Abstract

The discrete layer-peeling algorithm (DLPA) requires to discretize the continuous medium into discrete reflectors to synthesize nonuniform fiber Bragg gratings (FBG), and the discretization step of this discrete model should be sufficiently small for synthesis with high accuracy. However, the discretization step cannot be made arbitrarily small to decrease the discretization error, because the number of multiplications needed with the DLPA is proportional to the inverse square of the layer thickness. We propose a numerically extrapolated time domain DLPA (ETDLPA) to resolve this tradeoff between the numerical accuracy and the computational complexity. The accuracy of the proposed ETDLPA is higher than the conventional time domain DLPA (TDLPA) by an order of magnitude or more, with little computational overhead. To be specific, the computational efficiency of the ETDLPA is achieved through numerical extrapolation, and each addition of the extrapolation depth improves the order of accuracy by one. Therefore, the ETDLPA provides us with computationally more efficient and accurate methodology for the nonuniform FBG synthesis than the TDLPA.

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References

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    [CrossRef]
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    [CrossRef]
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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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2010

2009

2005

J. Bae, J. Chun, and T. Kailath, “The Schur algorithm applied to the design of optical multi-mirror structures,” Numer. Linear Algebra Appl. 12, 283–292 (2005).
[CrossRef]

2004

2003

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. Bae and J. Chun, “Numerical optimization approach for designing bandpass filters using fiber Bragg gratings,” Opt. Eng. 42, 23–29 (2003).
[CrossRef]

2001

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

1999

R. Feced, M. N. Zervas, and 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 and K. M. Risvik, “A genetic algorithm for the inverse problem in synthesis of fiber gratings,” J. Light-wave Technol. 16, 1928–1932 (1998).
[CrossRef]

1997

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

1987

A. M. Bruckstein and T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Review 29, 359–389 (1987).
[CrossRef]

1986

A. M. Bruckstein, I. Koltracht, and T. Kailath, “Inverse scattering with noisy data,” SIAM J. Sci. Stat. Comput. 7, 1331–1349 (1986).
[CrossRef]

1985

Aksnes, K.

Asghari, M. H.

Azana, J.

Bae, J.

J. Bae, J. Chun, and T. Kailath, “The Schur algorithm applied to the design of optical multi-mirror structures,” Numer. Linear Algebra Appl. 12, 283–292 (2005).
[CrossRef]

J. Bae and J. Chun, “Numerical optimization approach for designing bandpass filters using fiber Bragg gratings,” Opt. Eng. 42, 23–29 (2003).
[CrossRef]

Bland-Hawthorn, J.

Bruckstein, A. M.

A. M. Bruckstein and T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Review 29, 359–389 (1987).
[CrossRef]

A. M. Bruckstein, I. Koltracht, and T. Kailath, “Inverse scattering with noisy data,” SIAM J. Sci. Stat. Comput. 7, 1331–1349 (1986).
[CrossRef]

Buryak, A.

Chun, J.

J. Bae, J. Chun, and T. Kailath, “The Schur algorithm applied to the design of optical multi-mirror structures,” Numer. Linear Algebra Appl. 12, 283–292 (2005).
[CrossRef]

J. Bae and J. Chun, “Numerical optimization approach for designing bandpass filters using fiber Bragg gratings,” Opt. Eng. 42, 23–29 (2003).
[CrossRef]

Erdogan, T.

J. Skaar, L. 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.

R. Feced, M. N. Zervas, and 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]

Gong, Y.

Hayashi, J.

Haykin, S.

S. Haykin, Mordern Filters (Macmillan Publishing Company, 1990).

Horowitz, M.

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

Hu, X.

Kailath, T.

J. Bae, J. Chun, and T. Kailath, “The Schur algorithm applied to the design of optical multi-mirror structures,” Numer. Linear Algebra Appl. 12, 283–292 (2005).
[CrossRef]

A. M. Bruckstein and T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Review 29, 359–389 (1987).
[CrossRef]

A. M. Bruckstein, I. Koltracht, and T. Kailath, “Inverse scattering with noisy data,” SIAM J. Sci. Stat. Comput. 7, 1331–1349 (1986).
[CrossRef]

T. Kailath, “Signal processing applications of some moment problems,” in Proceedings of Symposia in Applied Mathematics (American Mathmatical Society, 1976), pp. 71–109.

Koltracht, I.

A. M. Bruckstein, I. Koltracht, and T. Kailath, “Inverse scattering with noisy data,” SIAM J. Sci. Stat. Comput. 7, 1331–1349 (1986).
[CrossRef]

Kumagai, T.

Li, H.

Li, M.

Lin, A.

Liu, X.

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. Quantum Electron . 35, 1105–1115 (1999).
[CrossRef]

Ogusu, K.

Risvik, K. M.

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

Rosenthal, A.

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

Shin, S. Y.

Sidi, A.

A. Sidi, Practical Extrapolation Methods (Cambridge University Press, 2003).
[CrossRef]

Skaar, J.

