Abstract

The problem of finding the dielectric constant of a slab from the reflection data is considered. It is shown that a recently developed method by Trantanella et al. [J. Opt. Soc. Am. A 12, 1469 (1995)], which is based on an interesting extension of the Born approximation, is intimately related to the Schwinger variational solution. This remarkable relation enables one to extend the class of profiles that can be reconstructed, evaluate analytically some of the quantities that were found approximately by Trantanella et al., thus simplifying the numerical computations, and also arrive at a systematic development of a sequence of more accurate extensions of the Born inversion method. To illustrate the proposed inversion procedures, an exactly solvable analytical example is presented.

© 1998 Optical Society of America

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References

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  1. K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, 2nd ed. (Springer-Verlag, New York, 1989).
  2. S. Coen, “Inverse scattering of a layered and dispersionless dielectric half-space. Part 1: Reflection data from plane waves at normal incidence,” IEEE Trans. Antennas Propag. AP-29, 726–732 (1981).
    [CrossRef]
  3. M. A. Hooshyar, L. S. Tamil, “Inverse scattering theory and the design of planar optical waveguides with the same propagation constants for different frequencies,” Inverse Probl. 9, 69–80 (1993).
    [CrossRef]
  4. J. Xia, A. K. Jordan, J. A. Kong, “Electromagnetic inverse scattering theory for inhomogeneous dielectrics: the local reflection model,” J. Opt. Soc. Am. A 11, 1081–1086 (1994).
    [CrossRef]
  5. R. Clayton, R. Stolt, “A Born-WKBJ inversion method for an acoustic reflection data,” Geophysics 46, 1559–1567 (1981).
    [CrossRef]
  6. R. G. Keys, A. B. Weglein, “Generalized linear inversion and the first Born theory for acoustic stratified media,” J. Math. Phys. (New York) 24, 1444–1449 (1983).
    [CrossRef]
  7. A. J. Devaney, M. L. Oristaglio, “Inversion procedure for inverse scattering within the distorted-wave Born approximation,” Phys. Rev. Lett. 51, 237–240 (1983).
    [CrossRef]
  8. H. D. Ladouceur, A. K. Jordan, “Renormalization of an inverse-scattering theory for inhomogeneous dielectrics,” J. Opt. Soc. Am. A 2, 1916–1921 (1985).
    [CrossRef]
  9. A. K. Jordan, H. D. Ladouceur, “Renormalization of an inverse-scattering theory for inhomogeneous dielectrics,” Phys. Rev. A 36, 4245–4253 (1987).
    [CrossRef] [PubMed]
  10. M. A. Hooshyar, T. H. Lam, M. Razavy, “Inverse scattering problem and the Schwinger approximation,” Can. J. Phys. 70, 282–288 (1992).
    [CrossRef]
  11. T. M. Habashy, R. W. Groom, B. Spies, “Beyond the Born and Rytov approximations: a non-linear approach to electromagnetic scattering approximation,” J. Geophys. Res. 98, 1759–1775 (1993).
    [CrossRef]
  12. C. J. Trantanella, D. G. Dudley, A. Nabulsi, “Beyond the Born approximation in one-dimensional profile reconstruction,” J. Opt. Soc. Am. A 12, 1469–1478 (1995).
    [CrossRef]
  13. C. Duneczky, R. E. Wyatt, “Lanczos recursion, continued fractions, Padé approximants, and variational principles in quantum scattering theory,” J. Chem. Phys. 89, 1448–1463 (1988).
    [CrossRef]
  14. J. R. Taylor, Scattering Theory (Wiley, New York, 1972).
  15. J. H. Mathews, Numerical Methods for Mathematics, Science, and Engineering, 2nd ed. (Prentice-Hall, London, 1992).
  16. H. E. Moses, “Calculation of the scattering potential from reflection coefficients,” Phys. Rev. 102, 559–567 (1952).
    [CrossRef]
  17. M. Razavy, “Determination of the wave velocity in an inhomogeneous medium from reflection coefficient,” J. Acoust. Soc. Am. 58, 956–963 (1975).
    [CrossRef]
  18. R. T. Prosser, “Formal solutions of inverse scattering problems. IV” J. Math. Phys. (New York) 23, 2127–2130 (1982).
    [CrossRef]
  19. A. J. Devaney, A. B. Weglein, “Inverse scattering using the Heitler equation,” Inverse Probl. 5, L49–L52 (1989).
    [CrossRef]
  20. A. K. Jordan, H. D. Ladouceur, “Renormalization of an inverse-scattering theory for Gaussian distributions,” in Multiple Scattering of Waves in Random Media and Random Rough Surfaces, V. K. Varadan, ed. (The Pennsylvania State University, University Park, Pa., 1985), pp. 233–240.

