Abstract

The worst-case error amplification factor in reconstructing a grating from its complex reflection spectrum is shown to be of the order 1/Tmin, where Tmin is the minimum transmissivity through the grating. For a uniform grating with coupling coefficient–length product κL, the error amplification is exp(2κL). The exponential dependence on the grating strength shows that spatial characterization of gratings from a measured reflection spectrum is impossible if the grating is sufficiently strong. For moderately strong gratings, a simple regularization technique is proposed to stabilize the solution of the inverse-scattering problem of computing the grating structure from the reflection spectrum.

© 2002 Optical Society of America

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References

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  1. R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
    [CrossRef]
  2. L. Poladian, “Simple grating synthesis algorithm,” Opt. Lett. 25, 787–789 (2000).
    [CrossRef]
  3. J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber gratings by layer-peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
    [CrossRef]
  4. J. Skaar, “Grating reconstruction from noisy reflection data,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides, 2001, Vol. 61 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), Paper BThB2.
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    [CrossRef]
  6. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
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  8. A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential method in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
    [CrossRef]
  9. E. I. Petermann, J. Skaar, B. E. Sahlgren, R. A. H. Stubbe, A. T. Friberg, “Characterization of fiber Bragg gratings by use of optical coherence-domain reflectometry,” J. Lightwave Technol. 17, 2371–2378 (1999).
    [CrossRef]
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    [CrossRef]
  11. 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]
  12. R. Feced, J. A. J. Fells, S. E. Kanellopoulos, P. J. Bennet, H. F. M. Priddle, “Impact of random phase errors on the performance of fiber grating dispersion compensators,” in Optical Fiber Communication Conference 2001, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), Paper WDD89.
  13. Rakesh, “A one-dimensional inverse problem for a hyperbolic system with complex coefficient,” Inverse Probl. 17, 1401–1417 (2001).
    [CrossRef]
  14. H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).
  15. A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, Dordrecht, The Netherlands, 1995).
  16. I. Koltracht, P. Lancaster, “Threshold algorithms for the prediction of reflection coefficients in a layered medium,” Geophysics 53, 908–919 (1988).
    [CrossRef]
  17. E. Brinkmeyer, G. Stolze, D. Johlen, “Optical space domain reflectometry (OSDR) for determination of strength and chirp distribution along optical fiber gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1997), pp. 33–35.
  18. P.-Y. Fonjallaz, P. Börjel, “Interferometric side diffraction technique for the characterisation of fibre gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides, 1999, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 230–232.

2001 (2)

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

Rakesh, “A one-dimensional inverse problem for a hyperbolic system with complex coefficient,” Inverse Probl. 17, 1401–1417 (2001).
[CrossRef]

2000 (2)

1999 (2)

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

E. I. Petermann, J. Skaar, B. E. Sahlgren, R. A. H. Stubbe, A. T. Friberg, “Characterization of fiber Bragg gratings by use of optical coherence-domain reflectometry,” J. Lightwave Technol. 17, 2371–2378 (1999).
[CrossRef]

1997 (1)

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

1995 (1)

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential method in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

1994 (1)

1988 (1)

I. Koltracht, P. Lancaster, “Threshold algorithms for the prediction of reflection coefficients in a layered medium,” Geophysics 53, 908–919 (1988).
[CrossRef]

1986 (1)

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

Bennet, P. J.

R. Feced, J. A. J. Fells, S. E. Kanellopoulos, P. J. Bennet, H. F. M. Priddle, “Impact of random phase errors on the performance of fiber grating dispersion compensators,” in Optical Fiber Communication Conference 2001, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), Paper WDD89.

Börjel, P.

P.-Y. Fonjallaz, P. Börjel, “Interferometric side diffraction technique for the characterisation of fibre gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides, 1999, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 230–232.

Brinkmeyer, E.

E. Brinkmeyer, G. Stolze, D. Johlen, “Optical space domain reflectometry (OSDR) for determination of strength and chirp distribution along optical fiber gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1997), pp. 33–35.

Bruckstein, A. M.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential method in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

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

Engl, H. W.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Erdogan, T.

J. Skaar, L. Wang, T. Erdogan, “On the synthesis of fiber 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, “Effects of random phase and amplitude errors in optical fiber Bragg gratings,” J. Lightwave Technol. 18, 90–101 (2000).
[CrossRef]

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

R. Feced, J. A. J. Fells, S. E. Kanellopoulos, P. J. Bennet, H. F. M. Priddle, “Impact of random phase errors on the performance of fiber grating dispersion compensators,” in Optical Fiber Communication Conference 2001, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), Paper WDD89.

Fells, J. A. J.

