Abstract

We develop a novel method that enables one to reconstruct the structure of highly reflecting fiber Bragg gratings from noisy reflection spectra. When the reflection spectrum is noisy and the grating reflectivity is high, noise in the Bragg zone of the reflection spectrum is amplified by the inverse scattering algorithms and prevents the reconstruction of the grating. Our method is based on regularizing the reflection spectrum in frequencies inside the Bragg zone by using the data on the grating spectrum outside the Bragg zone. The regularized reflection spectrum is used to reconstruct the grating structure by means of inverse scattering. Our method enables one to analyze gratings with a high reflectivity from a spectrum that contains a high level of noise. Such gratings could not be analyzed by using methods described in previous work [IEEE J. Quantum Electron. 39, 1238 (2003)].

© 2005 Optical Society of America

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  1. E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’fan-Levitan-Marchenko coupled equations,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
    [CrossRef]
  2. 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]
  3. L. Poladian, “Simple grating synthesis algorithm,” Opt. Lett. 25, 787–789 (2000).
    [CrossRef]
  4. J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Lightwave Technol. 37, 165–173 (2001).
  5. A. Rosenthal, M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
    [CrossRef]
  6. S. Keren, A. Rosenthal, M. Horowitz, “Measuring the structure of highly reflecting fiber Bragg gratings,” IEEE Photonics Technol. Lett. 15, 575–577 (2003).
    [CrossRef]
  7. A. M. Bruckstein, I. Koltracht, T. Kailath, “Inverse scattering with noisy data,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 7, 1331–1349 (1986).
    [CrossRef]
  8. J. Skaar, R. Feced, “Reconstruction of gratings from noisy reflection data,” J. Opt. Soc. Am. A 19, 2229–2237 (2002).
    [CrossRef]
  9. J. Skaar, O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
    [CrossRef]
  10. A. Rosenthal, M. Horowitz, “New technique to accurately interpolate the complex reflection spectrum of fiber Bragg gratings,” IEEE J. Quantum Electron. 40, 1099–1104 (2004).
    [CrossRef]
  11. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  12. L. Poladian, “Group-delay reconstruction for fiber Bragg gratings in reflection and transmission,” Opt. Lett. 22, 1571–1573 (1997).
    [CrossRef]
  13. A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962).
  14. M. J. Ablowitz, A. S. Fokas, Complex Variables (Cambridge U. Press, Cambridge, UK, 1997).
  15. L. Debnath, P. Mikusiéski, Introduction to Hilbert Spaces with Applications (Academic, San Diego, Calif., 1990).
  16. G. H. Golub, C. F. Van Ioan, Matrix Computations (The Johns Hopkins U. Press, Baltimore, Md., 1996).
  17. B. Porat, Digital Processing of Random Signals: Theory and Methods (Prentice Hall, Englewood Cliffs, N.J., 1994).
  18. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1966).
  19. M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Applied Mathematics, Philadelphia, Pa., 1981).
  20. L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
    [CrossRef]
  21. T. Kailath, Linear Systems (Prentice Hall, Englewood Cliffs, N.J., 1980).

2004 (1)

A. Rosenthal, M. Horowitz, “New technique to accurately interpolate the complex reflection spectrum of fiber Bragg gratings,” IEEE J. Quantum Electron. 40, 1099–1104 (2004).
[CrossRef]

2003 (3)

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

S. Keren, A. Rosenthal, M. Horowitz, “Measuring the structure of highly reflecting fiber Bragg gratings,” IEEE Photonics Technol. Lett. 15, 575–577 (2003).
[CrossRef]

J. Skaar, O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[CrossRef]

2002 (1)

2001 (1)

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Lightwave Technol. 37, 165–173 (2001).

2000 (1)

1999 (1)

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]

1997 (2)

1996 (1)

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’fan-Levitan-Marchenko coupled equations,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

1993 (1)

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
[CrossRef]

1986 (1)

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

Ablowitz, M. J.

M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Applied Mathematics, Philadelphia, Pa., 1981).

M. J. Ablowitz, A. S. Fokas, Complex Variables (Cambridge U. Press, Cambridge, UK, 1997).

Bruckstein, A. M.

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

Capmany, J.

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’fan-Levitan-Marchenko coupled equations,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1966).

Debnath, L.

L. Debnath, P. Mikusiéski, Introduction to Hilbert Spaces with Applications (Academic, San Diego, Calif., 1990).

Erdogan, T.

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Lightwave Technol. 37, 165–173 (2001).

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

Feced, R.

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

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]

Fokas, A. S.

M. J. Ablowitz, A. S. Fokas, Complex Variables (Cambridge U. Press, Cambridge, UK, 1997).

Golub, G. H.