1995 (1)

1994 (1)

1993 (2)

M. A. Hooshyar, L. S. Tamil, “Inverse scattering theory and the design of planar optical waveguides with the same propagation constants for different frequencies,” Inverse Probl. 9, 69–80 (1993).
[CrossRef]

T. M. Habashy, R. W. Groom, B. Spies, “Beyond the Born and Rytov approximations: a non-linear approach to electromagnetic scattering approximation,” J. Geophys. Res. 98, 1759–1775 (1993).
[CrossRef]

1992 (1)

M. A. Hooshyar, T. H. Lam, M. Razavy, “Inverse scattering problem and the Schwinger approximation,” Can. J. Phys. 70, 282–288 (1992).
[CrossRef]

1989 (1)

A. J. Devaney, A. B. Weglein, “Inverse scattering using the Heitler equation,” Inverse Probl. 5, L49–L52 (1989).
[CrossRef]

1988 (1)

C. Duneczky, R. E. Wyatt, “Lanczos recursion, continued fractions, Padé approximants, and variational principles in quantum scattering theory,” J. Chem. Phys. 89, 1448–1463 (1988).
[CrossRef]

1987 (1)

A. K. Jordan, H. D. Ladouceur, “Renormalization of an inverse-scattering theory for inhomogeneous dielectrics,” Phys. Rev. A 36, 4245–4253 (1987).
[CrossRef] [PubMed]

1985 (1)

1983 (2)

R. G. Keys, A. B. Weglein, “Generalized linear inversion and the first Born theory for acoustic stratified media,” J. Math. Phys. (New York) 24, 1444–1449 (1983).
[CrossRef]

A. J. Devaney, M. L. Oristaglio, “Inversion procedure for inverse scattering within the distorted-wave Born approximation,” Phys. Rev. Lett. 51, 237–240 (1983).
[CrossRef]

1982 (1)

R. T. Prosser, “Formal solutions of inverse scattering problems. IV” J. Math. Phys. (New York) 23, 2127–2130 (1982).
[CrossRef]

1981 (2)

S. Coen, “Inverse scattering of a layered and dispersionless dielectric half-space. Part 1: Reflection data from plane waves at normal incidence,” IEEE Trans. Antennas Propag. AP-29, 726–732 (1981).
[CrossRef]

R. Clayton, R. Stolt, “A Born-WKBJ inversion method for an acoustic reflection data,” Geophysics 46, 1559–1567 (1981).
[CrossRef]

1975 (1)

M. Razavy, “Determination of the wave velocity in an inhomogeneous medium from reflection coefficient,” J. Acoust. Soc. Am. 58, 956–963 (1975).
[CrossRef]

1952 (1)

H. E. Moses, “Calculation of the scattering potential from reflection coefficients,” Phys. Rev. 102, 559–567 (1952).
[CrossRef]

Chadan, K.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, 2nd ed. (Springer-Verlag, New York, 1989).

Clayton, R.

R. Clayton, R. Stolt, “A Born-WKBJ inversion method for an acoustic reflection data,” Geophysics 46, 1559–1567 (1981).
[CrossRef]

Coen, S.

S. Coen, “Inverse scattering of a layered and dispersionless dielectric half-space. Part 1: Reflection data from plane waves at normal incidence,” IEEE Trans. Antennas Propag. AP-29, 726–732 (1981).
[CrossRef]

Devaney, A. J.

A. J. Devaney, A. B. Weglein, “Inverse scattering using the Heitler equation,” Inverse Probl. 5, L49–L52 (1989).
[CrossRef]

A. J. Devaney, M. L. Oristaglio, “Inversion procedure for inverse scattering within the distorted-wave Born approximation,” Phys. Rev. Lett. 51, 237–240 (1983).
[CrossRef]

Dudley, D. G.