R. Feced, J. A. J. Fells, S. E. Kanellopoulos, P. J. Bennet, H. F. M. Priddle, “Impact of random phase errors on the performance of fiber grating dispersion compensators,” in Optical Fiber Communication Conference 2001, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), Paper WDD89.

Fonjallaz, P.-Y.

P.-Y. Fonjallaz, P. Börjel, “Interferometric side diffraction technique for the characterisation of fibre gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides, 1999, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 230–232.

Friberg, A. T.

Goncharsky, A. V.

A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, Dordrecht, The Netherlands, 1995).

Hanke, M.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Johlen, D.

E. Brinkmeyer, G. Stolze, D. Johlen, “Optical space domain reflectometry (OSDR) for determination of strength and chirp distribution along optical fiber gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1997), pp. 33–35.

Kailath, T.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential method in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

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

Kanellopoulos, S. E.

R. Feced, J. A. J. Fells, S. E. Kanellopoulos, P. J. Bennet, H. F. M. Priddle, “Impact of random phase errors on the performance of fiber grating dispersion compensators,” in Optical Fiber Communication Conference 2001, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), Paper WDD89.

Koltracht, I.

I. Koltracht, P. Lancaster, “Threshold algorithms for the prediction of reflection coefficients in a layered medium,” Geophysics 53, 908–919 (1988).
[CrossRef]

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

Lancaster, P.

I. Koltracht, P. Lancaster, “Threshold algorithms for the prediction of reflection coefficients in a layered medium,” Geophysics 53, 908–919 (1988).
[CrossRef]

Levy, B. C.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential method in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

Muriel, M. A.

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

Neubauer, A.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Petermann, E. I.

Poladian, L.

Priddle, H. F. M.

R. Feced, J. A. J. Fells, S. E. Kanellopoulos, P. J. Bennet, H. F. M. Priddle, “Impact of random phase errors on the performance of fiber grating dispersion compensators,” in Optical Fiber Communication Conference 2001, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), Paper WDD89.

Rakesh,

Rakesh, “A one-dimensional inverse problem for a hyperbolic system with complex coefficient,” Inverse Probl. 17, 1401–1417 (2001).
[CrossRef]

Sahlgren, B. E.

Skaar, J.

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

E. I. Petermann, J. Skaar, B. E. Sahlgren, R. A. H. Stubbe, A. T. Friberg, “Characterization of fiber Bragg gratings by use of optical coherence-domain reflectometry,” J. Lightwave Technol. 17, 2371–2378 (1999).
[CrossRef]

J. Skaar, “Grating reconstruction from noisy reflection data,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides, 2001, Vol. 61 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), Paper BThB2.

Song, G.-H.

Stepanov, V. V.

A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, Dordrecht, The Netherlands, 1995).

Stolze, G.

E. Brinkmeyer, G. Stolze, D. Johlen, “Optical space domain reflectometry (OSDR) for determination of strength and chirp distribution along optical fiber gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1997), pp. 33–35.

Stubbe, R. A. H.

Tikhonov, A. N.

A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, Dordrecht, The Netherlands, 1995).

Wang, L.

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

Yagola, A. G.

A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, Dordrecht, The Netherlands, 1995).

Zervas, M. N.

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]

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

Geophysics (1)

I. Koltracht, P. Lancaster, “Threshold algorithms for the prediction of reflection coefficients in a layered medium,” Geophysics 53, 908–919 (1988).
[CrossRef]

IEEE J. Quantum Electron. (2)

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

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

Inverse Probl. (1)

Rakesh, “A one-dimensional inverse problem for a hyperbolic system with complex coefficient,” Inverse Probl. 17, 1401–1417 (2001).
[CrossRef]

J. Lightwave Technol. (3)

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

Opt. Lett. (1)

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential method in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

SIAM J. Sci. Comput. (USA) (1)

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

Other (7)

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/ .

J. Skaar, “Grating reconstruction from noisy reflection data,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides, 2001, Vol. 61 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), Paper BThB2.

R. Feced, J. A. J. Fells, S. E. Kanellopoulos, P. J. Bennet, H. F. M. Priddle, “Impact of random phase errors on the performance of fiber grating dispersion compensators,” in Optical Fiber Communication Conference 2001, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), Paper WDD89.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).

A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, Dordrecht, The Netherlands, 1995).

E. Brinkmeyer, G. Stolze, D. Johlen, “Optical space domain reflectometry (OSDR) for determination of strength and chirp distribution along optical fiber gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1997), pp. 33–35.

P.-Y. Fonjallaz, P. Börjel, “Interferometric side diffraction technique for the characterisation of fibre gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides, 1999, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 230–232.

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Figures (6)

Fig. 1
Fig. 1

Solid curve, coupling function q(z) that ends at the reference point z=0. The perturbation δq(z) (dashed curve) starts at z=0.

