Abstract

We demonstrate an optimization approach for designing fiber Bragg gratings. A layer-peeling inverse-scattering algorithm is used to produce an initial solution, which is optimized numerically with an iterative optimization method. To avoid problems with local minima, we use merit functions that are zero for wavelengths for which the predefined demands (acceptance limits) are fulfilled, making it possible to alter the local minima under the optimization process without disturbing the global minimum. Because short gratings are difficult to design with inverse scattering, and because the time consume of the optimization increases rapidly with the grating length, the method is particularly useful for designing short gratings. The method is also useful when the demands are complex and difficult to handle with inverse-scattering methods. Design examples are given, including a dispersionless bandpass filter suitable for dense wavelength-division multiplexing and a filter with linear reflectivity.

© 2004 Optical Society of America

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References

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  1. J. Martin, F. Ouellette, “Novel writing technique of long and highly reflective in-fiber gratings,” Electron. Lett. 30, 811–812 (1994).
    [CrossRef]
  2. A. Asseh, H. Storøy, B. E. Sahlgren, S. Sandgren, R. Stubbe, “A writing technique for long fiber Bragg gratings with complex reflectivity profiles,” J. Lightwave Technol. 15, 1419–1423 (1997).
    [CrossRef]
  3. W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, R. I. Laming, “Complex grating structures with uniform phase masks based on the moving fiber-scanning beam technique,” Opt. Lett. 20, 2051–2053 (1995).
    [CrossRef] [PubMed]
  4. M. Ibsen, M. K. Durkin, M. J. Cole, R. I. Laming, “Optimised square passband fibre Bragg grating filter with in-band flat group delay response,” Electron. Lett. 34, 800–802 (1998).
    [CrossRef]
  5. K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
    [CrossRef]
  6. I. Petermann, B. Sahlgren, S. Helmfrid, A. Friberg, P. Fonjallaz, “Fabrication of advanced fiber Bragg gratings by use of sequential writing with a continuous-wave ultraviolet laser source,” Appl. Opt. 41, 1051–1056 (2002).
    [CrossRef] [PubMed]
  7. R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
    [CrossRef]
  8. J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
    [CrossRef]
  9. J. Skaar, K. M. Risvik, “A genetic algorithm for the inverse problem in synthesis of fiber gratings,” J. Lightwave Technol. 16, 1928–1932 (1998).
    [CrossRef]
  10. R. Feced, M. N. Zervas, “Effects of random phase and amplitude errors in optical fiber Bragg gratings,” J. Lightwave Technol. 18, 90–101 (2000).
    [CrossRef]
  11. J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
    [CrossRef]
  12. J. Skaar, R. Feced, “Reconstruction of gratings from noisy reflection data,” J. Opt. Soc. Am. A 19, 2229–2237 (2002).
    [CrossRef]
  13. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  14. J. Skaar, “Synthesis and characterization of fiber Bragg gratings,” Ph.D. dissertation (Norwegian University of Science and Technology, Trondheim, Norway, 2000), available online at http://www.fysel.ntnu.no/Department/Avhandlinger/dring/ .
  15. G.-H. Song, “Theory of symmetry in optical filter responses,” J. Opt. Soc. Am. A 11, 2027–2037 (1994).
    [CrossRef]
  16. L. Poladian, “Group-delay reconstruction for fiber Bragg gratings in reflection and transmission,” Opt. Lett. 22, 1571–1573 (1997).
    [CrossRef]

2002

2001

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

2000

1999

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

1998

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

M. Ibsen, M. K. Durkin, M. J. Cole, R. I. Laming, “Optimised square passband fibre Bragg grating filter with in-band flat group delay response,” Electron. Lett. 34, 800–802 (1998).
[CrossRef]

1997

K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

A. Asseh, H. Storøy, B. E. Sahlgren, S. Sandgren, R. Stubbe, “A writing technique for long fiber Bragg gratings with complex reflectivity profiles,” J. Lightwave Technol. 15, 1419–1423 (1997).
[CrossRef]

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

L. Poladian, “Group-delay reconstruction for fiber Bragg gratings in reflection and transmission,” Opt. Lett. 22, 1571–1573 (1997).
[CrossRef]

1995

1994

J. Martin, F. Ouellette, “Novel writing technique of long and highly reflective in-fiber gratings,” Electron. Lett. 30, 811–812 (1994).
[CrossRef]

G.-H. Song, “Theory of symmetry in optical filter responses,” J. Opt. Soc. Am. A 11, 2027–2037 (1994).
[CrossRef]

1965

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

Asseh, A.