G. H. Golub, C. F. Van Ioan, Matrix Computations (The Johns Hopkins U. Press, Baltimore, Md., 1996).

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1966).

Horowitz, M.

A. Rosenthal, M. Horowitz, “New technique to accurately interpolate the complex reflection spectrum of fiber Bragg gratings,” IEEE J. Quantum Electron. 40, 1099–1104 (2004).
[CrossRef]

S. Keren, A. Rosenthal, M. Horowitz, “Measuring the structure of highly reflecting fiber Bragg gratings,” IEEE Photonics Technol. Lett. 15, 575–577 (2003).
[CrossRef]

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

Kailath, T.

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

T. Kailath, Linear Systems (Prentice Hall, Englewood Cliffs, N.J., 1980).

Keren, S.

S. Keren, A. Rosenthal, M. Horowitz, “Measuring the structure of highly reflecting fiber Bragg gratings,” IEEE Photonics Technol. Lett. 15, 575–577 (2003).
[CrossRef]

Koltracht, I.

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

Marti, J.

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’fan-Levitan-Marchenko coupled equations,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

Mikusiéski, P.

L. Debnath, P. Mikusiéski, Introduction to Hilbert Spaces with Applications (Academic, San Diego, Calif., 1990).

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]

Papoulis, A.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962).

Peral, E.

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’fan-Levitan-Marchenko coupled equations,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

Poladian, L.

Porat, B.

B. Porat, Digital Processing of Random Signals: Theory and Methods (Prentice Hall, Englewood Cliffs, N.J., 1994).

Rosenthal, A.

A. Rosenthal, M. Horowitz, “New technique to accurately interpolate the complex reflection spectrum of fiber Bragg gratings,” IEEE J. Quantum Electron. 40, 1099–1104 (2004).
[CrossRef]

S. Keren, A. Rosenthal, M. Horowitz, “Measuring the structure of highly reflecting fiber Bragg gratings,” IEEE Photonics Technol. Lett. 15, 575–577 (2003).
[CrossRef]

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

Segur, H.

M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Applied Mathematics, Philadelphia, Pa., 1981).

Skaar, J.

J. Skaar, O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[CrossRef]

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,” J. Lightwave Technol. 37, 165–173 (2001).

Van Ioan, C. F.

G. H. Golub, C. F. Van Ioan, Matrix Computations (The Johns Hopkins U. Press, Baltimore, Md., 1996).

Waagaard, O. H.

J. Skaar, O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[CrossRef]

Wang, L.

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Lightwave Technol. 37, 165–173 (2001).

Zervas, M. N.

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]

IEEE J. Quantum Electron. (5)

J. Skaar, O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[CrossRef]

A. Rosenthal, M. Horowitz, “New technique to accurately interpolate the complex reflection spectrum of fiber Bragg gratings,” IEEE J. Quantum Electron. 40, 1099–1104 (2004).
[CrossRef]

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

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’fan-Levitan-Marchenko coupled equations,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

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]

IEEE Photonics Technol. Lett. (1)

S. Keren, A. Rosenthal, M. Horowitz, “Measuring the structure of highly reflecting fiber Bragg gratings,” IEEE Photonics Technol. Lett. 15, 575–577 (2003).
[CrossRef]

J. Lightwave Technol. (2)

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

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Lightwave Technol. 37, 165–173 (2001).

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

Opt. Lett. (2)

Phys. Rev. E (1)

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. (1)

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

Other (8)

T. Kailath, Linear Systems (Prentice Hall, Englewood Cliffs, N.J., 1980).

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962).

M. J. Ablowitz, A. S. Fokas, Complex Variables (Cambridge U. Press, Cambridge, UK, 1997).

L. Debnath, P. Mikusiéski, Introduction to Hilbert Spaces with Applications (Academic, San Diego, Calif., 1990).

G. H. Golub, C. F. Van Ioan, Matrix Computations (The Johns Hopkins U. Press, Baltimore, Md., 1996).

B. Porat, Digital Processing of Random Signals: Theory and Methods (Prentice Hall, Englewood Cliffs, N.J., 1994).

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1966).

M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Applied Mathematics, Philadelphia, Pa., 1981).

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Figures (6)

Fig. 1
Fig. 1

Reconstruction of a uniform grating with coupling coefficient q = 500 m-1 and length L = 1 cm from a noisy reflection spectrum. The figure compares the reconstruction obtained with the method developed in Ref. 9 (solid curve) with a direct reconstruction by the ILP algorithm (dashed curve) and with a reconstruction by the method presented in this paper (dotted curve). The reflection spectrum of the grating was sampled with a bandwidth of 10 nm and resolution of 0.002 nm. The standard deviation of the noise variables, added to the complex reflection spectrum, was equal to 5 × 10-5. The ILP algorithm as well as the algorithm presented in this paper have accurately reconstructed the grating profile.