Duneczky, C.

C. Duneczky, R. E. Wyatt, “Lanczos recursion, continued fractions, Padé approximants, and variational principles in quantum scattering theory,” J. Chem. Phys. 89, 1448–1463 (1988).
[CrossRef]

Groom, R. W.

T. M. Habashy, R. W. Groom, B. Spies, “Beyond the Born and Rytov approximations: a non-linear approach to electromagnetic scattering approximation,” J. Geophys. Res. 98, 1759–1775 (1993).
[CrossRef]

Habashy, T. M.

T. M. Habashy, R. W. Groom, B. Spies, “Beyond the Born and Rytov approximations: a non-linear approach to electromagnetic scattering approximation,” J. Geophys. Res. 98, 1759–1775 (1993).
[CrossRef]

Hooshyar, M. A.

M. A. Hooshyar, L. S. Tamil, “Inverse scattering theory and the design of planar optical waveguides with the same propagation constants for different frequencies,” Inverse Probl. 9, 69–80 (1993).
[CrossRef]

M. A. Hooshyar, T. H. Lam, M. Razavy, “Inverse scattering problem and the Schwinger approximation,” Can. J. Phys. 70, 282–288 (1992).
[CrossRef]

Jordan, A. K.

J. Xia, A. K. Jordan, J. A. Kong, “Electromagnetic inverse scattering theory for inhomogeneous dielectrics: the local reflection model,” J. Opt. Soc. Am. A 11, 1081–1086 (1994).
[CrossRef]

A. K. Jordan, H. D. Ladouceur, “Renormalization of an inverse-scattering theory for inhomogeneous dielectrics,” Phys. Rev. A 36, 4245–4253 (1987).
[CrossRef] [PubMed]

H. D. Ladouceur, A. K. Jordan, “Renormalization of an inverse-scattering theory for inhomogeneous dielectrics,” J. Opt. Soc. Am. A 2, 1916–1921 (1985).
[CrossRef]

A. K. Jordan, H. D. Ladouceur, “Renormalization of an inverse-scattering theory for Gaussian distributions,” in Multiple Scattering of Waves in Random Media and Random Rough Surfaces, V. K. Varadan, ed. (The Pennsylvania State University, University Park, Pa., 1985), pp. 233–240.

Keys, R. G.

R. G. Keys, A. B. Weglein, “Generalized linear inversion and the first Born theory for acoustic stratified media,” J. Math. Phys. (New York) 24, 1444–1449 (1983).
[CrossRef]

Kong, J. A.

Ladouceur, H. D.

A. K. Jordan, H. D. Ladouceur, “Renormalization of an inverse-scattering theory for inhomogeneous dielectrics,” Phys. Rev. A 36, 4245–4253 (1987).
[CrossRef] [PubMed]

H. D. Ladouceur, A. K. Jordan, “Renormalization of an inverse-scattering theory for inhomogeneous dielectrics,” J. Opt. Soc. Am. A 2, 1916–1921 (1985).
[CrossRef]

A. K. Jordan, H. D. Ladouceur, “Renormalization of an inverse-scattering theory for Gaussian distributions,” in Multiple Scattering of Waves in Random Media and Random Rough Surfaces, V. K. Varadan, ed. (The Pennsylvania State University, University Park, Pa., 1985), pp. 233–240.

Lam, T. H.

M. A. Hooshyar, T. H. Lam, M. Razavy, “Inverse scattering problem and the Schwinger approximation,” Can. J. Phys. 70, 282–288 (1992).
[CrossRef]

Mathews, J. H.

J. H. Mathews, Numerical Methods for Mathematics, Science, and Engineering, 2nd ed. (Prentice-Hall, London, 1992).

Moses, H. E.

H. E. Moses, “Calculation of the scattering potential from reflection coefficients,” Phys. Rev. 102, 559–567 (1952).
[CrossRef]

Nabulsi, A.

Oristaglio, M. L.

