Abstract

A fast, accurate numerical algorithm based on the Laplace transform is presented. By using this algorithm, the high-frequency structure present at the onset of the precursor field for both the delta-function pulse and the unit-step-function-modulated signal has been resolved. A comparison of the results with those obtained by modern asymptotic techniques demonstrates the ability of this numerical method to resolve the high-frequency structure associated with the onset of the Sommerfeld precursor field.

© 1989 Optical Society of America

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  1. O. Avenel, M. Rouff, E. Varoquaux, G. A. Williams, “Resonant pulse propagation of sound in superfluid He-B,” Phys. Rev. Lett. 50, 1591–1594 (1983).
    [CrossRef]
  2. R. E. McIntosh, A. Malaga, “Time dispersion of electromagnetic pulses by the ionosphere,” Radio Sci. 15, 645–654 (1980).
    [CrossRef]
  3. D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
    [CrossRef]
  4. C. M. Knop, “Pulsed electromagnetic wave propagation in dispersive media,”IEEE Trans. Antennas Propag. AP-12, 494–496 (1964).
    [CrossRef]
  5. G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
    [CrossRef]
  6. K. E. Oughstun, G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
    [CrossRef]
  7. D. Marcuse, “Pulse distortion in single-mode fibers,” Appl. Opt. 19, 1653–1660 (1980).
    [CrossRef] [PubMed]
  8. D. Marcuse, “Pulse distortion in single-mode fibers, part 2,” Appl. Opt. 19, 2969–2974 (1981).
    [CrossRef]
  9. A. Sommerfeld, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
    [CrossRef]
  10. L. Brillouin, “Über die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
    [CrossRef]
  11. L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
  12. G. C. Sherman, K. E. Oughstun, “Physical model of pulse dynamics in a linear dispersive medium with absorption (the Lorentz medium),” submitted to J. Opt. Soc. Am. B.
  13. K. E. Oughstun, P. T. Connors, G. C. Sherman, “Finite turn-on time effects on the transient phenomena in dispersive pulse propagation,” J. Opt. Soc. Am. A 3(13), P118 (1986).
  14. S. Shen, K. E. Oughstun, “Optical pulse propagation in a multiple resonant dispersive Lorentz medium,” J. Opt. Soc. Am. A 3(13), P67 (1986).
  15. T. Hosono, “Numerical inversion of Laplace transform and some applications to wave optics,” in Proceedings of the 1980 International Union of Radio Science International Symposium on Electromagnetic Waves (International Union of Radio Science, Munich, Federal Republic of Germany, 1980), paper 112, pp. C1–C4.
  16. K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1394–1419 (1989).
    [CrossRef]
  17. W. R. LePage, Complex Variables and the Laplace Transform for Engineers (Dover, New York, 1961), p. 325.
  18. T. J. I. Bromwich, An Introduction to the Theory of Infinite Series, 2nd ed. (Macmillan, London, 1926), pp. 62–65.
  19. R. A. Handelsman, N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mechan. Anal. 35, 267–283 (1969).
  20. K. E. Oughstun, P. Wyns, D. P. Foty, “Numerical determination of the signal velocity in dispersive pulse propagation,” J. Opt. Soc. Am. A 6, 1430–1440 (1989).
    [CrossRef]

1989 (2)

1988 (1)

1986 (2)

K. E. Oughstun, P. T. Connors, G. C. Sherman, “Finite turn-on time effects on the transient phenomena in dispersive pulse propagation,” J. Opt. Soc. Am. A 3(13), P118 (1986).

S. Shen, K. E. Oughstun, “Optical pulse propagation in a multiple resonant dispersive Lorentz medium,” J. Opt. Soc. Am. A 3(13), P67 (1986).

