## Numerically extrapolated discrete layer-peeling algorithm for synthesis of nonuniform fiber Bragg gratings |

Optics Express, Vol. 19, Issue 9, pp. 8254-8266 (2011)

http://dx.doi.org/10.1364/OE.19.008254

Acrobat PDF (896 KB)

### 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.

© 2011 OSA

## 1. Introduction

1. G. H. Song and S. Y. Shin, “Design of corrugated waveguide filters by the Gel’fand-Levitan-Marchenko inverse-scattering method,” J. Opt. Soc. Am. A **2**, 1905–1915 (1985). [CrossRef]

2. 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]

4. 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]

5. 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]

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

2. 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]

3. 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]

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

15. M. H. Asghari and J. Azana, “On the Design of Efficient and Accurate Arbitrary-Order Temporal Optical Integrators Using Fiber Bragg Gratings,” J. Lightwave Technol. **27**, 3888–3895 (2009). [CrossRef]

3. 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]

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

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

3. 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]

2. 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]

**37**, 165–173 (2001). [CrossRef]

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

## 2. ETDLPA

**35**, 1105–1115 (1999). [CrossRef]

*U*(

*z,δ*) is the forward propagating wave,

*V*(

*z,δ*) is the backward propagating wave,

*q*(

*z*) is the coupling coefficient at the grating position

*z*, and

*δ*is the detuning factor. The goal of the synthesis of nonuniform FBG is to find

*q*(

*z*) given the desired spectrum described by the scattering data

*U*(0

*,δ*) and

*V*(0

*,δ*).

**37**, 165–173 (2001). [CrossRef]

*h*. The error formula of the TDLPA with the layer thickness

*h*is given by the power series of

*h*, where

*q̂*(

*z,h*) is the coupling coefficient computed by the TDLPA with the layer thickness

*h*and

*a*

*is independent with*

_{l}*h*. A detailed derivation of Eq. (2) is given in appendix A. It is clear from Eq. (2) that the accuracy of the TDLPA is

*O*(

*h*). The key feature of the ETDLPA is to eliminate the remained error terms in Eq. (2) by combining the results of the TDLPA for various layer thickness values through a numerical extrapolation procedure. For example, the result of the TDLPA with the layer thickness 2

*h*can be written from Eq. (2) as follows,

*a*

_{1}

*h*of Eq. (2) is removed in Eq. (4), and thus, the accuracy order is increased from

*O*(

*h*) to

*O*(

*h*

^{2}). This observation shows that additional information for the coupling coefficient computed by the TDLPA with under-sampled scattering data can improve the accuracy of the TDLPA. Note that the scattering data with the layer thickness 2

*h*is just two times under-sampled data of the scattering data with the layer thickness

*h*. In the TDLPA, the accuracy

*O*(

*h*

^{2}) is obtained with the layer thickness of

*h*

^{2}. If the number of layers with the layer thickness

*h*is

*N*, the number of layers with the layer thickness

*h*

^{2}is

*N/h*. It is easy to become aware that the computational complexity of Eq. (4) can be remarkably lower than that of

*q̂*(

*z,h*

^{2}) to achieve the accuracy

*O*(

*h*

^{2}) if

*h*is sufficiently small. Note that the value of

*a*

*does not need for the extrapolation. The computational efficiency of the ETDLPA will be detailed at the end of this section.*

_{l}*P*-stage extrapolation procedure of the ETDLPA can be formulated using

*q̂*(

*z,h*)

*,q̂*(

*z,*2

*h*)

*,··· ,q̂*(

*z,*(

*P*+1)

*h*) to obtain the accuracy

*O*(

*h*

^{P}^{+1}). From Eq. (2), the result of the TDLPA with the layer thickness

*mh*can be written by Because the aim of the

*P*-stage extrapolation procedure is to eliminate

*a*

_{1}

*h, a*

_{2}

*h*

^{2}

*, ··· ,a*terms in Eq. (2) by combining

_{P}h^{P}*q̂*(

*z,h*)

*,q̂*(

*z,*2

*h*)

*,··· ,q̂*(

*z,*(

*P*+ 1)

*h*), we get where

*ω*is the weight to be determined and

_{m}*h*. Inserting Eq. (5) into the left-hand side of Eq. (6), we get

*ω*can be obtained by solving the following system of linear equations, where

_{m}*q̂*(

*z,mh,sh*) to denote the coupling coefficient computed by the TDLPA with the layer thickness