Fig. 2
Fig. 2

Error |q˜(z)Δz-q(z)Δz| of the reconstructed grating with coupling coefficient q(z)=κ. Solid curve, noisy input data; dotted curve, noise-free input data. The errors are compared with bound (15) with C=1/2 (dashed curve).

Fig. 3
Fig. 3

Error |q˜(z)Δz-q(z)Δz| of the reconstructed, apodized grating (solid curve). The error is plotted as a function of κz, with κ as the maximum value of the coupling coefficient, and compared with bound (15) with C=1/2 (dashed curve).

Fig. 4
Fig. 4

Error |q˜(z)Δz-q(z)Δz| of the reconstructed gratings. Solid curve, grating with a phase shift; dotted curve, chirped grating. The errors are compared with bound (15) with C=1/2 (dashed curve).

Fig. 5
Fig. 5

Resulting error amplification factor [Eq. (37)] as a function of grating strength κL for 2×100 numerical experiments with uniform gratings (circles) and chirped gratings (x marks, bottom curve). The amplification factors are compared with exp(2κL) (solid curve) and 1/Tmin (dashed curves) and the factor in Eq. (35) (dotted curves). The almost-straight lines correspond to the uniform gratings and the lower curves to the chirped gratings.

Fig. 6
Fig. 6

Reconstructed coupling coefficient for different values of the regularization parameter μ. The original grating was uniform with κL=5; for the grating in this figure κ=500 m-1 and L=0.01 m.

Equations (55)

Equations on this page are rendered with MathJax. Learn more.

hp(τ)=12π-r(β)exp(-iβτ)dβ.
h(j)=1Mmr(m)exp(-i2πjm/M),j=0, 1, 2 ,,
h(j)2Δzhp(j2Δz).
Δz=L/N=π/βw.
q(z)=-2hp*(2z).
ρj=-tanh(|qj|Δz)qj*/|qj|,j=0,1 ,, N-1.
u=[v0v1vN-1]=[100]
v=[u0u1uN-1]=[h(0)h(1)h(N-1)].
uv(1-|ρj|2)-1/2u-ρj*v-ρju+v.
q(jΔz)=-(1/Δz)arctanh(|ρj|)ρj*/|ρj|,
j=0,1 ,, N-1.
|h˜(j)-h(j)| forallj.
|h˜p(τ)-hp(τ)| h forallτ,
=h2Δz.
|r˜(m)-r(m)| r,forallm.
r/Mr.
r/M.
|ρ˜j-ρj| 2Ck=0j-11+|ρk|1-|ρk|+O(2),
0jN-1,
|q˜(z)Δz-q(z)Δz| 2C exp20z|q(z)|dz+O(2),
|q˜(z)Δz-q(z)Δz| 2C exp20z|q˜(z)|dz+O(2).
T˜=T2Tz0T1,
T=T2T1,
T=1/t*-r*/t*-r/t1/t.
rz0=-K*,
Tz0=(1-|K|2)-1/21KK*1.
r˜-r=t2t22 (K*+Kr22),
δr(β)r˜-r=-t2t22 [δq*(z)+δq(z)r22]dz.
δq2=-|δq(z)|2dz,δr2=-|δr(β)|2dβ.
δrδr/δqδq.
δr/δq=maxδrforallδq(z) suchthatδq=1.
δq=(δq/δr)δr.
δq/δr=1minδr forallδq(z) suchthatδq=1.
t(β)=t0(β)exp(iβd),
t2(β, z)=exp[iβ(d-z)],
r2(β, z)=0,
δr2=-|t0(β)|4|δQ(β)|2dβ,
δQ(β)=-δq*(z)exp(2iβz)dz.
δr2=-|δQ(β)|2dβ=πδq2
δq/δr=1π.
δr2|t0(β)|min4-|δQ(β)|2dβ=|Tmin(β)|2πδq2,
δq/δr=1π1Tmin.
δq/δr=1πηTmin2+1-η1/2,
(1-Rmax)-1=[exp(2κL)+2+exp(-2κL)]/4exp(2κL).
Aδρl2δhl2j=1N|ρ˜j-ρj|21/2j=1N|h˜(j)-h(j)|21/2,
Aπδqδr,
rˆ=(1-μ)r˜,
|rˆ-r| |rˆ-r˜|+|r˜-r| μ+r.
(r+μ)(1-max|rˆ|2)-1r(1-max|r˜|2)-1
=1+μr-2μ max|r˜|21-max|r˜|2+O(μ2).
C(I-H)-2,
(I-H)-1=I+H+  +HN,
(I-H)-2=I+2H+  +(N+1)HN
C12 (N+1)(N+2),
C(1-H)-2=1/(1-max{|h(j)|})2.

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