K. Aksnes and J. Skaar, “Design of short fiber Bragg gratings by use of optimization,” Appl. Opt. 43, 2226–2230 (2004).
[CrossRef] [PubMed]

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

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

Song, G. H.

Steblina, V.

Wang, L.

Yao, J.

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. Quantum Electron . 35, 1105–1115 (1999).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber bragg gratings,” IEEE J. Quantum Electron . 39, 1018–1026 (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. Quantum 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. Quantum Electron . 37, 165–173 (2001).
[CrossRef]

J. Light-wave Technol.

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

J. Lightwave Technol.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Numer. Linear Algebra Appl.

J. Bae, J. Chun, and T. Kailath, “The Schur algorithm applied to the design of optical multi-mirror structures,” Numer. Linear Algebra Appl. 12, 283–292 (2005).
[CrossRef]

Opt. Eng.

J. Bae and J. Chun, “Numerical optimization approach for designing bandpass filters using fiber Bragg gratings,” Opt. Eng. 42, 23–29 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

SIAM J. Sci. Stat. Comput.

A. M. Bruckstein, I. Koltracht, and T. Kailath, “Inverse scattering with noisy data,” SIAM J. Sci. Stat. Comput. 7, 1331–1349 (1986).
[CrossRef]

SIAM Review

A. M. Bruckstein and T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Review 29, 359–389 (1987).
[CrossRef]

Other

S. Haykin, Mordern Filters (Macmillan Publishing Company, 1990).

A. Sidi, Practical Extrapolation Methods (Cambridge University Press, 2003).
[CrossRef]

T. Kailath, “Signal processing applications of some moment problems,” in Proceedings of Symposia in Applied Mathematics (American Mathmatical Society, 1976), pp. 71–109.

J. Skaar, “Inverse scattering for one-dimensional periodic optical structures and application to design and characterization,” presented in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, Sydney, Australia, 4–8 July 2005.

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Figures (2)

Fig. 1
Fig. 1

Comparison of the ETDLPA and TDLPA for synthesis of a flat-top bandpass filter: (a) reflectivity, (b) enlarged plot of (a) marked by ellipsoid, (c) coupling coefficient, (d) enlarged plot of (a) marked by ellipsoid.

Fig. 2
Fig. 2

Coupling coefficient computed by ETDLPA(2, h) and TDLPA(h) for the third order Butterworth filter when the layer thickness h is 0.1.

Tables (2)

Tables Icon

Table 1 Comparison of the ETDLPA and TDLPA for Synthesis of the Third Order Butterworth Filter: Normalized Error and Involved Multiplications, h=0.1

Tables Icon

Table 2 Comparison of the ETDLPA and TDLPA for Synthesis of the Third Order Butterworth Filter: Normalized Error and Involved Multiplications, h=0.01

Equations (48)