*mh*and the starting position

*sh*. The reconstructed grating positions of

*q̂*(

*z,mh,sh*) are (

*s*+

*lm*)

*h,l*= 0

*,*1

*,*···. For convenience, we denote the

*P*-stage ETDLPA with the layer thickness

*h*and the TDLPA with the layer thickness

*h*by ETDLPA(

*P*,

*h*) and TDLPA(

*h*), respectively. As an example to explain the grating position problem, let us consider ETDLPA(2,

*h*). ETDLPA(2,

*h*) combines TDLPA(

*h*), TDLPA(2

*h*) and TDLPA(3

*h*) according to Eq. (6). The solution of Eq. (8) for

*P*= 2 is given by

*q̂*(

*z,h,*0),

*q̂*(

*z,*2

*h,*0), and

*q̂*(

*z,*3

*h,*0) are given by The functions

*q̂*(

*z,*2

*h,*0) and

*q̂*(

*z,*3

*h,*0) do not have values at the grating position of

*h*and we cannot get the extrapolated value at this grating position using

*q̂*(

*z,h,*0),

*q̂*(

*z,*2

*h,*0), and

*q̂*(

*z,*3

*h,*0). For grating positions except for the multiples of 6

*h*, we can not perform the extrapolation procedure because the coupling coefficient at these grating positions is not available for at least one layer thickness. Therefore, we can perform the extrapolation process only at the grating positions 6

*lh,l*= 0

*,*1

*,*···, denoted by the the bold face in Eqs. (11)–(13), using

*q̂*(

*z,h,*0),

*q̂*(

*z,*2

*h,*0), and

*q̂*(

*z,*3

*h,*0) as

*l*+ 1)

*h,l*= 0

*,*1

*,*2

*,···*, we need to move the starting position from 0 to

*h*. Similarly to the previous discussion about the starting position 0 to compute the extrapolated coupling coefficients at the grating positions 6

*lh,l*= 0

*,*1

*,···*, we can get the extrapolated value at the grating positions (6

*l*+ 1)

*h,l*= 0

*,*1

*,*2

*,···*using

*q̂*(

*z,h,h*),

*q̂*(

*z,*2

*h,h*), and

*q̂*(

*z,*3

*h,h*) as Note that the reconstructed grating positions of

*q̂*(

*z,h,h*),

*q̂*(

*z,*2

*h,h*), and

*q̂*(

*z,*3

*h,h*) are given by In general, the grating positions of

*q̂*(

*z,h,sh*)

*,q̂*(

*z,*2

*h,sh*)

*,··· ,q̂*(

*z,*(

*P*+ 1)

*h,sh*) coincide only at

*z*= (

*s*+

*lL*)

*h,l*= 0

*,*1

*,···*, where

*L*is the least common multiple of 1

*,*2

*,··· ,P*+ 1, i.e.,

*L*= LCM(1

*,*2

*,··· ,P*+ 1). Therefore, one execution of the extrapolation procedure in the ET-DLPA can provide the coupling coefficient at equally

*Lh*-spaced grating positions. For the reconstruction of the coupling coefficient at all grating positions, the extrapolation procedure of the ETDLPA should be performed

*L*-times, with the different starting positions spanning from 0 to (

*L*– 1)

*h*. The

*P*-stage extrapolation using

*q̂*(

*z,mh,sh*)

*,m*= 1

*,*2

*,··· ,P*+ 1 is given by with the accuracy

*O*(

*h*

^{P}^{+1}). Now ETDLPA(

*P*,

*h*) can be summarized as follows.

**Algorithm 1. ETDLPA(**

*P*

**,**

*h*

**)**

**Set**

*P*and

*h*;

*L*= LCM(1

*,*2

*,··· ,P*+ 1) ;

**For**s = 0,1,··· ,

*L*– 1

**For**

*m*= 1

*,*2

*,··· ,P*+ 1

*q̂*(

*z,mh,sh*);

**End**

**End**

*N*-layers involves

*N*

^{2}+

*N*multiplications. The

*P*-stage ETDLPA with

*N*-layers involves multiplications. In the left-hand side of Eq. (20), the first term and the second term are the number of the multiplications to compute

*q̂*(

*z,mh,sh*) and Eq. (19), respectively. If we compare the dominant

*N*

^{2}-term of Eq. (20) with that of the TDLPA, the number of the involved multiplications of the ETDLPA is increased by the factor of