A. Asseh, H. Storøy, B. E. Sahlgren, S. Sandgren, R. Stubbe, “A writing technique for long fiber Bragg gratings with complex reflectivity profiles,” J. Lightwave Technol. 15, 1419–1423 (1997).
[CrossRef]

Barcelos, S.

Cole, M. J.

M. Ibsen, M. K. Durkin, M. J. Cole, R. I. Laming, “Optimised square passband fibre Bragg grating filter with in-band flat group delay response,” Electron. Lett. 34, 800–802 (1998).
[CrossRef]

W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, R. I. Laming, “Complex grating structures with uniform phase masks based on the moving fiber-scanning beam technique,” Opt. Lett. 20, 2051–2053 (1995).
[CrossRef] [PubMed]

Durkin, M. K.

M. Ibsen, M. K. Durkin, M. J. Cole, R. I. Laming, “Optimised square passband fibre Bragg grating filter with in-band flat group delay response,” Electron. Lett. 34, 800–802 (1998).
[CrossRef]

Erdogan, T.

J. Skaar, L. Wang, 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.

Fonjallaz, P.

Friberg, A.

Helmfrid, S.

Hill, K. O.

K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

Ibsen, M.

M. Ibsen, M. K. Durkin, M. J. Cole, R. I. Laming, “Optimised square passband fibre Bragg grating filter with in-band flat group delay response,” Electron. Lett. 34, 800–802 (1998).
[CrossRef]

Laming, R. I.

M. Ibsen, M. K. Durkin, M. J. Cole, R. I. Laming, “Optimised square passband fibre Bragg grating filter with in-band flat group delay response,” Electron. Lett. 34, 800–802 (1998).
[CrossRef]

W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, R. I. Laming, “Complex grating structures with uniform phase masks based on the moving fiber-scanning beam technique,” Opt. Lett. 20, 2051–2053 (1995).
[CrossRef] [PubMed]

Loh, W. H.

Martin, J.

J. Martin, F. Ouellette, “Novel writing technique of long and highly reflective in-fiber gratings,” Electron. Lett. 30, 811–812 (1994).
[CrossRef]

Mead, R.

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

Meltz, G.

K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

Muriel, M. A.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Nelder, J. A.

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

Ouellette, F.

J. Martin, F. Ouellette, “Novel writing technique of long and highly reflective in-fiber gratings,” Electron. Lett. 30, 811–812 (1994).
[CrossRef]

Petermann, I.

Poladian, L.

Risvik, K. M.

Sahlgren, B.

Sahlgren, B. E.

A. Asseh, H. Storøy, B. E. Sahlgren, S. Sandgren, R. Stubbe, “A writing technique for long fiber Bragg gratings with complex reflectivity profiles,” J. Lightwave Technol. 15, 1419–1423 (1997).
[CrossRef]

Sandgren, S.

A. Asseh, H. Storøy, B. E. Sahlgren, S. Sandgren, R. Stubbe, “A writing technique for long fiber Bragg gratings with complex reflectivity profiles,” J. Lightwave Technol. 15, 1419–1423 (1997).
[CrossRef]

Skaar, J.

J. Skaar, R. Feced, “Reconstruction of gratings from noisy reflection data,” J. Opt. Soc. Am. A 19, 2229–2237 (2002).
[CrossRef]

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

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

J. Skaar, “Synthesis and characterization of fiber Bragg gratings,” Ph.D. dissertation (Norwegian University of Science and Technology, Trondheim, Norway, 2000), available online at http://www.fysel.ntnu.no/Department/Avhandlinger/dring/ .