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z = [ U ( z , δ ) V ( z , δ ) ] = [ i δ q ( z ) q * ( z ) i δ ] [ U ( z , δ ) V ( z , δ ) ] ,
q ^ ( z , h ) = q ( z ) + l = 1 a l h l ,
q ^ ( z , 2 h ) = q ( z ) + l = 1 a l 2 l h l .
2 q ^ ( z , h ) q ^ ( z , 2 h ) = q ( z ) 2 a 2 h 2 6 a 3 h 3 + .
q ^ ( z , m h ) = q ( z ) + l = 1 a l m l h l , 1 m P + 1.
m = 1 P + 1 ω m q ^ ( z , m h ) = q ( z ) + l = P + 1 a k h k ,
m = 1 P + 1 ω m q ^ ( z , m h ) = m = 1 P + 1 ω m ( q ( z ) + l = 1 a l m l h l ) = q ( z ) m = 1 P + 1 ω m + l = 1 P h l a l m = 1 P + 1 ω m m l + l = P + 1 h l a l m = 1 P + 1 ω m m l .
A P ω P = e 1 ,
A P = [ 1 1 1 1 1 2 1 3 1 ( P + 1 ) 2 2 3 2 ( P + 1 ) 2 2 P 3 P ( P + 1 ) P ] , ω P = [ ω 1 ω 2 ω 3 ω P + 1 ] , e 1 = [ 1 0 0 0 ] .
ω 2 T = [ ω 1 ω 2 ω 3 ] = [ 3 3 1 ]
q ^ ( z , h , 0 ) : 0 , h , 2 h , 3 h , 4 h , 5 h , 6 h , 7 h , 8 h , 9 h , 10 h , 11 h , 12 h , 13 h ,
q ^ ( z , 2 h , 0 ) : 0 , 2 h , 4 h , 6 h , 8 h , 10 h , 12 h , 14 h ,
q ^ ( z , 3 h , 0 ) : 0 , 3 h , 6 h , 9 h , 12 h , 15 h , .
q ( z ) | z = ( 6 l ) h = m = 1 3 ω m q ^ ( z , m h , 0 ) | z = ( 6 l ) h , l = 0 , 1 , .
q ( z ) | z = ( 6 l + 1 ) h = m = 1 3 ω m q ^ ( z , m h , h ) | z = ( 6 l + 1 ) h , l = 0 , 1 , .
q ^ ( z , h , h ) : h , 2 h , 3 h , 4 h , 5 h , 6 h , 7 h , 8 h , 8 h , 9 h , 10 h , 11 h , 12 h , 13 h , 14 h ,
q ^ ( z , 2 h , h ) : h , 3 h , 5 h , 7 h , 9 h , 11 h , 13 h , 15 h ,
q ^ ( z , 3 h , h ) : h , 4 h , 7 h , 10 h , 13 h , 16 h ,
q ( z ) | z = ( L l + s ) h = m = 1 P + 1 ω m q ^ ( z , m h , s h ) | z = ( L l + s ) h , l = 0 , 1 , ,
m = 1 P + 1 m ( N 2 m 2 + N m ) + L ( P + 1 ) N = N 2 m = 1 P + 1 1 m + ( L + 1 ) ( P + 1 ) N
g ( t ) = A sin ( 2 π B t ) π t , G ( δ ) = A   rect   ( δ B )
rect ( x ) = { 1 , for | x | 1 0 , for | x | > 1 ,
z | q ( z ) q ^ ( z ) | z | q ( z ) | ,
G ( s ) = r 0 l = 1 3 ( s α l ) , α 1 = exp [ i π 2 ( 1 + 2 l 1 3 ) ] .
[ U ( z + h , δ ) V ( z + h , δ ) ] = F [ U ( z , δ ) V ( z , δ ) ] ,
F = [ cosh ( γ h ) + i δ γ sinh ( γ h ) q * γ sinh ( γ h ) q γ sinh ( γ h ) cosh ( γ h ) i δ γ sinh ( γ h ) ]
F F ^ = T R ,
T = [ exp ( i δ h ) 0 0 exp ( i δ h ) ] , R = [ cosh ( | q | h ) q * | q | sinh ( | q | h ) q | q | sinh ( | q | h ) cosh ( | q | h ) ] .
F 11 = cosh ( γ h ) + i δ γ sinh ( γ h ) = ( 1 + γ 2 h 2 2 + O ( h 4 ) ) + i δ γ ( γ h + O ( h 3 ) ) = 1 + i δ h + γ 2 h 2 2 + O ( h 3 )
F ^ 11 = exp ( i δ h ) cosh ( | q | h ) = ( 1 + i δ h δ 2 h 2 2 + O ( h 3 ) ) ( 1 + | q | 2 h 2 2 + O ( h 4 ) ) = 1 + i δ h + γ 2 h 2 2 + O ( h 3 ) ,
F 21 = q γ sinh ( γ h ) = q γ ( γ h + O ( h 3 ) ) = q h + O ( h 3 ) ,
F ^ 21 = exp ( i δ h ) q | q | sinh ( | q | h ) = ( 1 + O ( h ) ) ( q | q | ( | q | h + O ( h 3 ) ) ) = q h + O ( h 2 ) ,
R = ( 1 | κ ( z ) | 2 ) 1 / 2 [ 1 κ * ( z ) κ ( z ) 1 ] ,
κ ( z ) = tanh ( h | q ( z ) | q * ( z ) | q ( z ) | .
κ ( z ) = h q * ( z ) + O ( h 3 ) .
[ U ^ h ( z + h , δ ) V ^ h ( z + h , δ ) ] = [ exp ( i δ h ) 0 0 exp ( i δ h ) ] ( 1 | h q ( z ) | 2 ) 1 / 2 [ 1 h q ( z ) h q * ( z ) 1 ] [ U ( z , δ ) V ( z , δ ) ] ,
[ u ^ h ( z + h , t + h ) v ^ h ( z + h , t h ) ] = ( 1 | h q ( z ) | 2 ) 1 / 2 [ 1 h q ( z ) h q * ( z ) 1 ] [ u ( z , t ) v ( z , t ) ] ,
u ^ h ( z + h , t + h ) = u ( z + h , t + h ) + l = 2 α l h l ,
v ^ h ( z + h , t + h ) = v ( z + h , t + h ) + l = 2 β l h l ,
h q ^ ( z + h , h ) = v ^ h ( z + h , z + h ) u ^ h ( z + h , z + h ) ,
h q ^ ( z + h , h ) = v ( z + h , t + h ) u ( z + h , t + h ) + l = 2 γ l h l ,
v ( z + h , t h ) = 0.
v ( z + h , t h ) = v ( z , t ) + h v ( 1 ) ( z , t ) + l = 2 h l l ! v ( l ) ( z , t ) ,
v ( n ) ( z , t ) = ( z t ) n v ( z , t ) .
v ( 1 ) ( z , t ) = ( z t ) v ( z , t ) = q ( z ) u ( z , t ) .
h q ( z ) = v ( z , t ) u ( z , t ) + l = 2 h l l ! v ( 1 ) ( z , t ) u ( z , t ) .
h q ( z + h ) = v ( z + h , t + h ) u ( z + h , t + h ) + l = 2 h l l ! v ( l ) ( z + h , t + h ) u ( z + h , t + h ) .
h q ^ ( z + h , h ) = h q ( z + h ) + l = 2 ξ l h l ,

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