefficient obtained using the reference grating (black) and IA (dotted).

Fig. 7
Fig. 7

Top and middle: amplitude and phase of coupling coefficient for a phase-shifted apodized grating calculated using the unchirped spectrum (gray), and the IA with 100 ° C (dotted), 130 ° C (×), and 160 ° C (+) temperature gradients. Bottom: reflection spectra of the unchirped grating (gray) and that calculated from the coupling coefficient obtained using the IA with 100 ° C (dotted), 130 ° C (×), and 160 ° C (+) temperature gradients.

Fig. 8
Fig. 8

Reflection spectra of the unchirped grating (gray) and that calculated from the coupling coefficient obtained using the IA with 100 ° C (dotted) and 130 ° C (×) temperature gradients. The inset graphs show expanded views of spectral regions.

Fig. 9
Fig. 9

Top and middle: amplitude and phase of coupling coefficient for a phase-shifted apodized grating calculated using the unchirped spectrum (gray), the reference grating (black), and the IA (dotted). Bottom: reflection spectra of the unchirped grating (gray) and that calculated from the coupling coefficient obtained using the reference grating (black) and IA (dotted).

Equations (7)

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δ n ( z ) = δ n d c ( z ) + δ n a c ( z ) cos ( 2 π Λ z + θ ( z ) ) ,
| κ ( z ) | = π η δ n a c ( z ) λ B , arg ( κ ( z ) ) = θ ( z ) 4 π η λ B 0 z δ n d c ( z ) d z + π 2 .
θ ( z ) = 4 π η n eff λ B 0 z ( λ B λ D ( z ) 1 ) d z ,
arg ( κ ( z ) ) Δ T = 4 π η n eff λ B 0 z ( 1 1 + α Δ T ( z ) 1 d n / d T n eff Δ T ( z ) ) d z + π 2 .
Δ T ( z ) = d arg ( κ ) d z λ B 4 π η n eff ( α + d n / d T n eff ) 1 .
error = all   λ ( R 1 ( λ ) R 2 ( λ ) ) 2 ,
Δ z = π 2 π n eff Δ λ λ 2 ,

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