Abstract

We compare different inverse scattering (IS) algorithms used to calculate profiles of fibre Bragg gratings (FBGs) and analyse their robustness, speed and implementation difficulties. We analyse sources of IS algorithm errors and discuss their relative importance. We discuss the optimal choice of IS algorithm for inverse-direct iterative optimisation schemes for grating design. We find that our time-domain layer-peeling method is an order of magnitude faster and more robust than the spectral domain algorithms considered here. We demonstrate that our method is essential to solving highly complex FBG designs demanded by astronomical applications.

© 2009 Optical Society of America

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  1. J. Bland-Hawthorn, M. England, and G. Edvell, "New approach to atmospheric OH suppression using an aperiodic fibre Bragg grating," Opt. Express 12, 5902-5909 (2004).
    [CrossRef] [PubMed]
  2. J. Bland-Hawthorn, A. Buryak, and K. Kolossovski, "Optimization algorithm for ultrabroadband multichannel aperiodic fiber Bragg grating filters," J. Opt. Soc. Am. A 25, 153-158 (2008).
    [CrossRef]
  3. A.V. Buryak, "Iterative scheme for the "mixed" scattering problem," in Proc. BGPP, 2003, MB3.
  4. K. Kolossovski, R. A. Sammut, A. V. Buryak, and D. Yu. Stepanov, " Three-step design optimization for multichannel fibre Bragg gratings," Opt. Express 11, 1029-1038 (2003).
    [CrossRef] [PubMed]
  5. R. Feced, M. N. Zervas, and M. A. Muriel, "An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings," IEEE J. Quant. Electron. 35,1105-1115 (1999).
    [CrossRef]
  6. L. Poladian, "Simple grating synthesis algorithm," Opt. Lett. 25, 787-789 (2000);L. Poladian, "Simple grating synthesis algorithm: errata," Opt. Lett. 25, 1400 (2000).
    [CrossRef]
  7. J. Skaar, L. Wang, and T. Erdogan, "On the synthesis of fiber Bragg gratings by layer peeling," IEEE J. Quant. Electron. 37, 165-173 (2001).
    [CrossRef]
  8. J. Skaar and R. Feced, "Reconstruction of gratings from noisy reflection data," J. Opt. Soc. Am. A 19, 2229-2237 (2002).
    [CrossRef]
  9. A. Rosenthal and M. Horowitz, "Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings," IEEE J. Quant. Electron. 39, 1018-1026 (2003).
    [CrossRef]
  10. K. Kolossovski, A. V. Buryak, and R. A. Sammut, "Optimised time-frequency domain layer-peeling algorithm to reconstruct fibre Bragg gratings," Electron. Lett. 40, 1046 (2004).
    [CrossRef]
  11. J. Skaar and O. H. Waagaard, "Design and characterization of finite-length fiber gratings," IEEE J. Quant. Electron. 39, 1238-1245 (2003).
    [CrossRef]
  12. L. Dong and S. Fortier, "Formulation of time-domain algorithm for fiber Bragg grating simulation and reconstruction," IEEE J. of Quant. Electron. 40, 1087-1098 (2004).
    [CrossRef]
  13. O. V. Belai, E. V. Podivilov, O. Ya. Schwarz, D. A. Shapiro, and L. L. Frumin, "Finite Bragg grating synthesis by numerical solution of Hermitian Gel’fand-Levitan-Marchenko equations," J. Opt. Soc. Am. B 23, 2040-2045 (2006).
    [CrossRef]
  14. O. V. Belai, L. L. Frumin, E. V. Podivilov, and D. A. Shapiro, "Efficient numerical method of the fiber Bragg grating synthesis," J. Opt. Soc. Am. B 24, 1451-1457 (2007).
    [CrossRef]
  15. C. K. Madsen and J. H. Zhao, "Optical flter design and analysis: a signal processing approach," Wiley & Sons, New York (1999).
  16. D. H. Jones, J. Bland-Hawthorn, & M.G. Burton, "Numerical evaluation of OH airglow suppression filters," Publ. Astron. Soc. Pac. 108, 929-938 (1996).
    [CrossRef]

2008 (1)

2007 (1)

2006 (1)

2004 (3)

J. Bland-Hawthorn, M. England, and G. Edvell, "New approach to atmospheric OH suppression using an aperiodic fibre Bragg grating," Opt. Express 12, 5902-5909 (2004).
[CrossRef] [PubMed]

K. Kolossovski, A. V. Buryak, and R. A. Sammut, "Optimised time-frequency domain layer-peeling algorithm to reconstruct fibre Bragg gratings," Electron. Lett. 40, 1046 (2004).
[CrossRef]