*P*. The increment of the computational cost of the ETDLPA compared to the TDLPA is negligible because the involved layer thickness values during the extrapolation procedure are larger than the given layer thickness

*h*. For example to illustrate the computationally efficient feature of the proposed ETDLPA, let us consider

*h*= 10

^{–3}. The layer thickness of the TDLPA should be

*h*

^{2}to achieve the accuracy

*O*(

*h*

^{2}), and then the number of layers of the TDLPA is increased to 10

^{3}

*N*. Therefore, we need 10

^{6}

*N*

^{2}+ 10

^{3}

*N*multiplications to get accuracy

*O*(

*h*

^{2}) using the TDLPA. However, the accuracy

*O*(

*h*

^{2}) can be obtained through ETDLPA(1,

*h*), and the involved multiplications of ETDLPA(1,

*h*) are just

*h*). The larger extrapolation stage better reveals the excellence of the ETDLPA. The involved multiplications of ETDLPA(2,

*h*) are

*O*(

*h*

^{3}). But TDLPA(

*h*

^{3}) requires 10

^{12}

*N*

^{2}+ 10

^{6}

*N*multiplications to get the accuracy

*O*(

*h*

^{3}).

## 3. Numerical examples

### 3.1. Nonuniform FBG synthesis example: flat-top bandpass filter

*g*(

*t*) and the spectral response

*G*(

*δ*), with 0 <

*A*≤ 1, where

*B*is the bandwidth. In this example, the grating length is 3.3 cm,

*A*= 0.9,

*B*= 1 nm, and

*h*= 0.005 cm. The impulse response was time-shifted and apodized by a Hann function as done in [2

**35**, 1105–1115 (1999). [CrossRef]

**37**, 165–173 (2001). [CrossRef]

*h*) with TDLPA(

*h*). The result of TDLPA(

*h/*1000) is also plotted to show the enhanced accuracy of the ETDLPA in Fig. 1. For a fair comparison, the coupling coefficient of TDLPA(

*h/*1000) is under-sampled 1,000-times and the reflectivity is plotted by using this under-sampled coupling coefficient. The part of Figs. 1(a) and 1(c) marked by ellipsoid enlarged to demonstrate clearly the improved accuracy of the ETDLPA at Figs. 1(b) and 1(d), respectively. The target reflectivity shown in Figs. 1(a) and 1(b) is plotted from the time-shifted and apodized impulse response. ETDLPA(2,

*h*) shows more accurate synthesis result than TDLPA(

*h*) as seen from Fig. 1(b). Also, the reflectivity of ETDLPA(2,

*h*) is even closer to the target reflectivity than TDLPA(

*h/*1000), but the involved multiplications of ETDLPA(2,

*h*) are only 2 × 10

^{–6}of that of TDLPA(

*h/*1000). Therefore, the ETDLPA can synthesize nonuniform FBG in a computationally efficient manner with high accuracy. Note that the computational complexity of ETDLPA(2,

*h*) is just two times of that of TDLPA(

*h*). In Figs. 1(c) and 1(d), we compare the accuracy of the synthesized coupling coefficient by using the result of the TDLPA with smaller layer thickness, although we can not know the exact coupling coefficient for the target reflectivity. It can be seen in Fig. 1(d) that the coupling coefficient reconstructed by ETDLPA(2,

*h*) is nearly same as that of TDLPA(

*h*/1,000). But it shows a gap up to 0.2 in the coupling coefficient between TDLPA(

*h*/1,000) and TDLPA(

*h*). The shape of the coupling coefficient is similar to the impulse response as mentioned in [1

1. G. H. Song and S. Y. Shin, “Design of corrugated waveguide filters by the Gel’fand-Levitan-Marchenko inverse-scattering method,” J. Opt. Soc. Am. A **2**, 1905–1915 (1985). [CrossRef]

### 3.2. Third order Butterworth filter

1. G. H. Song and S. Y. Shin, “Design of corrugated waveguide filters by the Gel’fand-Levitan-Marchenko inverse-scattering method,” J. Opt. Soc. Am. A **2**, 1905–1915 (1985). [CrossRef]

*q*(

*z*) is the exact coupling coefficient and

*q̂*(

*z*) is the coupling coefficient computed by the TDLPA or the ETDLPA.