1983 (1)

O. Avenel, M. Rouff, E. Varoquaux, G. A. Williams, “Resonant pulse propagation of sound in superfluid He-B,” Phys. Rev. Lett. 50, 1591–1594 (1983).
[CrossRef]

1982 (1)

D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
[CrossRef]

1981 (2)

D. Marcuse, “Pulse distortion in single-mode fibers, part 2,” Appl. Opt. 19, 2969–2974 (1981).
[CrossRef]

G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

1980 (2)

D. Marcuse, “Pulse distortion in single-mode fibers,” Appl. Opt. 19, 1653–1660 (1980).
[CrossRef] [PubMed]

R. E. McIntosh, A. Malaga, “Time dispersion of electromagnetic pulses by the ionosphere,” Radio Sci. 15, 645–654 (1980).
[CrossRef]

1969 (1)

R. A. Handelsman, N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mechan. Anal. 35, 267–283 (1969).

1964 (1)

C. M. Knop, “Pulsed electromagnetic wave propagation in dispersive media,”IEEE Trans. Antennas Propag. AP-12, 494–496 (1964).
[CrossRef]

1914 (2)

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[CrossRef]

L. Brillouin, “Über die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

Avenel, O.

O. Avenel, M. Rouff, E. Varoquaux, G. A. Williams, “Resonant pulse propagation of sound in superfluid He-B,” Phys. Rev. Lett. 50, 1591–1594 (1983).
[CrossRef]

Bleistein, N.

R. A. Handelsman, N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mechan. Anal. 35, 267–283 (1969).

Brillouin, L.

L. Brillouin, “Über die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).

Bromwich, T. J. I.

T. J. I. Bromwich, An Introduction to the Theory of Infinite Series, 2nd ed. (Macmillan, London, 1926), pp. 62–65.

Connors, P. T.

K. E. Oughstun, P. T. Connors, G. C. Sherman, “Finite turn-on time effects on the transient phenomena in dispersive pulse propagation,” J. Opt. Soc. Am. A 3(13), P118 (1986).

Foty, D. P.

Handelsman, R. A.

R. A. Handelsman, N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mechan. Anal. 35, 267–283 (1969).

Hosono, T.

T. Hosono, “Numerical inversion of Laplace transform and some applications to wave optics,” in Proceedings of the 1980 International Union of Radio Science International Symposium on Electromagnetic Waves (International Union of Radio Science, Munich, Federal Republic of Germany, 1980), paper 112, pp. C1–C4.

Knop, C. M.

C. M. Knop, “Pulsed electromagnetic wave propagation in dispersive media,”IEEE Trans. Antennas Propag. AP-12, 494–496 (1964).
[CrossRef]

LePage, W. R.

W. R. LePage, Complex Variables and the Laplace Transform for Engineers (Dover, New York, 1961), p. 325.

Malaga, A.

R. E. McIntosh, A. Malaga, “Time dispersion of electromagnetic pulses by the ionosphere,” Radio Sci. 15, 645–654 (1980).
[CrossRef]

Marcuse, D.

D. Marcuse, “Pulse distortion in single-mode fibers, part 2,” Appl. Opt. 19, 2969–2974 (1981).
[CrossRef]

D. Marcuse, “Pulse distortion in single-mode fibers,” Appl. Opt. 19, 1653–1660 (1980).
[CrossRef] [PubMed]

McIntosh, R. E.

R. E. McIntosh, A. Malaga, “Time dispersion of electromagnetic pulses by the ionosphere,” Radio Sci. 15, 645–654 (1980).
[CrossRef]

Oughstun, K. E.

K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1394–1419 (1989).
[CrossRef]

K. E. Oughstun, P. Wyns, D. P. Foty, “Numerical determination of the signal velocity in dispersive pulse propagation,” J. Opt. Soc. Am. A 6, 1430–1440 (1989).
[CrossRef]

K. E. Oughstun, G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
[CrossRef]

S. Shen, K. E. Oughstun, “Optical pulse propagation in a multiple resonant dispersive Lorentz medium,” J. Opt. Soc. Am. A 3(13), P67 (1986).

K. E. Oughstun, P. T. Connors, G. C. Sherman, “Finite turn-on time effects on the transient phenomena in dispersive pulse propagation,” J. Opt. Soc. Am. A 3(13), P118 (1986).

G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

G. C. Sherman, K. E. Oughstun, “Physical model of pulse dynamics in a linear dispersive medium with absorption (the Lorentz medium),” submitted to J. Opt. Soc. Am. B.

Rouff, M.

O. Avenel, M. Rouff, E. Varoquaux, G. A. Williams, “Resonant pulse propagation of sound in superfluid He-B,” Phys. Rev. Lett. 50, 1591–1594 (1983).
[CrossRef]

Shen, S.