A. J. Devaney, M. L. Oristaglio, “Inversion procedure for inverse scattering within the distorted-wave Born approximation,” Phys. Rev. Lett. 51, 237–240 (1983).
[CrossRef]

Prosser, R. T.

R. T. Prosser, “Formal solutions of inverse scattering problems. IV” J. Math. Phys. (New York) 23, 2127–2130 (1982).
[CrossRef]

Razavy, M.

M. A. Hooshyar, T. H. Lam, M. Razavy, “Inverse scattering problem and the Schwinger approximation,” Can. J. Phys. 70, 282–288 (1992).
[CrossRef]

M. Razavy, “Determination of the wave velocity in an inhomogeneous medium from reflection coefficient,” J. Acoust. Soc. Am. 58, 956–963 (1975).
[CrossRef]

Sabatier, P. C.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, 2nd ed. (Springer-Verlag, New York, 1989).

Spies, B.

T. M. Habashy, R. W. Groom, B. Spies, “Beyond the Born and Rytov approximations: a non-linear approach to electromagnetic scattering approximation,” J. Geophys. Res. 98, 1759–1775 (1993).
[CrossRef]

Stolt, R.

R. Clayton, R. Stolt, “A Born-WKBJ inversion method for an acoustic reflection data,” Geophysics 46, 1559–1567 (1981).
[CrossRef]

Tamil, L. S.

M. A. Hooshyar, L. S. Tamil, “Inverse scattering theory and the design of planar optical waveguides with the same propagation constants for different frequencies,” Inverse Probl. 9, 69–80 (1993).
[CrossRef]

Taylor, J. R.

J. R. Taylor, Scattering Theory (Wiley, New York, 1972).

Trantanella, C. J.

Weglein, A. B.

A. J. Devaney, A. B. Weglein, “Inverse scattering using the Heitler equation,” Inverse Probl. 5, L49–L52 (1989).
[CrossRef]

R. G. Keys, A. B. Weglein, “Generalized linear inversion and the first Born theory for acoustic stratified media,” J. Math. Phys. (New York) 24, 1444–1449 (1983).
[CrossRef]

Wyatt, R. E.

C. Duneczky, R. E. Wyatt, “Lanczos recursion, continued fractions, Padé approximants, and variational principles in quantum scattering theory,” J. Chem. Phys. 89, 1448–1463 (1988).
[CrossRef]

Xia, J.

Can. J. Phys. (1)

M. A. Hooshyar, T. H. Lam, M. Razavy, “Inverse scattering problem and the Schwinger approximation,” Can. J. Phys. 70, 282–288 (1992).
[CrossRef]

Geophysics (1)

R. Clayton, R. Stolt, “A Born-WKBJ inversion method for an acoustic reflection data,” Geophysics 46, 1559–1567 (1981).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

S. Coen, “Inverse scattering of a layered and dispersionless dielectric half-space. Part 1: Reflection data from plane waves at normal incidence,” IEEE Trans. Antennas Propag. AP-29, 726–732 (1981).
[CrossRef]

Inverse Probl. (2)

M. A. Hooshyar, L. S. Tamil, “Inverse scattering theory and the design of planar optical waveguides with the same propagation constants for different frequencies,” Inverse Probl. 9, 69–80 (1993).
[CrossRef]

A. J. Devaney, A. B. Weglein, “Inverse scattering using the Heitler equation,” Inverse Probl. 5, L49–L52 (1989).
[CrossRef]

J. Acoust. Soc. Am. (1)

M. Razavy, “Determination of the wave velocity in an inhomogeneous medium from reflection coefficient,” J. Acoust. Soc. Am. 58, 956–963 (1975).
[CrossRef]

J. Chem. Phys. (1)

C. Duneczky, R. E. Wyatt, “Lanczos recursion, continued fractions, Padé approximants, and variational principles in quantum scattering theory,” J. Chem. Phys. 89, 1448–1463 (1988).
[CrossRef]

J. Geophys. Res. (1)

T. M. Habashy, R. W. Groom, B. Spies, “Beyond the Born and Rytov approximations: a non-linear approach to electromagnetic scattering approximation,” J. Geophys. Res. 98, 1759–1775 (1993).
[CrossRef]