*g*(

*t*) can be computed by the inverse Laplace transform of

*G*(

*s*). The length of the grating is 6 m and

*h*= 0.1 are summarized in Table 1. TDLPA(

*h*

^{2}) and ETDLPA(1,

*h*) have the same accuracy

*O*(

*h*

^{2}) and are grouped in Table 1. Similarly, TDLPA(

*h*

^{3}) and ETDLPA(2,

*h*) are grouped in Table 1 because those have the same accuracy

*O*(

*h*

^{3}). The validity of Eq. (2) is verified indirectly in this table by observing the trend of the normalized error of TDLPA(

*h*), TDLPA(

*h*

^{2}), and TDLPA(

*h*

^{3}). The normalized error of the TDLPA is reduced exactly by the decreasing factor of the layer thickness, which corresponds to Eq. (2). For example, the layer thickness and the normalized error of TDLPA(

*h*

^{2}) are 1/10 of those of TDLPA(

*h*), respectively. Also, it is observed in Table 1 that the accuracy order of the ETDLPA is properly increased according to Eq. (6). The normalized error of ETDLPA(1,

*h*) is reduced to 1.1 × 10

^{–2}from 7.1 × 10

^{–2}which is the normalized error of DLPA(

*h*), and the normalized error of ETDLPA(2,

*h*) is reduced to 3.2 × 10

^{–3}from 1.1 × 10

^{–2}which is the normalized error of ETDLPA(1,

*h*). Although we can conclude in this table that ETDLPA(

*P*,

*h*) and TDLPA(

*h*

^{P}^{+1}) have the same accuracy, it can be seen that the normalized error of ETDLPA(1,

*h*) and ETDLPA(2,

*h*) are slightly larger than that of TDLPA(

*h*

^{2}) and TDLPA(

*h*

^{3}), respectively. This differences are resulted from the different representation of the remained error terms. For example, the dominant error term of TDLPA(

*h*

^{2}) and ETDLPA(1,

*h*) are

*a*

_{1}

*h*

^{2}and −2

*a*

_{2}

*h*

^{2}, respectively. The computational efficiency of the ETDLPA is cleared by comparing the involved multiplications of the ETDLPA and the TDLPA in Table 1. The multiplications of ETDLPA(1,

*h*) are less than those of TDLPA(

*h*

^{2}) by the factor of 1/64, in spite of the fact that ETDLPA(1,

*h*) and TDLPA(

*h*

^{2}) have the same accuracy. This computational efficiency of the ETDLPA is more increased for higher

*P*as seen by comparing the multiplications of ETDLPA(2,

*h*) and TDLPA(

*h*

^{3}) in Table 1. The coupling coefficient computed by ETDLPA(2,

*h*) and TDLPA(

*h*) are plotted in Fig. 2 for the illustrative purpose. We can see the large gap between the TDLPA(

*h*) and the exact value, but the result of ETDLPA(2,

*h*) can not be discriminated from the exact value with the naked eye in this figure. Also, we can see, again, the similar shape between the computed coupling coefficients and the impulse response as mentioned in [1

**2**, 1905–1915 (1985). [CrossRef]

*h*) using the latest PC is less than one second, but the running time of TDLPA(

*h*

^{3}) is greater than 168 hours.

## 4. Conclusions

## Appendix A Error formula of the TDLPA

**35**, 1105–1115 (1999). [CrossRef]

**37**, 165–173 (2001). [CrossRef]

*h*is given by [2

**35**, 1105–1115 (1999). [CrossRef]

**37**, 165–173 (2001). [CrossRef]

*F*is and

*γ*

^{2}= |

*q*|

^{2}–

*δ*

^{2}. To discretize Eq. (1), the approximation of

*F*is given by [2

**35**, 1105–1115 (1999). [CrossRef]

*A*is the (

_{ij}*i, j*) entry of the matrix

*A*. Therefore, the accuracy of Eq. (27) is

*O*(

*h*

^{2}). The matrix

*R*can be rewritten into the familiar hyperbolic rotation matrix [3

**37**, 165–173 (2001). [CrossRef]

*R*explains the interaction of the waves using the reflection coefficient

*κ*(

*z*) at the grating position

*z*, and the matrix

*T*describes the propagation of the waves in the homogeneous interval (

*z,z*+

*h*). The reflection coefficient

*κ*(

*z*) can be approximated using the Taylor series expansion of the hyperbolic tangent function in Eq. (34) as follows, Note that the accuracy of Eq. (35) does not degrade the accuracy of Eq. (27) because the accuracy order of Eq. (35) is higher than that of Eq. (27).