L. Dong and S. Fortier, "Formulation of time-domain algorithm for fiber Bragg grating simulation and reconstruction," IEEE J. of Quant. Electron. 40, 1087-1098 (2004).
[CrossRef]

2003 (3)

J. Skaar and O. H. Waagaard, "Design and characterization of finite-length fiber gratings," IEEE J. Quant. Electron. 39, 1238-1245 (2003).
[CrossRef]

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

K. Kolossovski, R. A. Sammut, A. V. Buryak, and D. Yu. Stepanov, " Three-step design optimization for multichannel fibre Bragg gratings," Opt. Express 11, 1029-1038 (2003).
[CrossRef] [PubMed]

2002 (1)

2001 (1)

J. Skaar, L. Wang, and T. Erdogan, "On the synthesis of fiber Bragg gratings by layer peeling," IEEE J. Quant. Electron. 37, 165-173 (2001).
[CrossRef]

2000 (1)

1999 (1)

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

1996 (1)

D. H. Jones, J. Bland-Hawthorn, & M.G. Burton, "Numerical evaluation of OH airglow suppression filters," Publ. Astron. Soc. Pac. 108, 929-938 (1996).
[CrossRef]

Belai, O. V.

Bland-Hawthorn, J.

Burton, M.G.

D. H. Jones, J. Bland-Hawthorn, & M.G. Burton, "Numerical evaluation of OH airglow suppression filters," Publ. Astron. Soc. Pac. 108, 929-938 (1996).
[CrossRef]

Buryak, A.

Buryak, A. V.

K. Kolossovski, A. V. Buryak, and R. A. Sammut, "Optimised time-frequency domain layer-peeling algorithm to reconstruct fibre Bragg gratings," Electron. Lett. 40, 1046 (2004).
[CrossRef]

K. Kolossovski, R. A. Sammut, A. V. Buryak, and D. Yu. Stepanov, " Three-step design optimization for multichannel fibre Bragg gratings," Opt. Express 11, 1029-1038 (2003).
[CrossRef] [PubMed]

Dong, L.

L. Dong and S. Fortier, "Formulation of time-domain algorithm for fiber Bragg grating simulation and reconstruction," IEEE J. of Quant. Electron. 40, 1087-1098 (2004).
[CrossRef]

Edvell, G.

England, M.

Erdogan, T.

J. Skaar, L. Wang, and T. Erdogan, "On the synthesis of fiber Bragg gratings by layer peeling," IEEE J. Quant. Electron. 37, 165-173 (2001).
[CrossRef]

Feced, R.

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

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

Fortier, S.

L. Dong and S. Fortier, "Formulation of time-domain algorithm for fiber Bragg grating simulation and reconstruction," IEEE J. of Quant. Electron. 40, 1087-1098 (2004).
[CrossRef]

Frumin, L. L.

Horowitz, M.

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

Jones, D. H.

D. H. Jones, J. Bland-Hawthorn, & M.G. Burton, "Numerical evaluation of OH airglow suppression filters," Publ. Astron. Soc. Pac. 108, 929-938 (1996).
[CrossRef]

Kolossovski, K.

Muriel, M. A.

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

Podivilov, E. V.

Poladian, L.

Rosenthal, A.

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

Sammut, R. A.

K. Kolossovski, A. V. Buryak, and R. A. Sammut, "Optimised time-frequency domain layer-peeling algorithm to reconstruct fibre Bragg gratings," Electron. Lett. 40, 1046 (2004).
[CrossRef]

K. Kolossovski, R. A. Sammut, A. V. Buryak, and D. Yu. Stepanov, " Three-step design optimization for multichannel fibre Bragg gratings," Opt. Express 11, 1029-1038 (2003).
[CrossRef] [PubMed]

Schwarz, O. Ya.

Shapiro, D. A.

Skaar, J.

J. Skaar and O. H. Waagaard, "Design and characterization of finite-length fiber gratings," IEEE J. Quant. Electron. 39, 1238-1245 (2003).
[CrossRef]

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

J. Skaar, L. Wang, and T. Erdogan, "On the synthesis of fiber Bragg gratings by layer peeling," IEEE J. Quant. Electron. 37, 165-173 (2001).
[CrossRef]

Stepanov, D. Yu.

Waagaard, O. H.

J. Skaar and O. H. Waagaard, "Design and characterization of finite-length fiber gratings," IEEE J. Quant. Electron. 39, 1238-1245 (2003).
[CrossRef]

Wang, L.