S. Shen, K. E. Oughstun, “Optical pulse propagation in a multiple resonant dispersive Lorentz medium,” J. Opt. Soc. Am. A 3(13), P67 (1986).

Sherman, G. C.

K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1394–1419 (1989).
[CrossRef]

K. E. Oughstun, G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
[CrossRef]

K. E. Oughstun, P. T. Connors, G. C. Sherman, “Finite turn-on time effects on the transient phenomena in dispersive pulse propagation,” J. Opt. Soc. Am. A 3(13), P118 (1986).

G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

G. C. Sherman, K. E. Oughstun, “Physical model of pulse dynamics in a linear dispersive medium with absorption (the Lorentz medium),” submitted to J. Opt. Soc. Am. B.

Sommerfeld, A.

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[CrossRef]

Trizna, D. B.

D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
[CrossRef]

Varoquaux, E.

O. Avenel, M. Rouff, E. Varoquaux, G. A. Williams, “Resonant pulse propagation of sound in superfluid He-B,” Phys. Rev. Lett. 50, 1591–1594 (1983).
[CrossRef]

Weber, T. A.

D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
[CrossRef]

Williams, G. A.

O. Avenel, M. Rouff, E. Varoquaux, G. A. Williams, “Resonant pulse propagation of sound in superfluid He-B,” Phys. Rev. Lett. 50, 1591–1594 (1983).
[CrossRef]

Wyns, P.

Ann. Phys. (2)

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[CrossRef]

L. Brillouin, “Über die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

Appl. Opt. (2)

D. Marcuse, “Pulse distortion in single-mode fibers,” Appl. Opt. 19, 1653–1660 (1980).
[CrossRef] [PubMed]

D. Marcuse, “Pulse distortion in single-mode fibers, part 2,” Appl. Opt. 19, 2969–2974 (1981).
[CrossRef]

Arch. Ration. Mechan. Anal. (1)

R. A. Handelsman, N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mechan. Anal. 35, 267–283 (1969).

IEEE Trans. Antennas Propag. (1)

C. M. Knop, “Pulsed electromagnetic wave propagation in dispersive media,”IEEE Trans. Antennas Propag. AP-12, 494–496 (1964).
[CrossRef]

J. Opt. Soc. Am. A (4)

K. E. Oughstun, P. Wyns, D. P. Foty, “Numerical determination of the signal velocity in dispersive pulse propagation,” J. Opt. Soc. Am. A 6, 1430–1440 (1989).
[CrossRef]

K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1394–1419 (1989).
[CrossRef]

K. E. Oughstun, P. T. Connors, G. C. Sherman, “Finite turn-on time effects on the transient phenomena in dispersive pulse propagation,” J. Opt. Soc. Am. A 3(13), P118 (1986).

S. Shen, K. E. Oughstun, “Optical pulse propagation in a multiple resonant dispersive Lorentz medium,” J. Opt. Soc. Am. A 3(13), P67 (1986).

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

Phys. Rev. Lett. (2)

G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

O. Avenel, M. Rouff, E. Varoquaux, G. A. Williams, “Resonant pulse propagation of sound in superfluid He-B,” Phys. Rev. Lett. 50, 1591–1594 (1983).
[CrossRef]

Radio Sci. (2)

R. E. McIntosh, A. Malaga, “Time dispersion of electromagnetic pulses by the ionosphere,” Radio Sci. 15, 645–654 (1980).
[CrossRef]

D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
[CrossRef]

Other (5)

L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).

G. C. Sherman, K. E. Oughstun, “Physical model of pulse dynamics in a linear dispersive medium with absorption (the Lorentz medium),” submitted to J. Opt. Soc. Am. B.

T. Hosono, “Numerical inversion of Laplace transform and some applications to wave optics,” in Proceedings of the 1980 International Union of Radio Science International Symposium on Electromagnetic Waves (International Union of Radio Science, Munich, Federal Republic of Germany, 1980), paper 112, pp. C1–C4.

W. R. LePage, Complex Variables and the Laplace Transform for Engineers (Dover, New York, 1961), p. 325.

T. J. I. Bromwich, An Introduction to the Theory of Infinite Series, 2nd ed. (Macmillan, London, 1926), pp. 62–65.

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Figures (6)

Fig. 1
Fig. 1

Propagated field evolution for an input delta-function pulse at a propagation distance of z = 1 × 10−6 m.