*Û*and

_{h}*V̂*are used to discriminate between the approximated wave variables computed by Eq. (36) with the layer thickness

_{h}*h*and the exact wave variables

*U*and

*V*, and the accuracy of this discretization is

*O*(

*h*

^{2}).

**37**, 165–173 (2001). [CrossRef]

*u*,

*v*,

*û*and

_{h}*v̂*are the inverse fourier transform of

_{h}*U*,

*V*,

*Û*and

_{h}*V̂*, respectively, and the accuracy of this discretization is also

_{h}*O*(

*h*

^{2}). Using Eq. (37), we can compute the discretized wave variables

*û*and

_{h}*v̂*with the accuracy

_{h}*O*(

*h*

^{2}), and thus, we can write as follows, where

*α*and

_{l}*β*are independent with

_{l}*h*. These power series representations follow from the Taylor series expansion used for the derivation of Eq. (36) and Eq. (37). From the causality of the wave propagation, the reflection coefficient can be computed from Eq. (37) as follows [17, 18

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

*q̂*(

*z*+

*h,h*) is the coupling coefficient reconstructed by the TDLPA with the layer thickness

*h*at the position

*z*+

*h*. Inserting Eq. (38) and Eq. (39) into Eq. (40), it is easy to see that we can rewrite Eq. (40) as follows, where

*γ*is independent with

_{l}*h*.

*q*(

*z*) in Eq. (41). If the arrival time of the right-propagating wave

*u*at position

*z*is

*t*, the left-propagating wave

*v*cannot exist at position

*z*+

*h*for a time lower than

*t*due to the causality of the wave propagation. Therefore, we get The Taylor series for

*v*(

*z*+

*h,t – h*) at (

*z,t*) is where From the space-time representation of Eq. (1), we know that Inserting Eq. (42) and Eq. (45) into Eq. (43), Eq. (43) can be rewritten as Therefore, In Eq. (46), it is interesting to note that

*v*(

*z,t*) to the incident wave

*u*(

*z,t*), and thus

*hq*(

*z*) approximates the reflection coefficient with accuracy

*O*(

*h*

^{2}). Inserting Eq. (47) into Eq. (41), we get where

*ξ*is independent with

_{l}*h*. Dividing the both sides of Eq. (48) by

*h*, we can obtain Eq. (2). Note that the explicit representation of

*α*and

_{l},β_{l},γ_{l}*ξ*are not detailed because these values do not need for a numerical extrapolation. It is clear to see that Eq. (2) also holds for the piecewise uniform model by the concatenation of the transfer matrices. The TDLPA is consisted of Eq. (37) and Eq. (40). The accuracy of

_{l}*F̂*is

*O*(

*h*

^{3}) if the discrete reflector is placed at the center of the layer as described in [2

**35**, 1105–1115 (1999). [CrossRef]

*F̂*is increased from

*O*(

*h*

^{2}) to

*O*(

*h*

^{3}), it cannot increase the accuracy order of Eq. (2) because the accuracy of Eq. (46) is

*O*(

*h*

^{2}). Moreover, Eq. (37) is appropriate for the general identification problem of the local reflectivity in spite of the lower accuracy order by one.

*O*(

*N*) computing time units with

*N*-processors [17–19], where

*N*is the total number of layers. See the Chapter 10 of [19] for the detailed procedure of the parallel implementation of the TDLPA.

## Acknowledgments

## References and links

1. | G. H. Song and S. Y. Shin, “Design of corrugated waveguide filters by the Gel’fand-Levitan-Marchenko inverse-scattering method,” J. Opt. Soc. Am. A |

2. | 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 . |

3. | J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber bragg gratings by layer peeling,” IEEE J. Quantum Electron . |

4. | J. Bae, J. Chun, and T. Kailath, “The Schur algorithm applied to the design of optical multi-mirror structures,” Numer. Linear Algebra Appl. |

5. | J. Skaar and K. M. Risvik, “A genetic algorithm for the inverse problem in synthesis of fiber gratings,” J. Light-wave Technol. |

6. | J. Bae and J. Chun, “Numerical optimization approach for designing bandpass filters using fiber Bragg gratings,” Opt. Eng. |

7. | K. Aksnes and J. Skaar, “Design of short fiber Bragg gratings by use of optimization,” Appl. Opt. |

8. | A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber bragg gratings,” IEEE J. Quantum Electron . |