J. Skaar, L. Wang, and T. Erdogan, "On the synthesis of fiber Bragg gratings by layer peeling," IEEE J. Quant. Electron. 37, 165-173 (2001).
[CrossRef]

Zervas, M. N.

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

Electron. Lett. (1)

K. Kolossovski, A. V. Buryak, and R. A. Sammut, "Optimised time-frequency domain layer-peeling algorithm to reconstruct fibre Bragg gratings," Electron. Lett. 40, 1046 (2004).
[CrossRef]

IEEE J. of Quant. Electron. (1)

L. Dong and S. Fortier, "Formulation of time-domain algorithm for fiber Bragg grating simulation and reconstruction," IEEE J. of Quant. Electron. 40, 1087-1098 (2004).
[CrossRef]

IEEE J. Quant. Electron. (4)

J. Skaar and O. H. Waagaard, "Design and characterization of finite-length fiber gratings," IEEE J. Quant. Electron. 39, 1238-1245 (2003).
[CrossRef]

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

J. Skaar, L. Wang, and T. Erdogan, "On the synthesis of fiber Bragg gratings by layer peeling," IEEE J. Quant. Electron. 37, 165-173 (2001).
[CrossRef]

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

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

Opt. Express (2)

Opt. Lett. (1)

Publ. Astron. Soc. Pac. (1)

D. H. Jones, J. Bland-Hawthorn, & M.G. Burton, "Numerical evaluation of OH airglow suppression filters," Publ. Astron. Soc. Pac. 108, 929-938 (1996).
[CrossRef]

Other (2)

A.V. Buryak, "Iterative scheme for the "mixed" scattering problem," in Proc. BGPP, 2003, MB3.

C. K. Madsen and J. H. Zhao, "Optical flter design and analysis: a signal processing approach," Wiley & Sons, New York (1999).

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Figures (5)

Fig. 1.
Fig. 1.

An example of a small ripple development in the tail of a strong FBG design (see around z = 30 region).

Fig. 2.
Fig. 2.

The maximum high-quality convergence for the method of [7]: Top value for transmission reduction is limited to -31dB (for accepted T ripple of ±0.1 dB).

Fig. 3.
Fig. 3.

The maximum high-quality convergence for the method of [5] with modifications of [10]: Top value for transmission reduction is limited to -40dB (for accepted T ripple of ±0.1 dB).

Fig. 4.
Fig. 4.

The maximum high-quality convergence for the method of [11, 12]: Top value for transmission reduction is -102dB. For lower values of Tmin the tail ripple starts to exceed 0.001 level. It is important to note that this tail ripple does note manifest itself in substantial T and group delay ripples for Tmin > -120dB.

Fig. 5.
Fig. 5.

An example of 147-channel FBG filter design with 400nm bandwidth (obtained with algorithm V). Algorithms II and III failed to converge on this design.

Tables (2)

Tables Icon

Table 1. A summary of popular IS algorithms including references and information about the required number of operations M needed for one IS cycle. Only the fastest methods MN 2 are potentially suitable for the iterative mixed scattering scheme as in Ref. [2].

Tables Icon

Table 2. The results for comparative speed analysis of algorithms II, III and V. All data represent CPU clock time in seconds for different algorithms and different number of grid points.

Equations (11)

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E b z + E b q ( z ) E f = 0 ,
E f z E f q * ( z ) E b = 0 ,
[ E b ( 0 ) E f ( 0 ) ] = [ T 11 T 12 T 21 T 22 ] [ E b ( L ) E f ( L ) ] ,
T ̂ = [ 1 t * r 1 t r 2 t 1 t ] ,
M T ( Δ , β ) = M S ( Δ ) M P ( Δ , β ) ,
M T ( Δ , β ) = M P ( Δ / 2 , β ) M S ( Δ ) M P ( Δ / 2 , β ) ,
u ( 1 ) = [ u 1 ( 1 ) , u 2 ( 1 ) , u N ( 1 ) ] = [ 1,0 , 0 ]
v ( 1 ) = [ v 1 ( 1 ) , v 2 ( 1 ) , v N ( 1 ) ] = [ h ( 1 ) , h ( 2 ) , h ( N ) ] ,
v k ( j + 1 ) = v k + 1 ( j ) ρ j u k + 1 ( j ) ,
u k + 1 ( j + 1 ) = u k + 1 ( j ) ρ j * v k + 1 ( j ) ,
R ( λ ) = ( 1 T min ) / cosh [ ( λ λ i Δ λ ) n ] ,

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