J. Math. Phys. (New York) (2)

R. T. Prosser, “Formal solutions of inverse scattering problems. IV” J. Math. Phys. (New York) 23, 2127–2130 (1982).
[CrossRef]

R. G. Keys, A. B. Weglein, “Generalized linear inversion and the first Born theory for acoustic stratified media,” J. Math. Phys. (New York) 24, 1444–1449 (1983).
[CrossRef]

J. Opt. Soc. Am. A (3)

Phys. Rev. (1)

H. E. Moses, “Calculation of the scattering potential from reflection coefficients,” Phys. Rev. 102, 559–567 (1952).
[CrossRef]

Phys. Rev. A (1)

A. K. Jordan, H. D. Ladouceur, “Renormalization of an inverse-scattering theory for inhomogeneous dielectrics,” Phys. Rev. A 36, 4245–4253 (1987).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

A. J. Devaney, M. L. Oristaglio, “Inversion procedure for inverse scattering within the distorted-wave Born approximation,” Phys. Rev. Lett. 51, 237–240 (1983).
[CrossRef]

Other (4)

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, 2nd ed. (Springer-Verlag, New York, 1989).

J. R. Taylor, Scattering Theory (Wiley, New York, 1972).

J. H. Mathews, Numerical Methods for Mathematics, Science, and Engineering, 2nd ed. (Prentice-Hall, London, 1992).

A. K. Jordan, H. D. Ladouceur, “Renormalization of an inverse-scattering theory for Gaussian distributions,” in Multiple Scattering of Waves in Random Media and Random Rough Surfaces, V. K. Varadan, ed. (The Pennsylvania State University, University Park, Pa., 1985), pp. 233–240.

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Equations (60)