9. | 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. |

10. | H. Li, T. Kumagai, and K. Ogusu, “Advanced design of a multichannel fiber Bragg grating based on a layer-peeling method,” J. Opt. Soc. Am. B |

11. | M. Li, J. Hayashi, and H. Li, “Advanced design of a complex fiber Bragg grating for a multichannel asymmetrical triangular filter,” J. Opt. Soc. Am. B |

12. | A. Buryak, J. Bland-Hawthorn, and V. Steblina, “Comparison of Inverse Scattering Algorithms for Designing Ultrabroadband Fibre Bragg Gratings,” Opt. Express |

13. | Y. Gong, A. Lin, X. Hu, L. Wang, and X. Liu, “Optimal method of designing triangular-spectrum fiber Bragg gratings with low index modulation and chirp-free structure,” J. Opt. Soc. Am. B |

14. | M. Li and J. Yao, “All-fiber temporal photonic fractional Hilbert transformer based on a directly designed fiber Bragg grating,” Opt. Lett. |

15. | M. H. Asghari and J. Azana, “On the Design of Efficient and Accurate Arbitrary-Order Temporal Optical Integrators Using Fiber Bragg Gratings,” J. Lightwave Technol. |

16. | A. M. Bruckstein, I. Koltracht, and T. Kailath, “Inverse scattering with noisy data,” SIAM J. Sci. Stat. Comput. |

17. | T. Kailath, “Signal processing applications of some moment problems,” in |

18. | A. M. Bruckstein and T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Review |

19. | S. Haykin, |

20. | T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. |

21. | A. Sidi, |

**OCIS Codes**

(290.3200) Scattering : Inverse scattering

(070.2025) Fourier optics and signal processing : Discrete optical signal processing

(060.3735) Fiber optics and optical communications : Fiber Bragg gratings

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: February 17, 2011

Revised Manuscript: March 19, 2011

Manuscript Accepted: March 26, 2011

Published: April 14, 2011

**Citation**

Youngchol Choi, Joohwan Chun, and Jinho Bae, "Numerically extrapolated discrete layer-peeling algorithm for synthesis of nonuniform fiber Bragg gratings," Opt. Express **19**, 8254-8266 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8254

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### References

- G. H. Song and S. Y. Shin, “Design of corrugated waveguide filters by the Gel’fand-Levitan-Marchenko inverse-scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985). [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. 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. 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. Bae and J. Chun, “Numerical optimization approach for designing bandpass filters using fiber Bragg gratings,” Opt. Eng. 42, 23–29 (2003). [CrossRef]
- K. Aksnes and J. Skaar, “Design of short fiber Bragg gratings by use of optimization,” Appl. Opt. 43, 2226–2230 (2004). [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]
- 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.
- H. Li, T. Kumagai, and K. Ogusu, “Advanced design of a multichannel fiber Bragg grating based on a layer-peeling method,” J. Opt. Soc. Am. B 21, 1929–1938 (2004). [CrossRef]
- M. Li, J. Hayashi, and H. Li, “Advanced design of a complex fiber Bragg grating for a multichannel asymmetrical triangular filter,” J. Opt. Soc. Am. B 26228–234 (2009). [CrossRef]
- A. Buryak, J. Bland-Hawthorn, and V. Steblina, “Comparison of Inverse Scattering Algorithms for Designing Ultrabroadband Fibre Bragg Gratings,” Opt. Express 17, 1995–2004 (2009). [CrossRef] [PubMed]
- Y. Gong, A. Lin, X. Hu, L. Wang, and X. Liu, “Optimal method of designing triangular-spectrum fiber Bragg gratings with low index modulation and chirp-free structure,” J. Opt. Soc. Am. B 26, 1042–1048 (2009). [CrossRef]
- M. Li and J. Yao, “All-fiber temporal photonic fractional Hilbert transformer based on a directly designed fiber Bragg grating,” Opt. Lett. 35, 223–225 (2010). [CrossRef] [PubMed]
- M. H. Asghari and J. Azana, “On the Design of Efficient and Accurate Arbitrary-Order Temporal Optical Integrators Using Fiber Bragg Gratings,” J. Lightwave Technol. 27, 3888–3895 (2009). [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.
- A. M. Bruckstein and T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Review 29, 359–389 (1987). [CrossRef]
- S. Haykin, Mordern Filters (Macmillan Publishing Company, 1990).
- T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997). [CrossRef]
- A. Sidi, Practical Extrapolation Methods (Cambridge University Press, 2003). [CrossRef]

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