Song, G.-H.

Storøy, H.

A. Asseh, H. Storøy, B. E. Sahlgren, S. Sandgren, R. Stubbe, “A writing technique for long fiber Bragg gratings with complex reflectivity profiles,” J. Lightwave Technol. 15, 1419–1423 (1997).
[CrossRef]

Stubbe, R.

A. Asseh, H. Storøy, B. E. Sahlgren, S. Sandgren, R. Stubbe, “A writing technique for long fiber Bragg gratings with complex reflectivity profiles,” J. Lightwave Technol. 15, 1419–1423 (1997).
[CrossRef]

Wang, L.

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

Zervas, M. N.

Appl. Opt.

Comput. J.

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

Electron. Lett.

J. Martin, F. Ouellette, “Novel writing technique of long and highly reflective in-fiber gratings,” Electron. Lett. 30, 811–812 (1994).
[CrossRef]

M. Ibsen, M. K. Durkin, M. J. Cole, R. I. Laming, “Optimised square passband fibre Bragg grating filter with in-band flat group delay response,” Electron. Lett. 34, 800–802 (1998).
[CrossRef]

IEEE J. Quantum Electron.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

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

J. Lightwave Technol.

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

R. Feced, M. N. Zervas, “Effects of random phase and amplitude errors in optical fiber Bragg gratings,” J. Lightwave Technol. 18, 90–101 (2000).
[CrossRef]

K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

A. Asseh, H. Storøy, B. E. Sahlgren, S. Sandgren, R. Stubbe, “A writing technique for long fiber Bragg gratings with complex reflectivity profiles,” J. Lightwave Technol. 15, 1419–1423 (1997).
[CrossRef]

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

J. Opt. Soc. Am. A

Opt. Lett.

Other

J. Skaar, “Synthesis and characterization of fiber Bragg gratings,” Ph.D. dissertation (Norwegian University of Science and Technology, Trondheim, Norway, 2000), available online at http://www.fysel.ntnu.no/Department/Avhandlinger/dring/ .

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Figures (7)

Fig. 1
Fig. 1

Top-hat filter (2 cm) with 45.5-GHz bandwidth and 4.5-GHz guard bandwidth designed by optimization. Solid lines correspond to reflection, dotted lines to transmission, and dashed lines to the predefined limits. Note that the lower bound for in-band power reflection is represented as an upper bound for in-band transmission.

Fig. 2
Fig. 2

Coupling function of the filter shown in Fig. 1. The staircase shape indicates the piecewise-uniform representation of the grating. Note that the coupling function is restricted to be real; i.e., the associated power reflection and group delay spectrum are symmetric.15

Fig. 3
Fig. 3

WDM filter suitable for 50-GHz channel separation designed by optimization (2 cm). Solid lines correspond to reflection, dotted lines to transmission, and dashed lines to the predefined limits. Note that the lower bound for in-band power reflection is represented as an upper bound for in-band transmission. Also note that the in-band dispersion is the dispersion in reflection, whereas the out-of-band dispersion is in transmission.

Fig. 4
Fig. 4

Coupling function of the filter shown in Fig. 3.

Fig. 5
Fig. 5

WDM filters designed with the discrete layer-peeling method. Solid lines represent a 20-cm-long grating, dotted lines a 2-cm grating, and dashed lines the predefined limits as in Fig. 3. Transmission and reflection are plotted with the same lines. Note that the dispersion curves on the sides represent the transmission dispersion, whereas the curve in the middle shows the reflection dispersion.

Fig. 6
Fig. 6

The power reflection spectrum of a 2-cm-long grating designed by optimization. Dotted lines show the linear target. The maximum deviation from the target is 0.00017.

Fig. 7
Fig. 7

Coupling function of the triangular filter shown in Fig. 6.

Equations (3)

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E= j in-band cjej+ j out-of-band cjej,
ej= Rib-Rj,if Rib-Rj>00,else
ej= Rj-Rob,if Rj-Rob>00,else

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