Fig. 2
Fig. 2

Expanded views of the initial evolution of the Sommerfeld precursor for an input delta-function pulse at a propagation distance of z = 10−6 m for the following ranges of θ: (a) 1.00–1.3, (b) 1.00–1.050, and (c) 1.00–1.010. The dotted curves represent the numerically determined behavior, and the solid curves represent the results of the uniform asymptotic theory.

Fig. 3
Fig. 3

Remnant of the input delta-function pulse at a propagation distance of z = 10−6 m. The precursor field evolution for θ > 1 is unobservable in the vertical scale of (a), in which the dotted and the solid curves represent numerical results and asymptotic-expansion results, respectively. The dependence of this field structure on the initial sum parameter k, which is a measure of the spectral domain modeled, is depicted in (b).

Fig. 4
Fig. 4

Initial evolution of the Sommerfeld precursor for an input delta-function pulse at a propagation distance of z = 10−6 m for several values of the initial sum index k. The dotted curve represents results for k = 8 × 105, and the dashed curve represents results for k = 1 × 106; the solid curve represents the results of the uniform asymptotic theory.

Fig. 5
Fig. 5

Expanded views of the initial evolution of the Sommerfeld precursor for an input step-function-modulated signal of carrier frequency ωc = 1016/sec at a propagation distance of z = 10−6 m for the following ranges of θ: (a) 1.000–1.300, (b) 1.000–1.050, and (c) 1.000–1.010. The dotted curves depict the numerically determined behavior, and the solid curves represent the results of the uniform asymptotic theory.

Fig. 6
Fig. 6

Evolution of the instantaneous angular frequency of oscillation of the propagated field for an input delta-function pulse. The solid the angular frequencies ωS and ωB for the Sommerfeld and Brillouin precursor fields, respectively, as predicted by the asymptotic curves depict dashed curves depict the angular frequency of oscillation ωE obtained from the energy-transport velocity in the dispersive theory. The medium. The triangular data points are obtained from the numerical calculation of the field evolution.

Equations (45)