Fig. 2
Fig. 2

Reconstruction of a uniform grating with length of L = 4 mm, coupling coefficient q = 2.5 × 103 (m-1), and maximum reflectivity 1 – 10-8 from a noisy reflection spectrum. The reconstruction was performed by using the method described in this paper (solid curve) and was compared with the original profile (dashed curve) and with a direct reconstruction of the grating from the noisy reflection spectrum (dotted curve). The standard deviation of the noise-variable amplitude added to the complex reflection spectrum was equal to 0.02.

Fig. 3
Fig. 3

Reconstruction of a Gaussian grating with length L = 4 mm, coupling coefficient q(z) = {1900 exp[-6.2 × 105(z - L/2)2]} m-1, and maximum reflectivity 0.999. The reflection spectrum of the grating was sampled with a bandwidth of 20 nm and a resolution of 0.02 nm. The standard deviation of the noise-variable amplitude added to the complex reflection spectrum was equal to 10-3. Curve definitions as in Fig. 2.

Fig. 4
Fig. 4

Noisy reflectivity of a uniform grating with length 4 mm, coupling coefficient q=1320 (m-1), and maximum reflectivity 0.9999. The standard deviation of the noise variable amplitude added to the complex reflection spectrum was equal to 0.1.

Fig. 5
Fig. 5

Reconstruction of the uniform grating with the reflectivity shown in Fig. 4. Curve definitions as in Fig. 2.

Fig. 6
Fig. 6

Reconstruction of a uniform grating with length 4 mm, coupling coefficient of q=1320 (m-1), and maximum reflectivity 0.9999. We added to each point in the reflection spectrum a Gaussian random variable with zero mean. The noise variables were generated by a Gaussian ARMA(1, 0) process with a covariance matrix given in Eq. (9) with a parameter ρ=0.9. Curve definitions as in Fig. 2.

Equations (18)

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du1(k, z)dz+iku1(k, z)=q(z)u2(k, z), du2(k, z)dz-iku2(k, z)=q*(z)u1(k, z),
r(k)=b(k)a(k), t(k)=1a(k).
a(k)=exp(-ikL)+-LLα(τ)exp(ikτ)dτ, b(k)=-LLβ(τ)exp(ikτ)dτ.
α(τ=-L)=-12 0L|q(z)|2dz,  α(τ=L)=0, β(τ=-L)=-q(z=0)2, β(τ=L)=-q(z=L)2.
|r(k)|2+|t(k)|2=1.
|a(k)|2=1+|b(k)|2=11-|r(k)|2.
ΔB(τ)n=1Ncn exp(iknτ),
ΔB(τ)n=1Ndnfn(τ),
cov[n(ki), n(kj)]=2×10-2ρ|i-j|
U(k, z)=01exp[ik(z-L)]+z-LL-zF(τ, z)exp(ikτ)dτ,  0zL,
df1(τ, z)dz-df1(τ, z)dτ=q(z)f2(τ, z), df2(τ, z)dz+df2(τ, z)dτ=q*(z)f1(τ, z),
f1(z-L, z)=-q(z)2, f2(L-z, z)=0.
f1(L-z, z)=-q(z=L)2, f2(z-L, z)=-12 zL|q(z)|2dz.
U(k, z) =cosh(γΔ)+ikγsinh(γΔ)-qγsinh(γΔ)-qγsinh(γΔ)cosh(γΔ)-ikγsinh(γΔ)×U(k, z=L)
ddz Δu1(k, z)+ikΔu1(k, z)=qΔu2(k, z)+Δq(z)u2(z), ddz Δu2(k, z)-ikΔu2(k, z)=qΔu1(k, z)+{Δq(z)}*u1(z).
Δv1(k, z=0)=-q2γ2 cosh(γL)+k2γ2 cosh[γ(L-2z0)]-i kγ               sinh[γ(L-2z0)], Δv2(k, z=0)=qγ               sinh(γL)-i kγcosh(γL)+i kγcosh[γ(L-2z0)].
ΔU(k, z=0)=0L[Δq(z0)Δv1(k,z0),{Δq(z0)}*Δv2(k, z0)]dz0.
Δa(k)/a(k)Δq(k)/q(k)qL, Δb(k)/b(k)Δq(k)/q(k)qL, Δ|r|(k)/|r(k)|Re{Δq(k)}/q(k)t(k)2qL.

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