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Ey(z)=exp(-jkbz)+k02jb0dq(z)Ey(z)×exp(-jkb|z-z|)dz,
-(1+|z|)|q(z)|dz<.
Ey(z)
=exp(-jkbz)+R(kb)exp(jkbz)forz-T(kb)exp(-jkbz)forz+,
R(kb)=k02jb0dq(z)Ey(z)exp(-jkbz)dz,
T(kb)=1+k02jb0dq(z)Ey(z)exp(jkbz)dz.
Q(2kb)=0dq(z)exp(-2jkbz)dzD(kb),
D(kb)=2jbk0R(kb)
Ey(z)=A exp(-jkbz)+Bk02jb0dq(z)×exp(-jkbz)×exp(-jkb|z-z|)dzforz[0, d],
Ey(z)=exp(-jkbz)+k02jb0dq(z)×exp(-jkb|z-z|)A exp(-jkbz)+Bk02jb0dq(z)exp(-jkbz)×exp(-jkb|z-z|)dzdz
forz[0, d],
Ey(z)=A+Bk02jbQ1(z)exp(-jkbz)+Bk02jbQ2(kb, z)exp(jkbz)
forz[0, d],
Q1(z)=0zq(z)dz,
Q2(kb, z)=zdq(z)exp(-2jkbz)dz.
R(kb)k02jb0dq(z)A+Bk02jbQ1(z)×exp(-2jkbz)dz+Bk02jb0dq(z)Q2(kb, z)dz.
B=M0(kb)M0(kb)-M1(kb),A=1,
M0(kb)Q(2kb)=Q2(kb, 0),
M1(kb)k02jb0d[Q1(z)exp(-2jkbz)+Q2(kb, z)]q(z)dz.
D(kb)M02(kb)M0(kb)-M1(kb).
M1(kb)
=k020d0dG(z, z, kb)q(z)exp(-jkbz)dzq(z)
×exp(-jkbz)dz,
G(z, z, kb)=12jkbexp(-jkb|z-z|)
2z2G(z, z, kb)+kb2 G(z, z, kb)=-δ(z-z).
G(z, z, kb)=12πlim0 - exp[-j(z-z)q]q2-kb2+jdq.
M1(kb)=k022πlim0 - M0kb+q2M0kb-q2q2-kb2+jdq.
M02(kb)-D˜(kb)M0(kb)+D(kb)[M1(kb)-M0(kb)]
+D˜(kb)M0(kb)0,
D˜(kb)=D(kb)1-k02jbM0(0).
M0(kb)D˜(kb)/2+{D˜2(kb)/4-D(kb)[M1(kb)-M0(kb)]-D˜(kb)M0(kb)}1/2.
M1(i)(kb)=k022πlim0 -M0(i)kb+q2M0(i)kb-q2q2-kb2+jdq.
-|M0(i+1)(kb)-M0(i)(kb)|dkb<,
q(z)=λδ(z-z0),
Ey(z)=exp(-jkbz)+λk02jbEy(z0)exp(-jkb|z-z0|).
Ey(z)=exp(-jkbz)+λk02jb-λk0×exp[-jkb(|z-z0|+z0)].
R(kb)=λk02jb-λk0exp(-2jkbz0);
D(kb)=2jλb2jb-λkbexp(-2jkbz0).
q(z)=1π-M0(kb)exp(2jkbz)dkb.
qB(z)=4b exp[-4b(z-z0)/λ],2b,0,z>z0z=z0z<z0,
M0(kb)=λ exp(-2jkbz0).
M1(kb)=k02λ22πlim0 -×exp[-j(kb+q)z0]exp[-j(kb-q)z0]q2-kb2+jdq=kbλ22jbexp(-2jkbz0).
qS(z)=1π-λ exp[2jkb(z-z0)]dkb=λδ(z-z0).
S[Ψ]=Ψ0, k02qΨ+Ψ(-), k02qΨ0-Ψ(-), k02(q-qGk02q)Ψ,
S[Ψtrial]=k02{2M0(kb)c0(kb)-c02(kb)[M0(kb)-M1(kb)]}.
c0(kb)=M0(kb)M0(kb)-M1(kb).
R(kb)=k02jb0dq(z)Ψtrial exp(-jkbz)dz.
R(kb)k02jb[AQ2(kb, 0)+BM1(kb)]=k02jbM0(kb)+M0(kb)M0(kb)-M1(kb)M1(kb)=k02jbM02(kb)M0(kb)-M1(kb)=k02jbc0(kb)M0(kb).
Ψtrial=[c0(kb)+c1(kb)(Gk02q)++cn(kb)×(Gk02q)n]Ψ0,
(Gk02q)Ψ0=k02-G(z, z, kb)q(z)Ψ0(kb, z)dz.
D(kb)M02(kb)M0(kb)-M1(kb)forn=1(Schwinger)M0(kb)+M12(kb)M1(kb)-M2(kb)forn=2(Newton)M0(kb)+M1(kb)++Mn-2(kb)+Mn-12(kb)Mn-1(kb)-Mn(kb)forn>2,
Mm(kb)=Ψ0,(k02qG)mqΨ0.
Mm(kb)=k022πm lim1,,m0 -- M0kb+q12M0q2-q12M0qm-qm-12M0kb-qm2(q12-kb2+j1)(qm-12-kb2+jm-1)(qm2-kb2+jm)dq1dqm.
Mm(i)(kb)=k022πmlim1,,m0 -- M0(i)kb+q12M0(i)q2-q12M0(i)qm-qm-12M0(i)kb-qm2(q12-kb2+j1)(qm-12-kb2+jm-1)(qm2-kb2+jm)dq1dqm.
M0(i+1)(kb)=D(kb)-[M1(i)(kb)]2M1(i)(kb)-M2(i)(kb)forn=2D(kb)-M1(i)(kb)Mn-2(i)(kb)-[Mn-1(i)(kb)]2Mn-1(i)(kb)-Mn(i)(kb)forn>2.
Mm(kb)=λλk022πmlim1,,m0 --exp[-j(kb+q1)]×exp[-j(q2-q1)]exp[-j(qm-qm-1)]exp[-j(kb-qm)](q12-kb2+j1)(qm-12-kb2+jm-1)(qm2-kb2+jm)dq1dqm.
Mm(kb)=λλkb2jbm exp(-2jkbz0).
λ1-λkb2jbexp(-2jkbz0)
=λ exp(-2jkbz0)1+λkb2jb+λkb2jb2+λkb2jbn-2+λkb2jbn-11-λkb2jb=D(kb).
11-x=1+x1-x=1+x+x21-x=1+x+x2+xn-2+xn-11-x.

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