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F ( z , i ω ) = - + f ( z , t ) e i ω t d t ,
f ( z , t ) = 1 2 π - + F ( z , i ω ) e - i ω t d ω ,
F ( z , s ) = - + f ( z , t ) e - σ t e i ω t d t ,
f ( z , t ) e - σ t = 1 2 π - + F ( z , s ) e - i ω t d ω ,
f ( z , t ) = 1 2 π - + F ( z , s ) e s t d ω .
f ( z , t ) = - i 2 π - i + i F ( z , s ) e s t d s .
F ( z , s ) = 0 + f ( z , t ) e - σ t e i ω t d t ,
f ( z , t ) = - i 2 π Br F ( z , s ) e s t d s ,
[ 2 + k 2 ( ω ) ] F ( z , i ω ) = 0 ,
F ( z , i ω ) = F + ( i ω ) exp [ i k ( ω ) z ] + F - ( i ω ) exp [ - i k ( ω ) z ] .
f ( z , t ) = - i 2 π Br F + ( s ) exp [ i k ( i s ) z ] e s t d s .
f ( 0 , t ) = - i 2 π Br F + ( s ) e s t d s ,
f ( z , t ) = - i 2 π Br F ( 0 , s ) exp [ i k ( i s ) z ] e s t d s ,
e s t E ( a , s t ) = e a 2 cosh ( a - s t ) .
E ( a , s t ) = e a 2 cosh ( a - s t ) = e s t 1 + exp [ - 2 ( a - s t ) ] .
exp [ - 2 ( a - s t ) ] exp ( 2 i s t ) = - 1 ,
s t = a ,             2 s t = - π + 2 n π             ( n Z ) ,
E ( a , s t ) = n = - + h n s t - [ a + i ( n - ½ ) π ] .
h n = lim s s p n e s t 1 + exp [ - 2 ( a - s t ) ] { s t - [ a + i ( n - ½ ) π ] } = i ( - 1 ) n e a 2 ,
E ( a , s t ) = i e a 2 n = - + ( - 1 ) n s t - [ a + i ( n - ½ ) π ] .
f app ( z , t ) = i e a 2 n = - + 1 2 π i Br ( - 1 ) n F ( 0 , s ) × exp [ i k ( i s ) z ] s t - [ a + i ( n - ½ ) π ] ds = i e a 2 t n = - + ( - 1 ) n F [ 0 , a + i ( n - ½ ) π t ] × exp [ i k ( i s p n t ) z ] = i e a 2 t n = 1 + ( - 1 ) n × { F [ z , a + i ( n - ½ ) π t ] - F [ z , a - i ( n - ½ ) π t ] } .
f app ( z , t ) = e a t n = 1 + ( - 1 ) n + 1 Im { F [ z , a + i ( n - ½ ) π t ] } = e a t n = 1 + F n .
f app ( z , t ) = e a t ( n = 1 m ( - 1 ) n + 1 + Im { F [ z , a + i ( n - ½ ) π t ] } + n = m + 1 + F n ) .
f app ( z , t ) = e a t [ n = 1 k - 1 F n + n = 0 + ( D n F k ) 1 2 n + 1 ] ,
D v n = v n - v n + 1 , D 2 v n = D v n - D v n + 1 = v n - 2 v n + 1 + v n + 2 .
n = 0 m D n F k 1 2 n + 1 = 1 2 m + 1 n = 0 m A m , n F k + n ,
A m , m = 1 , A m , n - 1 = A m , n + ( m + 1 n ) ,
f app ( z , t ) = e a t ( n = 1 k - 1 F n + 1 2 m + 1 n = 0 m A m , n F k + n ) ,
k ( ω ) = ω c n ( ω ) = ω c ( 1 - ω p 2 ω 2 + 2 i δ ω - ω 0 2 ) 1 / 2 ,
k ( i s ) = i s c ( 1 + ω p 2 s 2 + 2 δ s + ω 0 2 ) 1 / 2 ,
F ( z , s ) = F ( 0 , s ) exp [ - s z c ( 1 + ω p 2 s 2 + 2 δ s + ω 0 2 ) 1 / 2 ] .
exp [ - 2 ( a - s t ) ] = exp { - 2 [ a - Re ( s ) t ] } 1
E ( a , s t ) = e s t n = 0 + ( - 1 ) n exp [ - 2 ( a - s t ) n ] = e s t { 1 - exp [ - 2 ( a - s t ) ] + exp [ - 4 ( a - s t ) ] + . } .
δ app = E ( a - s t ) - e s t = | e - 2 a e 3 s t 1 + exp [ - 2 ( a - s t ) ] | e - 2 a .
| e a t ( n = m + 1 + D n F k 1 2 n + 1 ) | = | 1 2 m + 1 n = 0 + ( - 1 ) n D m + 1 F k + n | < 1 2 m + 1 D m + 1 F k .
ω 0 = 4.0 × 10 16 / sec , b 2 = 20.0 × 10 32 / sec 2 , δ = 0.28 × 10 16 / sec
R ( k , m ) = i = 0 m D i F k 1 2 i + 1 .
D 0 F k = F k , D 1 F k = F k - F k + 1 , D 2 F k = F k - 2 F k + 1 + F k + 2 , D i F k = j = 0 i ( - 1 ) j ( j i ) F k + j ,
R ( k , m ) = 1 2 m + 1 [ j = 0 m j = 0 i ( - 1 ) j ( j i ) 2 m - i F k + j ] .
i = j m ( j i ) 2 m - i = A k + 1 , m + 1 .
A i , j - 1 = A i , j + ( j - 1 i ) ,
A i , j = k = j i ( k i ) .
A 1 , 1 = 1 ,             A 2 , 2 = 1 , A 2 , 1 = ( 1 2 ) + ( 2 2 ) = A 1 , 0 + A 1 , 1 = 3 ,
A i - 1 , j - 1 + A i - 1 , j = k = j - 1 i - 1 ( k i - 1 ) + k = j i - 1 ( k i - 1 ) = k = j i ( k - 1 i - 1 ) + k = j i - 1 ( k i - 1 ) = ( i - 1 i - 1 ) + k = j i - 1 [ ( k - 1 i - 1 ) + ( k i - 1 ) ] = ( i i ) + k = j i - 1 ( k i ) = k = j i ( k i ) = A i , j .
column row 0 1 2 3 4 0 1 1 2 1 2 4 3 1 3 8 7 4 1 4 16 15 11 5 1 . . . . . . . n 2 n . . . . .

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